Period lattices of elliptic curves and related functions - Elliptic curves

Skip to content
Back to top
View this page
Edit this page

# Period lattices of elliptic curves and related functions

Let \(E\) be an elliptic curve defined over a number field \(K\) (including \(\QQ\)). We attach a period lattice (a discrete rank 2 subgroup of \(\CC\)) to each embedding of \(K\) into \(\CC\).

In the case of real embeddings, the lattice is stable under complex conjugation and is called a real lattice. These have two types: rectangular, (the real curve has two connected components and positive discriminant) or non-rectangular (one connected component, negative discriminant).

The periods are computed to arbitrary precision using the AGM (Gauss’s Arithmetic-Geometric Mean).

EXAMPLES:

Sage

    sage: x = polygen(ZZ, 'x')
    sage: K.<a> = NumberField(x^3 - 2)                                                  # needs sage.rings.number_field
    sage: E = EllipticCurve([0,1,0,a,a])                                                # needs sage.rings.number_field

Python

    >>> from sage.all import *
    >>> x = polygen(ZZ, 'x')
    >>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)# needs sage.rings.number_field
    >>> E = EllipticCurve([Integer(0),Integer(1),Integer(0),a,a])                                                # needs sage.rings.number_field

First we try a real embedding:

Sage

    sage: emb = K.embeddings(RealField())[0]                                            # needs sage.rings.number_field
    sage: L = E.period_lattice(emb); L                                                  # needs sage.rings.number_field
    Period lattice associated to Elliptic Curve defined by y^2 = x^3 + x^2 + a*x + a
     over Number Field in a with defining polynomial x^3 - 2
     with respect to the embedding Ring morphism:
       From: Number Field in a with defining polynomial x^3 - 2
       To:   Algebraic Real Field
       Defn: a |--> 1.259921049894873?

[Python]

    >>> from sage.all import *
    >>> emb = K.embeddings(RealField())[Integer(0)]                                            # needs sage.rings.number_field
    >>> L = E.period_lattice(emb); L                                                  # needs sage.rings.number_field
    Period lattice associated to Elliptic Curve defined by y^2 = x^3 + x^2 + a*x + a
     over Number Field in a with defining polynomial x^3 - 2
     with respect to the embedding Ring morphism:
       From: Number Field in a with defining polynomial x^3 - 2
       To:   Algebraic Real Field
       Defn: a |--> 1.259921049894873?

The first basis period is real:

Sage

    sage: L.basis()                                                                     # needs sage.rings.number_field
    (3.81452977217855, 1.90726488608927 + 1.34047785962440*I)
    sage: L.is_real()                                                                   # needs sage.rings.number_field
    True

Python

    >>> from sage.all import *
    >>> L.basis()                                                                     # needs sage.rings.number_field
    (3.81452977217855, 1.90726488608927 + 1.34047785962440*I)
    >>> L.is_real()                                                                   # needs sage.rings.number_field
    True

For a basis \(\omega_1,\omega_2\) normalised so that \(\omega_1/\omega_2\) is in the fundamental region of the upper half-plane, use the method `normalised_basis()` instead:

Sage

    sage: L.normalised_basis()                                                          # needs sage.rings.number_field
    (1.90726488608927 - 1.34047785962440*I, -1.90726488608927 - 1.34047785962440*I)

[Python]

    >>> from sage.all import *
    >>> L.normalised_basis()                                                          # needs sage.rings.number_field
    (1.90726488608927 - 1.34047785962440*I, -1.90726488608927 - 1.34047785962440*I)

Next a complex embedding:

Sage

    sage: emb = K.embeddings(ComplexField())[0]                                         # needs sage.rings.number_field
    sage: L = E.period_lattice(emb); L                                                  # needs sage.rings.number_field
    Period lattice associated to Elliptic Curve defined by y^2 = x^3 + x^2 + a*x + a
     over Number Field in a with defining polynomial x^3 - 2
     with respect to the embedding Ring morphism:
       From: Number Field in a with defining polynomial x^3 - 2
       To:   Algebraic Field
       Defn: a |--> -0.6299605249474365? - 1.091123635971722?*I

Python

    >>> from sage.all import *
    >>> emb = K.embeddings(ComplexField())[Integer(0)]                                         # needs sage.rings.number_field
    >>> L = E.period_lattice(emb); L                                                  # needs sage.rings.number_field
    Period lattice associated to Elliptic Curve defined by y^2 = x^3 + x^2 + a*x + a
     over Number Field in a with defining polynomial x^3 - 2
     with respect to the embedding Ring morphism:
       From: Number Field in a with defining polynomial x^3 - 2
       To:   Algebraic Field
       Defn: a |--> -0.6299605249474365? - 1.091123635971722?*I

In this case, the basis \(\omega_1\), \(\omega_2\) is always normalised so that \(\tau = \omega_1/\omega_2\) is in the fundamental region in the upper half plane:

Sage

    sage: # needs sage.rings.number_field
    sage: w1, w2 = L.basis(); w1, w2
    (-1.37588604166076 - 2.58560946624443*I, -2.10339907847356 + 0.428378776460622*I)
    sage: L.is_real()
    False
    sage: tau = w1/w2; tau
    0.387694505032876 + 1.30821088214407*I
    sage: L.normalised_basis()
    (-1.37588604166076 - 2.58560946624443*I, -2.10339907847356 + 0.428378776460622*I)

[Python]

    >>> from sage.all import *
    >>> # needs sage.rings.number_field
    >>> w1, w2 = L.basis(); w1, w2
    (-1.37588604166076 - 2.58560946624443*I, -2.10339907847356 + 0.428378776460622*I)
    >>> L.is_real()
    False
    >>> tau = w1/w2; tau
    0.387694505032876 + 1.30821088214407*I
    >>> L.normalised_basis()
    (-1.37588604166076 - 2.58560946624443*I, -2.10339907847356 + 0.428378776460622*I)

We test that bug Issue #8415 (caused by a PARI bug fixed in v2.3.5) is OK:

Sage

    sage: # needs sage.rings.number_field
    sage: E = EllipticCurve('37a')
    sage: K.<a> = QuadraticField(-7)
    sage: EK = E.change_ring(K)
    sage: EK.period_lattice(K.complex_embeddings()[0])
    Period lattice associated to Elliptic Curve defined by y^2 + y = x^3 + (-1)*x
     over Number Field in a with defining polynomial x^2 + 7
      with a = 2.645751311064591?*I
     with respect to the embedding Ring morphism:
       From: Number Field in a with defining polynomial x^2 + 7
             with a = 2.645751311064591?*I
       To:   Algebraic Field
       Defn: a |--> -2.645751311064591?*I

Python

    >>> from sage.all import *
    >>> # needs sage.rings.number_field
    >>> E = EllipticCurve('37a')
    >>> K = QuadraticField(-Integer(7), names=('a',)); (a,) = K._first_ngens(1)
    >>> EK = E.change_ring(K)
    >>> EK.period_lattice(K.complex_embeddings()[Integer(0)])
    Period lattice associated to Elliptic Curve defined by y^2 + y = x^3 + (-1)*x
     over Number Field in a with defining polynomial x^2 + 7
      with a = 2.645751311064591?*I
     with respect to the embedding Ring morphism:
       From: Number Field in a with defining polynomial x^2 + 7
             with a = 2.645751311064591?*I
       To:   Algebraic Field
       Defn: a |--> -2.645751311064591?*I

REFERENCES:

- [CT2013]

AUTHORS:

- ?: initial version.

- John Cremona:

  - Adapted to handle real embeddings of number fields, September 2008.

  - Added basis_matrix function, November 2008

  - Added support for complex embeddings, May 2009.

  - Added complex elliptic logs, March 2010; enhanced, October 2010.

class sage.schemes.elliptic_curves.period_lattice.PeriodLattice(base_ring, rank, degree, sparse=False, coordinate_ring=None, category=None)

  Bases: `FreeModule_generic_pid`

  The class for the period lattice of an algebraic variety.

class sage.schemes.elliptic_curves.period_lattice.PeriodLattice_ell(E, embedding=None)

  Bases: `PeriodLattice`

  The class for the period lattice of an elliptic curve.

  Currently supported are elliptic curves defined over \(\QQ\), and elliptic curves defined over a number field with a real or complex embedding, where the lattice constructed depends on that embedding.

  basis(prec=None, algorithm='sage')

    Return a basis for this period lattice as a 2-tuple.

    INPUT:

    - `prec` – (default: `None`) precision in bits (default precision if `None`)

    - `algorithm` – string (default: `'sage'`); choice of implementation (for real embeddings only) between `'sage'` (native Sage implementation) or `'pari'` (use the PARI library: only available for real embeddings)

    OUTPUT:

    (tuple of Complex) \((\omega_1,\omega_2)\) where the lattice is \(\ZZ\omega_1 + \ZZ\omega_2\). If the lattice is real then \(\omega_1\) is real and positive, \(\Im(\omega_2)>0\) and \(\Re(\omega_2/\omega_1)\) is either \(0\) (for rectangular lattices) or \(\frac{1}{2}\) (for non-rectangular lattices). Otherwise, \(\omega_1/\omega_2\) is in the fundamental region of the upper half-plane. If the latter normalisation is required for real lattices, use the method `normalised_basis()` instead.

    EXAMPLES:

    Sage

        sage: E = EllipticCurve('37a')
        sage: E.period_lattice().basis()
        (2.99345864623196, 2.45138938198679*I)

    Python

        >>> from sage.all import *
        >>> E = EllipticCurve('37a')
        >>> E.period_lattice().basis()
        (2.99345864623196, 2.45138938198679*I)

    This shows that the issue reported at Issue #3954 is fixed:

    Sage

        sage: E = EllipticCurve('37a')
        sage: b1 = E.period_lattice().basis(prec=30)
        sage: b2 = E.period_lattice().basis(prec=30)
        sage: b1 == b2
        True

    [Python]

        >>> from sage.all import *
        >>> E = EllipticCurve('37a')
        >>> b1 = E.period_lattice().basis(prec=Integer(30))
        >>> b2 = E.period_lattice().basis(prec=Integer(30))
        >>> b1 == b2
        True

    This shows that the issue reported at Issue #4064 is fixed:

    Sage

        sage: E = EllipticCurve('37a')
        sage: E.period_lattice().basis(prec=30)[0].parent()
        Real Field with 30 bits of precision
        sage: E.period_lattice().basis(prec=100)[0].parent()
        Real Field with 100 bits of precision

    Python

        >>> from sage.all import *
        >>> E = EllipticCurve('37a')
        >>> E.period_lattice().basis(prec=Integer(30))[Integer(0)].parent()
        Real Field with 30 bits of precision
        >>> E.period_lattice().basis(prec=Integer(100))[Integer(0)].parent()
        Real Field with 100 bits of precision

    Sage

        sage: # needs sage.rings.number_field
        sage: x = polygen(ZZ, 'x')
        sage: K.<a> = NumberField(x^3 - 2)
        sage: emb = K.embeddings(RealField())[0]
        sage: E = EllipticCurve([0,1,0,a,a])
        sage: L = E.period_lattice(emb)
        sage: L.basis(64)
        (3.81452977217854509, 1.90726488608927255 + 1.34047785962440202*I)

        sage: # needs sage.rings.number_field
        sage: emb = K.embeddings(ComplexField())[0]
        sage: L = E.period_lattice(emb)
        sage: w1, w2 = L.basis(); w1, w2
        (-1.37588604166076 - 2.58560946624443*I, -2.10339907847356 + 0.428378776460622*I)
        sage: L.is_real()
        False
        sage: tau = w1/w2; tau
        0.387694505032876 + 1.30821088214407*I

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> x = polygen(ZZ, 'x')
        >>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
        >>> emb = K.embeddings(RealField())[Integer(0)]
        >>> E = EllipticCurve([Integer(0),Integer(1),Integer(0),a,a])
        >>> L = E.period_lattice(emb)
        >>> L.basis(Integer(64))
        (3.81452977217854509, 1.90726488608927255 + 1.34047785962440202*I)

        >>> # needs sage.rings.number_field
        >>> emb = K.embeddings(ComplexField())[Integer(0)]
        >>> L = E.period_lattice(emb)
        >>> w1, w2 = L.basis(); w1, w2
        (-1.37588604166076 - 2.58560946624443*I, -2.10339907847356 + 0.428378776460622*I)
        >>> L.is_real()
        False
        >>> tau = w1/w2; tau
        0.387694505032876 + 1.30821088214407*I

  basis_matrix(prec=None, normalised=False)

    Return the basis matrix of this period lattice.

    INPUT:

    - `prec` – integer or `None` (default); real precision in bits (default real precision if `None`)

    - `normalised` – boolean (default: `False`); if `True` and the embedding is real, use the normalised basis (see `normalised_basis()`) instead of the default

    OUTPUT:

    A \(2\times 2\) real matrix whose rows are the lattice basis vectors, after identifying \(\CC\) with \(\RR^2\).

    EXAMPLES:

    Sage

        sage: E = EllipticCurve('37a')
        sage: E.period_lattice().basis_matrix()
        [ 2.99345864623196 0.000000000000000]
        [0.000000000000000  2.45138938198679]

    Python

        >>> from sage.all import *
        >>> E = EllipticCurve('37a')
        >>> E.period_lattice().basis_matrix()
        [ 2.99345864623196 0.000000000000000]
        [0.000000000000000  2.45138938198679]

    Sage

        sage: # needs sage.rings.number_field
        sage: x = polygen(ZZ, 'x')
        sage: K.<a> = NumberField(x^3 - 2)
        sage: emb = K.embeddings(RealField())[0]
        sage: E = EllipticCurve([0,1,0,a,a])
        sage: L = E.period_lattice(emb)
        sage: L.basis_matrix(64)
        [ 3.81452977217854509 0.000000000000000000]
        [ 1.90726488608927255  1.34047785962440202]

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> x = polygen(ZZ, 'x')
        >>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
        >>> emb = K.embeddings(RealField())[Integer(0)]
        >>> E = EllipticCurve([Integer(0),Integer(1),Integer(0),a,a])
        >>> L = E.period_lattice(emb)
        >>> L.basis_matrix(Integer(64))
        [ 3.81452977217854509 0.000000000000000000]
        [ 1.90726488608927255  1.34047785962440202]

    See Issue #4388:

    Sage

        sage: L = EllipticCurve('11a1').period_lattice()
        sage: L.basis_matrix()
        [ 1.26920930427955 0.000000000000000]
        [0.634604652139777  1.45881661693850]
        sage: L.basis_matrix(normalised=True)
        [-0.634604652139777  -1.45881661693850]
        [ -1.26920930427955  0.000000000000000]

    Python

        >>> from sage.all import *
        >>> L = EllipticCurve('11a1').period_lattice()
        >>> L.basis_matrix()
        [ 1.26920930427955 0.000000000000000]
        [0.634604652139777  1.45881661693850]
        >>> L.basis_matrix(normalised=True)
        [-0.634604652139777  -1.45881661693850]
        [ -1.26920930427955  0.000000000000000]

    Sage

        sage: L = EllipticCurve('389a1').period_lattice()
        sage: L.basis_matrix()
        [ 2.49021256085505 0.000000000000000]
        [0.000000000000000  1.97173770155165]
        sage: L.basis_matrix(normalised=True)
        [ 2.49021256085505 0.000000000000000]
        [0.000000000000000 -1.97173770155165]

    [Python]

        >>> from sage.all import *
        >>> L = EllipticCurve('389a1').period_lattice()
        >>> L.basis_matrix()
        [ 2.49021256085505 0.000000000000000]
        [0.000000000000000  1.97173770155165]
        >>> L.basis_matrix(normalised=True)
        [ 2.49021256085505 0.000000000000000]
        [0.000000000000000 -1.97173770155165]

  complex_area(prec=None)

    Return the area of a fundamental domain for the period lattice of the elliptic curve.

    INPUT:

    - `prec` – integer or `None` (default); real precision in bits (default real precision if `None`)

    EXAMPLES:

    Sage

        sage: E = EllipticCurve('37a')
        sage: E.period_lattice().complex_area()
        7.33813274078958

    Python

        >>> from sage.all import *
        >>> E = EllipticCurve('37a')
        >>> E.period_lattice().complex_area()
        7.33813274078958

    Sage

        sage: # needs sage.rings.number_field
        sage: x = polygen(ZZ, 'x')
        sage: K.<a> = NumberField(x^3 - 2)
        sage: embs = K.embeddings(ComplexField())
        sage: E = EllipticCurve([0,1,0,a,a])
        sage: [E.period_lattice(emb).is_real() for emb in K.embeddings(CC)]
        [False, False, True]
        sage: [E.period_lattice(emb).complex_area() for emb in embs]
        [6.02796894766694, 6.02796894766694, 5.11329270448345]

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> x = polygen(ZZ, 'x')
        >>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
        >>> embs = K.embeddings(ComplexField())
        >>> E = EllipticCurve([Integer(0),Integer(1),Integer(0),a,a])
        >>> [E.period_lattice(emb).is_real() for emb in K.embeddings(CC)]
        [False, False, True]
        >>> [E.period_lattice(emb).complex_area() for emb in embs]
        [6.02796894766694, 6.02796894766694, 5.11329270448345]

  coordinates(z, rounding=None)

    Return the coordinates of a complex number w.r.t. the lattice basis.

    INPUT:

    - `z` – complex number

    - `rounding` – (default: `None`) whether and how to round the output (see below)

    OUTPUT:

    When `rounding` is `None` (the default), returns a tuple of reals \(x\), \(y\) such that \(z=xw_1+yw_2\) where \(w_1\), \(w_2\) are a basis for the lattice (normalised in the case of complex embeddings).

    When `rounding` is `'round'`, returns a tuple of integers \(n_1\), \(n_2\) which are the closest integers to the \(x\), \(y\) defined above. If \(z\) is in the lattice these are the coordinates of \(z\) with respect to the lattice basis.

    When `rounding` is `'floor'`, returns a tuple of integers \(n_1\), \(n_2\) which are the integer parts to the \(x\), \(y\) defined above. These are used in `reduce()`

    EXAMPLES:

    Sage

        sage: E = EllipticCurve('389a')
        sage: L = E.period_lattice()
        sage: w1, w2 = L.basis(prec=100)
        sage: P = E([-1,1])
        sage: zP = P.elliptic_logarithm(precision=100); zP  # abs tol 1e-28
        0.47934825019021931612953301006 + 0.98586885077582410221120384908*I
        sage: L.coordinates(zP)  # abs tol 1e-28
        (0.19249290511394227352563996419, 0.50000000000000000000000000000)
        sage: sum([x*w for x, w in zip(L.coordinates(zP), L.basis(prec=100))])  # abs tol 1e-28
        0.47934825019021931612953301006 + 0.98586885077582410221120384908*I

        sage: L.coordinates(12*w1 + 23*w2)  # abs tol 1e-28
        (12.000000000000000000000000000, 23.000000000000000000000000000)
        sage: L.coordinates(12*w1 + 23*w2, rounding='floor')
        (11, 22)
        sage: L.coordinates(12*w1 + 23*w2, rounding='round')
        (12, 23)

    Python

        >>> from sage.all import *
        >>> E = EllipticCurve('389a')
        >>> L = E.period_lattice()
        >>> w1, w2 = L.basis(prec=Integer(100))
        >>> P = E([-Integer(1),Integer(1)])
        >>> zP = P.elliptic_logarithm(precision=Integer(100)); zP  # abs tol 1e-28
        0.47934825019021931612953301006 + 0.98586885077582410221120384908*I
        >>> L.coordinates(zP)  # abs tol 1e-28
        (0.19249290511394227352563996419, 0.50000000000000000000000000000)
        >>> sum([x*w for x, w in zip(L.coordinates(zP), L.basis(prec=Integer(100)))])  # abs tol 1e-28
        0.47934825019021931612953301006 + 0.98586885077582410221120384908*I

        >>> L.coordinates(Integer(12)*w1 + Integer(23)*w2)  # abs tol 1e-28
        (12.000000000000000000000000000, 23.000000000000000000000000000)
        >>> L.coordinates(Integer(12)*w1 + Integer(23)*w2, rounding='floor')
        (11, 22)
        >>> L.coordinates(Integer(12)*w1 + Integer(23)*w2, rounding='round')
        (12, 23)

  curve()

    Return the elliptic curve associated with this period lattice.

    EXAMPLES:

    Sage

        sage: E = EllipticCurve('37a')
        sage: L = E.period_lattice()
        sage: L.curve() is E
        True

    Python

        >>> from sage.all import *
        >>> E = EllipticCurve('37a')
        >>> L = E.period_lattice()
        >>> L.curve() is E
        True

    Sage

        sage: # needs sage.rings.number_field
        sage: x = polygen(ZZ, 'x')
        sage: K.<a> = NumberField(x^3 - 2)
        sage: E = EllipticCurve([0,1,0,a,a])
        sage: L = E.period_lattice(K.embeddings(RealField())[0])
        sage: L.curve() is E
        True

        sage: L = E.period_lattice(K.embeddings(ComplexField())[0])                 # needs sage.rings.number_field
        sage: L.curve() is E                                                        # needs sage.rings.number_field
        True

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> x = polygen(ZZ, 'x')
        >>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
        >>> E = EllipticCurve([Integer(0),Integer(1),Integer(0),a,a])
        >>> L = E.period_lattice(K.embeddings(RealField())[Integer(0)])
        >>> L.curve() is E
        True

        >>> L = E.period_lattice(K.embeddings(ComplexField())[Integer(0)])                 # needs sage.rings.number_field
        >>> L.curve() is E                                                        # needs sage.rings.number_field
        True

  e_log_RC(xP, yP, prec=None, reduce=True)

    Return the elliptic logarithm of a real or complex point.

    INPUT:

    - `xP`, `yP` (real or complex) – Coordinates of a point on the embedded elliptic curve associated with this period lattice.

    - `prec` – (default: `None`) real precision in bits (default real precision if `None`)

    - `reduce` – boolean (default: `True`); if `True`, the result is reduced with respect to the period lattice basis

    OUTPUT:

    (complex number) The elliptic logarithm of the point \((xP,yP)\) with respect to this period lattice. If \(E\) is the elliptic curve and \(\sigma:K\to\CC\) the embedding, the returned value \(z\) is such that \(z\pmod{L}\) maps to \((xP,yP)=\sigma(P)\) under the standard Weierstrass isomorphism from \(\CC/L\) to \(\sigma(E)\). If `reduce` is `True`, the output is reduced so that it is in the fundamental period parallelogram with respect to the normalised lattice basis.

    ALGORITHM:

    Uses the complex AGM. See [CT2013] for details.

    EXAMPLES:

    Sage

        sage: E = EllipticCurve('389a')
        sage: L = E.period_lattice()
        sage: P = E([-1,1])
        sage: xP, yP = [RR(c) for c in P.xy()]

    Python

        >>> from sage.all import *
        >>> E = EllipticCurve('389a')
        >>> L = E.period_lattice()
        >>> P = E([-Integer(1),Integer(1)])
        >>> xP, yP = [RR(c) for c in P.xy()]

    The elliptic log from the real coordinates:

    Sage

        sage: L.e_log_RC(xP, yP)  # abs tol 1e-15
        0.479348250190219 + 0.985868850775824*I

    [Python]

        >>> from sage.all import *
        >>> L.e_log_RC(xP, yP)  # abs tol 1e-15
        0.479348250190219 + 0.985868850775824*I

    The same elliptic log from the algebraic point:

    Sage

        sage: L(P)  # abs tol 1e-15
        0.479348250190219 + 0.985868850775824*I

    Python

        >>> from sage.all import *
        >>> L(P)  # abs tol 1e-15
        0.479348250190219 + 0.985868850775824*I

    A number field example:

    Sage

        sage: # needs sage.rings.number_field
        sage: x = polygen(ZZ, 'x')
        sage: K.<a> = NumberField(x^3 - 2)
        sage: E = EllipticCurve([0,0,0,0,a])
        sage: v = K.real_places()[0]
        sage: L = E.period_lattice(v)
        sage: P = E.lift_x(1/3*a^2 + a + 5/3)
        sage: L(P)  # abs tol 1e-15
        3.51086196882538
        sage: xP, yP = [v(c) for c in P.xy()]
        sage: L.e_log_RC(xP, yP)  # abs tol 1e-15
        3.51086196882538

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> x = polygen(ZZ, 'x')
        >>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
        >>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),Integer(0),a])
        >>> v = K.real_places()[Integer(0)]
        >>> L = E.period_lattice(v)
        >>> P = E.lift_x(Integer(1)/Integer(3)*a**Integer(2) + a + Integer(5)/Integer(3))
        >>> L(P)  # abs tol 1e-15
        3.51086196882538
        >>> xP, yP = [v(c) for c in P.xy()]
        >>> L.e_log_RC(xP, yP)  # abs tol 1e-15
        3.51086196882538

    Elliptic logs of real points which do not come from algebraic points:

    Sage

        sage: # needs sage.rings.number_field
        sage: ER = EllipticCurve([v(ai) for ai in E.a_invariants()])
        sage: P = ER.lift_x(12.34)
        sage: xP, yP = P.xy()
        sage: xP, yP  # abs tol 1e-15
        (12.3400000000000, -43.3628968710567)
        sage: L.e_log_RC(xP, yP)  # abs tol 1e-15
        0.284656841192041
        sage: xP, yP = ER.lift_x(0).xy()  # abs tol 1e-15
        sage: L.e_log_RC(xP, yP)
        1.34921304541057

    Python

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> ER = EllipticCurve([v(ai) for ai in E.a_invariants()])
        >>> P = ER.lift_x(RealNumber('12.34'))
        >>> xP, yP = P.xy()
        >>> xP, yP  # abs tol 1e-15
        (12.3400000000000, -43.3628968710567)
        >>> L.e_log_RC(xP, yP)  # abs tol 1e-15
        0.284656841192041
        >>> xP, yP = ER.lift_x(Integer(0)).xy()  # abs tol 1e-15
        >>> L.e_log_RC(xP, yP)
        1.34921304541057

    Elliptic logs of complex points:

    Sage

        sage: # needs sage.rings.number_field
        sage: v = K.complex_embeddings()[0]
        sage: L = E.period_lattice(v)
        sage: P = E.lift_x(1/3*a^2 + a + 5/3)
        sage: L(P)  # abs tol 1e-15
        1.68207104397706 - 1.87873661686704*I
        sage: xP, yP = [v(c) for c in P.xy()]
        sage: L.e_log_RC(xP, yP)  # abs tol 1e-15
        1.68207104397706 - 1.87873661686704*I
        sage: EC = EllipticCurve([v(ai) for ai in E.a_invariants()])
        sage: xP, yP = EC.lift_x(0).xy()
        sage: L.e_log_RC(xP, yP)  # abs tol 1e-15
        2.06711431204080 - 1.73451485683471*I

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> v = K.complex_embeddings()[Integer(0)]
        >>> L = E.period_lattice(v)
        >>> P = E.lift_x(Integer(1)/Integer(3)*a**Integer(2) + a + Integer(5)/Integer(3))
        >>> L(P)  # abs tol 1e-15
        1.68207104397706 - 1.87873661686704*I
        >>> xP, yP = [v(c) for c in P.xy()]
        >>> L.e_log_RC(xP, yP)  # abs tol 1e-15
        1.68207104397706 - 1.87873661686704*I
        >>> EC = EllipticCurve([v(ai) for ai in E.a_invariants()])
        >>> xP, yP = EC.lift_x(Integer(0)).xy()
        >>> L.e_log_RC(xP, yP)  # abs tol 1e-15
        2.06711431204080 - 1.73451485683471*I

  ei()

    Return the x-coordinates of the 2-division points of the elliptic curve associated with this period lattice, as elements of `QQbar`.

    EXAMPLES:

    Sage

        sage: E = EllipticCurve('37a')
        sage: L = E.period_lattice()
        sage: L.ei()
        [-1.107159871688768?, 0.2695944364054446?, 0.8375654352833230?]

    Python

        >>> from sage.all import *
        >>> E = EllipticCurve('37a')
        >>> L = E.period_lattice()
        >>> L.ei()
        [-1.107159871688768?, 0.2695944364054446?, 0.8375654352833230?]

    In the following example, we should have one purely real 2-division point coordinate, and two conjugate purely imaginary coordinates.

    Sage

        sage: # needs sage.rings.number_field
        sage: x = polygen(ZZ, 'x')
        sage: K.<a> = NumberField(x^3 - 2)
        sage: E = EllipticCurve([0,1,0,a,a])
        sage: L = E.period_lattice(K.embeddings(RealField())[0])
        sage: x1,x2,x3 = L.ei()
        sage: abs(x1.real()) + abs(x2.real()) < 1e-14
        True
        sage: x1.imag(), x2.imag(), x3
        (-1.122462048309373?, 1.122462048309373?, -1.000000000000000?)

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> x = polygen(ZZ, 'x')
        >>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
        >>> E = EllipticCurve([Integer(0),Integer(1),Integer(0),a,a])
        >>> L = E.period_lattice(K.embeddings(RealField())[Integer(0)])
        >>> x1,x2,x3 = L.ei()
        >>> abs(x1.real()) + abs(x2.real()) < RealNumber('1e-14')
        True
        >>> x1.imag(), x2.imag(), x3
        (-1.122462048309373?, 1.122462048309373?, -1.000000000000000?)

    Sage

        sage: L = E.period_lattice(K.embeddings(ComplexField())[0])                 # needs sage.rings.number_field
        sage: L.ei()                                                                # needs sage.rings.number_field
        [-1.000000000000000? + 0.?e-1...*I,
         -0.9720806486198328? - 0.561231024154687?*I,
         0.9720806486198328? + 0.561231024154687?*I]

    Python

        >>> from sage.all import *
        >>> L = E.period_lattice(K.embeddings(ComplexField())[Integer(0)])                 # needs sage.rings.number_field
        >>> L.ei()                                                                # needs sage.rings.number_field
        [-1.000000000000000? + 0.?e-1...*I,
         -0.9720806486198328? - 0.561231024154687?*I,
         0.9720806486198328? + 0.561231024154687?*I]

  elliptic_exponential(z, to_curve=True)

    Return the elliptic exponential of a complex number.

    INPUT:

    - `z` – complex number (viewed modulo this period lattice)

    - `to_curve` – boolean (default: `True`); see below

    OUTPUT:

    - If `to_curve` is False, a 2-tuple of real or complex numbers representing the point \((x,y) = (\wp(z),\wp'(z))\) where \(\wp\) denotes the Weierstrass \(\wp\)-function with respect to this lattice.

    - If `to_curve` is True, the point \((X,Y) = (x-b_2/12,y-(a_1(x-b_2/12)-a_3)/2)\) as a point in \(E(\RR)\) or \(E(\CC)\), with \((x,y) = (\wp(z),\wp'(z))\) as above, where \(E\) is the elliptic curve over \(\RR\) or \(\CC\) whose period lattice this is.

    - If the lattice is real and \(z\) is also real then the output is a pair of real numbers if `to_curve` is `True`, or a point in \(E(\RR)\) if `to_curve` is `False`.

    Note

    The precision is taken from that of the input `z`.

    EXAMPLES:

    Sage

        sage: E = EllipticCurve([1,1,1,-8,6])
        sage: P = E(1, -2)
        sage: L = E.period_lattice()
        sage: z = L(P); z
        1.17044757240090
        sage: L.elliptic_exponential(z)
        (0.999999999999999 : -2.00000000000000 : 1.00000000000000)
        sage: _.curve()
        Elliptic Curve defined by y^2 + 1.00000000000000*x*y + 1.00000000000000*y
         = x^3 + 1.00000000000000*x^2 - 8.00000000000000*x + 6.00000000000000
         over Real Field with 53 bits of precision
        sage: L.elliptic_exponential(z,to_curve=False)
        (1.41666666666667, -2.00000000000000)
        sage: z = L(P, prec=201); z
        1.17044757240089592298992188482371493504472561677451007994189
        sage: L.elliptic_exponential(z)
        (1.00000000000000000000000000000000000000000000000000000000000
         : -2.00000000000000000000000000000000000000000000000000000000000
         : 1.00000000000000000000000000000000000000000000000000000000000)

    Python

        >>> from sage.all import *
        >>> E = EllipticCurve([Integer(1),Integer(1),Integer(1),-Integer(8),Integer(6)])
        >>> P = E(Integer(1), -Integer(2))
        >>> L = E.period_lattice()
        >>> z = L(P); z
        1.17044757240090
        >>> L.elliptic_exponential(z)
        (0.999999999999999 : -2.00000000000000 : 1.00000000000000)
        >>> _.curve()
        Elliptic Curve defined by y^2 + 1.00000000000000*x*y + 1.00000000000000*y
         = x^3 + 1.00000000000000*x^2 - 8.00000000000000*x + 6.00000000000000
         over Real Field with 53 bits of precision
        >>> L.elliptic_exponential(z,to_curve=False)
        (1.41666666666667, -2.00000000000000)
        >>> z = L(P, prec=Integer(201)); z
        1.17044757240089592298992188482371493504472561677451007994189
        >>> L.elliptic_exponential(z)
        (1.00000000000000000000000000000000000000000000000000000000000
         : -2.00000000000000000000000000000000000000000000000000000000000
         : 1.00000000000000000000000000000000000000000000000000000000000)

    Examples over number fields:

    Sage

        sage: # needs sage.rings.number_field
        sage: x = polygen(QQ)
        sage: K.<a> = NumberField(x^3 - 2)
        sage: embs = K.embeddings(CC)
        sage: E = EllipticCurve('37a')
        sage: EK = E.change_ring(K)
        sage: Li = [EK.period_lattice(e) for e in embs]
        sage: P = EK(-1, -1)
        sage: Q = EK(a - 1, 1 - a^2)
        sage: zi = [L.elliptic_logarithm(P) for L in Li]
        sage: [c.real() for c in Li[0].elliptic_exponential(zi[0])]
        [-1.00000000000000, -1.00000000000000, 1.00000000000000]
        sage: [c.real() for c in Li[0].elliptic_exponential(zi[1])]
        [-1.00000000000000, -1.00000000000000, 1.00000000000000]
        sage: [c.real() for c in Li[0].elliptic_exponential(zi[2])]
        [-1.00000000000000, -1.00000000000000, 1.00000000000000]

        sage: # needs sage.rings.number_field
        sage: zi = [L.elliptic_logarithm(Q) for L in Li]
        sage: Li[0].elliptic_exponential(zi[0])
        (-1.62996052494744 - 1.09112363597172*I
         : 1.79370052598410 - 1.37472963699860*I
         : 1.00000000000000)
        sage: [embs[0](c) for c in Q]
        [-1.62996052494744 - 1.09112363597172*I,
         1.79370052598410 - 1.37472963699860*I,
         1.00000000000000]
        sage: Li[1].elliptic_exponential(zi[1])
        (-1.62996052494744 + 1.09112363597172*I
         : 1.79370052598410 + 1.37472963699860*I
         : 1.00000000000000)
        sage: [embs[1](c) for c in Q]
        [-1.62996052494744 + 1.09112363597172*I,
         1.79370052598410 + 1.37472963699860*I,
         1.00000000000000]
        sage: [c.real() for c in Li[2].elliptic_exponential(zi[2])]
        [0.259921049894873, -0.587401051968199, 1.00000000000000]
        sage: [embs[2](c) for c in Q]
        [0.259921049894873, -0.587401051968200, 1.00000000000000]

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> x = polygen(QQ)
        >>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
        >>> embs = K.embeddings(CC)
        >>> E = EllipticCurve('37a')
        >>> EK = E.change_ring(K)
        >>> Li = [EK.period_lattice(e) for e in embs]
        >>> P = EK(-Integer(1), -Integer(1))
        >>> Q = EK(a - Integer(1), Integer(1) - a**Integer(2))
        >>> zi = [L.elliptic_logarithm(P) for L in Li]
        >>> [c.real() for c in Li[Integer(0)].elliptic_exponential(zi[Integer(0)])]
        [-1.00000000000000, -1.00000000000000, 1.00000000000000]
        >>> [c.real() for c in Li[Integer(0)].elliptic_exponential(zi[Integer(1)])]
        [-1.00000000000000, -1.00000000000000, 1.00000000000000]
        >>> [c.real() for c in Li[Integer(0)].elliptic_exponential(zi[Integer(2)])]
        [-1.00000000000000, -1.00000000000000, 1.00000000000000]

        >>> # needs sage.rings.number_field
        >>> zi = [L.elliptic_logarithm(Q) for L in Li]
        >>> Li[Integer(0)].elliptic_exponential(zi[Integer(0)])
        (-1.62996052494744 - 1.09112363597172*I
         : 1.79370052598410 - 1.37472963699860*I
         : 1.00000000000000)
        >>> [embs[Integer(0)](c) for c in Q]
        [-1.62996052494744 - 1.09112363597172*I,
         1.79370052598410 - 1.37472963699860*I,
         1.00000000000000]
        >>> Li[Integer(1)].elliptic_exponential(zi[Integer(1)])
        (-1.62996052494744 + 1.09112363597172*I
         : 1.79370052598410 + 1.37472963699860*I
         : 1.00000000000000)
        >>> [embs[Integer(1)](c) for c in Q]
        [-1.62996052494744 + 1.09112363597172*I,
         1.79370052598410 + 1.37472963699860*I,
         1.00000000000000]
        >>> [c.real() for c in Li[Integer(2)].elliptic_exponential(zi[Integer(2)])]
        [0.259921049894873, -0.587401051968199, 1.00000000000000]
        >>> [embs[Integer(2)](c) for c in Q]
        [0.259921049894873, -0.587401051968200, 1.00000000000000]

    Test to show that Issue #8820 is fixed:

    Sage

        sage: # needs sage.rings.number_field
        sage: E = EllipticCurve('37a')
        sage: K.<a> = QuadraticField(-5)
        sage: L = E.change_ring(K).period_lattice(K.places()[0])
        sage: L.elliptic_exponential(CDF(.1,.1))  # abs tol 1e-15
        (0.0000142854026029 - 49.9960001066650*I
         : 249.520141250950 + 250.019855549131*I : 1.00000000000000)
        sage: L.elliptic_exponential(CDF(.1,.1), to_curve=False)
        (0.0000142854026029447 - 49.9960001066650*I,
         500.040282501900 + 500.039711098263*I)

    Python

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> E = EllipticCurve('37a')
        >>> K = QuadraticField(-Integer(5), names=('a',)); (a,) = K._first_ngens(1)
        >>> L = E.change_ring(K).period_lattice(K.places()[Integer(0)])
        >>> L.elliptic_exponential(CDF(RealNumber('.1'),RealNumber('.1')))  # abs tol 1e-15
        (0.0000142854026029 - 49.9960001066650*I
         : 249.520141250950 + 250.019855549131*I : 1.00000000000000)
        >>> L.elliptic_exponential(CDF(RealNumber('.1'),RealNumber('.1')), to_curve=False)
        (0.0000142854026029447 - 49.9960001066650*I,
         500.040282501900 + 500.039711098263*I)

    \(z=0\) is treated as a special case:

    Sage

        sage: E = EllipticCurve([1,1,1,-8,6])
        sage: L = E.period_lattice()
        sage: L.elliptic_exponential(0)
        (0.000000000000000 : 1.00000000000000 : 0.000000000000000)
        sage: L.elliptic_exponential(0, to_curve=False)
        (+infinity, +infinity)

    [Python]

        >>> from sage.all import *
        >>> E = EllipticCurve([Integer(1),Integer(1),Integer(1),-Integer(8),Integer(6)])
        >>> L = E.period_lattice()
        >>> L.elliptic_exponential(Integer(0))
        (0.000000000000000 : 1.00000000000000 : 0.000000000000000)
        >>> L.elliptic_exponential(Integer(0), to_curve=False)
        (+infinity, +infinity)

    Sage

        sage: # needs sage.rings.number_field
        sage: E = EllipticCurve('37a')
        sage: K.<a> = QuadraticField(-5)
        sage: L = E.change_ring(K).period_lattice(K.places()[0])
        sage: P = L.elliptic_exponential(0); P
        (0.000000000000000 : 1.00000000000000 : 0.000000000000000)
        sage: P.parent()
        Abelian group of points on Elliptic Curve defined by
         y^2 + 1.00000000000000*y = x^3 + (-1.00000000000000)*x
         over Complex Field with 53 bits of precision

    Python

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> E = EllipticCurve('37a')
        >>> K = QuadraticField(-Integer(5), names=('a',)); (a,) = K._first_ngens(1)
        >>> L = E.change_ring(K).period_lattice(K.places()[Integer(0)])
        >>> P = L.elliptic_exponential(Integer(0)); P
        (0.000000000000000 : 1.00000000000000 : 0.000000000000000)
        >>> P.parent()
        Abelian group of points on Elliptic Curve defined by
         y^2 + 1.00000000000000*y = x^3 + (-1.00000000000000)*x
         over Complex Field with 53 bits of precision

    Very small \(z\) are handled properly (see Issue #8820):

    Sage

        sage: # needs sage.rings.number_field
        sage: K.<a> = QuadraticField(-1)
        sage: E = EllipticCurve([0,0,0,a,0])
        sage: L = E.period_lattice(K.complex_embeddings()[0])
        sage: L.elliptic_exponential(1e-100)
        (0.000000000000000 : 1.00000000000000 : 0.000000000000000)

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> K = QuadraticField(-Integer(1), names=('a',)); (a,) = K._first_ngens(1)
        >>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),a,Integer(0)])
        >>> L = E.period_lattice(K.complex_embeddings()[Integer(0)])
        >>> L.elliptic_exponential(RealNumber('1e-100'))
        (0.000000000000000 : 1.00000000000000 : 0.000000000000000)

    The elliptic exponential of \(z\) is returned as \((0 : 1 : 0)\) if the coordinates of \(z\) with respect to the period lattice are approximately integral:

    Sage

        sage: (100/log(2.0,10))/0.8
        415.241011860920
        sage: L.elliptic_exponential((RealField(415)(1e-100))).is_zero()            # needs sage.rings.number_field
        True
        sage: L.elliptic_exponential((RealField(420)(1e-100))).is_zero()            # needs sage.rings.number_field
        False

    Python

        >>> from sage.all import *
        >>> (Integer(100)/log(RealNumber('2.0'),Integer(10)))/RealNumber('0.8')
        415.241011860920
        >>> L.elliptic_exponential((RealField(Integer(415))(RealNumber('1e-100')))).is_zero()            # needs sage.rings.number_field
        True
        >>> L.elliptic_exponential((RealField(Integer(420))(RealNumber('1e-100')))).is_zero()            # needs sage.rings.number_field
        False

  elliptic_logarithm(P, prec=None, reduce=True)

    Return the elliptic logarithm of a point.

    INPUT:

    - `P` – point on the elliptic curve associated with this period lattice

    - `prec` – (default: `None`) real precision in bits (default real precision if `None`)

    - `reduce` – boolean (default: `True`); if `True`, the result is reduced with respect to the period lattice basis

    OUTPUT:

    (complex number) The elliptic logarithm of the point \(P\) with respect to this period lattice. If \(E\) is the elliptic curve and \(\sigma:K\to\CC\) the embedding, the returned value \(z\) is such that \(z\pmod{L}\) maps to \(\sigma(P)\) under the standard Weierstrass isomorphism from \(\CC/L\) to \(\sigma(E)\). If `reduce` is `True`, the output is reduced so that it is in the fundamental period parallelogram with respect to the normalised lattice basis.

    ALGORITHM:

    Uses the complex AGM. See [CT2013] for details.

    EXAMPLES:

    Sage

        sage: E = EllipticCurve('389a')
        sage: L = E.period_lattice()
        sage: E.discriminant() > 0
        True
        sage: L.real_flag
        1
        sage: P = E([-1,1])
        sage: P.is_on_identity_component ()
        False
        sage: L.elliptic_logarithm(P, prec=96)
        0.4793482501902193161295330101 + 0.9858688507758241022112038491*I
        sage: Q=E([3,5])
        sage: Q.is_on_identity_component()
        True
        sage: L.elliptic_logarithm(Q, prec=96)
        1.931128271542559442488585220

    Python

        >>> from sage.all import *
        >>> E = EllipticCurve('389a')
        >>> L = E.period_lattice()
        >>> E.discriminant() > Integer(0)
        True
        >>> L.real_flag
        1
        >>> P = E([-Integer(1),Integer(1)])
        >>> P.is_on_identity_component ()
        False
        >>> L.elliptic_logarithm(P, prec=Integer(96))
        0.4793482501902193161295330101 + 0.9858688507758241022112038491*I
        >>> Q=E([Integer(3),Integer(5)])
        >>> Q.is_on_identity_component()
        True
        >>> L.elliptic_logarithm(Q, prec=Integer(96))
        1.931128271542559442488585220

    Note that this is actually the inverse of the Weierstrass isomorphism:

    Sage

        sage: L.elliptic_exponential(_)  # abs tol 1e-26
        (3.000000000000000000000000000 : 5.000000000000000000000000000 : 1.000000000000000000000000000)

    [Python]

        >>> from sage.all import *
        >>> L.elliptic_exponential(_)  # abs tol 1e-26
        (3.000000000000000000000000000 : 5.000000000000000000000000000 : 1.000000000000000000000000000)

    An example with negative discriminant, and a torsion point:

    Sage

        sage: E = EllipticCurve('11a1')
        sage: L = E.period_lattice()
        sage: E.discriminant() < 0
        True
        sage: L.real_flag
        -1
        sage: P = E([16,-61])
        sage: L.elliptic_logarithm(P)
        0.253841860855911
        sage: L.real_period() / L.elliptic_logarithm(P)
        5.00000000000000

    Python

        >>> from sage.all import *
        >>> E = EllipticCurve('11a1')
        >>> L = E.period_lattice()
        >>> E.discriminant() < Integer(0)
        True
        >>> L.real_flag
        -1
        >>> P = E([Integer(16),-Integer(61)])
        >>> L.elliptic_logarithm(P)
        0.253841860855911
        >>> L.real_period() / L.elliptic_logarithm(P)
        5.00000000000000

    An example where precision is problematic:

    Sage

        sage: E = EllipticCurve([1, 0, 1, -85357462, 303528987048])  #18074g1
        sage: P = E([4458713781401/835903744, -64466909836503771/24167649046528, 1])
        sage: L = E.period_lattice()
        sage: L.ei()
        [5334.003952567705? - 1.964393150436?e-6*I,
         5334.003952567705? + 1.964393150436?e-6*I,
         -10668.25790513541?]
        sage: L.elliptic_logarithm(P,prec=100)
        0.27656204014107061464076203097

    [Python]

        >>> from sage.all import *
        >>> E = EllipticCurve([Integer(1), Integer(0), Integer(1), -Integer(85357462), Integer(303528987048)])  #18074g1
        >>> P = E([Integer(4458713781401)/Integer(835903744), -Integer(64466909836503771)/Integer(24167649046528), Integer(1)])
        >>> L = E.period_lattice()
        >>> L.ei()
        [5334.003952567705? - 1.964393150436?e-6*I,
         5334.003952567705? + 1.964393150436?e-6*I,
         -10668.25790513541?]
        >>> L.elliptic_logarithm(P,prec=Integer(100))
        0.27656204014107061464076203097

    Some complex examples, taken from the paper by Cremona and Thongjunthug:

    Sage

        sage: # needs sage.rings.number_field
        sage: K.<i> = QuadraticField(-1)
        sage: a4 = 9*i - 10
        sage: a6 = 21 - i
        sage: E = EllipticCurve([0,0,0,a4,a6])
        sage: e1 = 3 - 2*i; e2 = 1 + i; e3 = -4 + i
        sage: emb = K.embeddings(CC)[1]
        sage: L = E.period_lattice(emb)
        sage: P = E(2 - i, 4 + 2*i)

    Python

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)
        >>> a4 = Integer(9)*i - Integer(10)
        >>> a6 = Integer(21) - i
        >>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),a4,a6])
        >>> e1 = Integer(3) - Integer(2)*i; e2 = Integer(1) + i; e3 = -Integer(4) + i
        >>> emb = K.embeddings(CC)[Integer(1)]
        >>> L = E.period_lattice(emb)
        >>> P = E(Integer(2) - i, Integer(4) + Integer(2)*i)

    By default, the output is reduced with respect to the normalised lattice basis, so that its coordinates with respect to that basis lie in the interval [0,1):

    Sage

        sage: z = L.elliptic_logarithm(P, prec=100); z                              # needs sage.rings.number_field
        0.70448375537782208460499649302 - 0.79246725643650979858266018068*I
        sage: L.coordinates(z)                                                      # needs sage.rings.number_field
        (0.46247636364807931766105406092, 0.79497588726808704200760395829)

    [Python]

        >>> from sage.all import *
        >>> z = L.elliptic_logarithm(P, prec=Integer(100)); z                              # needs sage.rings.number_field
        0.70448375537782208460499649302 - 0.79246725643650979858266018068*I
        >>> L.coordinates(z)                                                      # needs sage.rings.number_field
        (0.46247636364807931766105406092, 0.79497588726808704200760395829)

    Using `reduce=False` this step can be omitted. In this case the coordinates are usually in the interval [-0.5,0.5), but this is not guaranteed. This option is mainly for testing purposes:

    Sage

        sage: z = L.elliptic_logarithm(P, prec=100, reduce=False); z                # needs sage.rings.number_field
        0.57002153834710752778063503023 + 0.46476340520469798857457031393*I
        sage: L.coordinates(z)                                                      # needs sage.rings.number_field
        (0.46247636364807931766105406092, -0.20502411273191295799239604171)

    Python

        >>> from sage.all import *
        >>> z = L.elliptic_logarithm(P, prec=Integer(100), reduce=False); z                # needs sage.rings.number_field
        0.57002153834710752778063503023 + 0.46476340520469798857457031393*I
        >>> L.coordinates(z)                                                      # needs sage.rings.number_field
        (0.46247636364807931766105406092, -0.20502411273191295799239604171)

    The elliptic logs of the 2-torsion points are half-periods:

    Sage

        sage: L.elliptic_logarithm(E(e1, 0), prec=100)                              # needs sage.rings.number_field
        0.64607575874356525952487867052 + 0.22379609053909448304176885364*I
        sage: L.elliptic_logarithm(E(e2, 0), prec=100)                              # needs sage.rings.number_field
        0.71330686725892253793705940192 - 0.40481924028150941053684639367*I
        sage: L.elliptic_logarithm(E(e3, 0), prec=100)                              # needs sage.rings.number_field
        0.067231108515357278412180731396 - 0.62861533082060389357861524731*I

    [Python]

        >>> from sage.all import *
        >>> L.elliptic_logarithm(E(e1, Integer(0)), prec=Integer(100))                              # needs sage.rings.number_field
        0.64607575874356525952487867052 + 0.22379609053909448304176885364*I
        >>> L.elliptic_logarithm(E(e2, Integer(0)), prec=Integer(100))                              # needs sage.rings.number_field
        0.71330686725892253793705940192 - 0.40481924028150941053684639367*I
        >>> L.elliptic_logarithm(E(e3, Integer(0)), prec=Integer(100))                              # needs sage.rings.number_field
        0.067231108515357278412180731396 - 0.62861533082060389357861524731*I

    We check this by doubling and seeing that the resulting coordinates are integers:

    Sage

        sage: L.coordinates(2*L.elliptic_logarithm(E(e1, 0), prec=100))             # needs sage.rings.number_field
        (1.0000000000000000000000000000, 0.00000000000000000000000000000)
        sage: L.coordinates(2*L.elliptic_logarithm(E(e2, 0), prec=100))             # needs sage.rings.number_field
        (1.0000000000000000000000000000, 1.0000000000000000000000000000)
        sage: L.coordinates(2*L.elliptic_logarithm(E(e3, 0), prec=100))             # needs sage.rings.number_field
        (0.00000000000000000000000000000, 1.0000000000000000000000000000)

    Python

        >>> from sage.all import *
        >>> L.coordinates(Integer(2)*L.elliptic_logarithm(E(e1, Integer(0)), prec=Integer(100)))             # needs sage.rings.number_field
        (1.0000000000000000000000000000, 0.00000000000000000000000000000)
        >>> L.coordinates(Integer(2)*L.elliptic_logarithm(E(e2, Integer(0)), prec=Integer(100)))             # needs sage.rings.number_field
        (1.0000000000000000000000000000, 1.0000000000000000000000000000)
        >>> L.coordinates(Integer(2)*L.elliptic_logarithm(E(e3, Integer(0)), prec=Integer(100)))             # needs sage.rings.number_field
        (0.00000000000000000000000000000, 1.0000000000000000000000000000)

    Sage

        sage: # needs sage.rings.number_field
        sage: a4 = -78*i + 104
        sage: a6 = -216*i - 312
        sage: E = EllipticCurve([0,0,0,a4,a6])
        sage: emb = K.embeddings(CC)[1]
        sage: L = E.period_lattice(emb)
        sage: P = E(3 + 2*i, 14 - 7*i)
        sage: L.elliptic_logarithm(P)
        0.297147783912228 - 0.546125549639461*I
        sage: L.coordinates(L.elliptic_logarithm(P))
        (0.628653378040238, 0.371417754610223)
        sage: e1 = 1 + 3*i; e2 = -4 - 12*i; e3 = -e1 - e2
        sage: L.coordinates(L.elliptic_logarithm(E(e1, 0)))
        (0.500000000000000, 0.500000000000000)
        sage: L.coordinates(L.elliptic_logarithm(E(e2, 0)))
        (1.00000000000000, 0.500000000000000)
        sage: L.coordinates(L.elliptic_logarithm(E(e3, 0)))
        (0.500000000000000, 0.000000000000000)

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> a4 = -Integer(78)*i + Integer(104)
        >>> a6 = -Integer(216)*i - Integer(312)
        >>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),a4,a6])
        >>> emb = K.embeddings(CC)[Integer(1)]
        >>> L = E.period_lattice(emb)
        >>> P = E(Integer(3) + Integer(2)*i, Integer(14) - Integer(7)*i)
        >>> L.elliptic_logarithm(P)
        0.297147783912228 - 0.546125549639461*I
        >>> L.coordinates(L.elliptic_logarithm(P))
        (0.628653378040238, 0.371417754610223)
        >>> e1 = Integer(1) + Integer(3)*i; e2 = -Integer(4) - Integer(12)*i; e3 = -e1 - e2
        >>> L.coordinates(L.elliptic_logarithm(E(e1, Integer(0))))
        (0.500000000000000, 0.500000000000000)
        >>> L.coordinates(L.elliptic_logarithm(E(e2, Integer(0))))
        (1.00000000000000, 0.500000000000000)
        >>> L.coordinates(L.elliptic_logarithm(E(e3, Integer(0))))
        (0.500000000000000, 0.000000000000000)

  gens(prec=None, algorithm='sage')

    Return a basis for this period lattice as a 2-tuple.

    This is an alias for `basis()`. See the docstring there for a more in-depth explanation and further examples.

    INPUT:

    - `prec` – (default: `None`) precision in bits (default precision if `None`)

    - `algorithm` – string (default: `'sage'`); choice of implementation (for real embeddings only) between `'sage'` (native Sage implementation) or `'pari'` (use the PARI library: only available for real embeddings)

    OUTPUT:

    (tuple of Complex) \((\omega_1,\omega_2)\) where the lattice is \(\ZZ\omega_1 + \ZZ\omega_2\). If the lattice is real then \(\omega_1\) is real and positive, \(\Im(\omega_2)>0\) and \(\Re(\omega_2/\omega_1)\) is either \(0\) (for rectangular lattices) or \(\frac{1}{2}\) (for non-rectangular lattices). Otherwise, \(\omega_1/\omega_2\) is in the fundamental region of the upper half-plane. If the latter normalisation is required for real lattices, use the method `normalised_basis()` instead.

    EXAMPLES:

    Sage

        sage: E = EllipticCurve('37a')
        sage: E.period_lattice().gens()
        (2.99345864623196, 2.45138938198679*I)

        sage: E.period_lattice().gens(prec=100)
        (2.9934586462319596298320099794, 2.4513893819867900608542248319*I)

    Python

        >>> from sage.all import *
        >>> E = EllipticCurve('37a')
        >>> E.period_lattice().gens()
        (2.99345864623196, 2.45138938198679*I)

        >>> E.period_lattice().gens(prec=Integer(100))
        (2.9934586462319596298320099794, 2.4513893819867900608542248319*I)

  property is_approximate

    `self.is_approximate` is deprecated, use `not self.curve().is_exact()` instead.

  is_real()

    Return `True` if this period lattice is real.

    EXAMPLES:

    Sage

        sage: f = EllipticCurve('11a')
        sage: f.period_lattice().is_real()
        True

    Python

        >>> from sage.all import *
        >>> f = EllipticCurve('11a')
        >>> f.period_lattice().is_real()
        True

    Sage

        sage: # needs sage.rings.number_field
        sage: K.<i> = QuadraticField(-1)
        sage: E = EllipticCurve(K, [0,0,0,i,2*i])
        sage: emb = K.embeddings(ComplexField())[0]
        sage: L = E.period_lattice(emb)
        sage: L.is_real()
        False

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)
        >>> E = EllipticCurve(K, [Integer(0),Integer(0),Integer(0),i,Integer(2)*i])
        >>> emb = K.embeddings(ComplexField())[Integer(0)]
        >>> L = E.period_lattice(emb)
        >>> L.is_real()
        False

    Sage

        sage: x = polygen(ZZ, 'x')
        sage: K.<a> = NumberField(x^3 - 2)                                          # needs sage.rings.number_field
        sage: E = EllipticCurve([0,1,0,a,a])                                        # needs sage.rings.number_field
        sage: [E.period_lattice(emb).is_real() for emb in K.embeddings(CC)]         # needs sage.rings.number_field
        [False, False, True]

    Python

        >>> from sage.all import *
        >>> x = polygen(ZZ, 'x')
        >>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)# needs sage.rings.number_field
        >>> E = EllipticCurve([Integer(0),Integer(1),Integer(0),a,a])                                        # needs sage.rings.number_field
        >>> [E.period_lattice(emb).is_real() for emb in K.embeddings(CC)]         # needs sage.rings.number_field
        [False, False, True]

    ALGORITHM:

    The lattice is real if it is associated to a real embedding; such lattices are stable under conjugation.

  is_rectangular()

    Return `True` if this period lattice is rectangular.

    Note

    Only defined for real lattices; a `RuntimeError` is raised for non-real lattices.

    EXAMPLES:

    Sage

        sage: f = EllipticCurve('11a')
        sage: f.period_lattice().basis()
        (1.26920930427955, 0.634604652139777 + 1.45881661693850*I)
        sage: f.period_lattice().is_rectangular()
        False

    Python

        >>> from sage.all import *
        >>> f = EllipticCurve('11a')
        >>> f.period_lattice().basis()
        (1.26920930427955, 0.634604652139777 + 1.45881661693850*I)
        >>> f.period_lattice().is_rectangular()
        False

    Sage

        sage: f = EllipticCurve('37b')
        sage: f.period_lattice().basis()
        (1.08852159290423, 1.76761067023379*I)
        sage: f.period_lattice().is_rectangular()
        True

    [Python]

        >>> from sage.all import *
        >>> f = EllipticCurve('37b')
        >>> f.period_lattice().basis()
        (1.08852159290423, 1.76761067023379*I)
        >>> f.period_lattice().is_rectangular()
        True

    ALGORITHM:

    The period lattice is rectangular precisely if the discriminant of the Weierstrass equation is positive, or equivalently if the number of real components is 2.

  normalised_basis(prec=None, algorithm='sage')

    Return a normalised basis for this period lattice as a 2-tuple.

    INPUT:

    - `prec` – (default: `None`) precision in bits (default precision if `None`)

    - `algorithm` – string (default: `'sage'`); choice of implementation (for real embeddings only) between `'sage'` (native Sage implementation) or `'pari'` (use the PARI library: only available for real embeddings)

    OUTPUT:

    (tuple of Complex) \((\omega_1,\omega_2)\) where the lattice has the form \(\ZZ\omega_1 + \ZZ\omega_2\). The basis is normalised so that \(\omega_1/\omega_2\) is in the fundamental region of the upper half-plane. For an alternative normalisation for real lattices (with the first period real), use the method `basis()` instead.

    EXAMPLES:

    Sage

        sage: E = EllipticCurve('37a')
        sage: E.period_lattice().normalised_basis()
        (2.99345864623196, -2.45138938198679*I)

    Python

        >>> from sage.all import *
        >>> E = EllipticCurve('37a')
        >>> E.period_lattice().normalised_basis()
        (2.99345864623196, -2.45138938198679*I)

    Sage

        sage: # needs sage.rings.number_field
        sage: x = polygen(ZZ, 'x')
        sage: K.<a> = NumberField(x^3 - 2)
        sage: emb = K.embeddings(RealField())[0]
        sage: E = EllipticCurve([0,1,0,a,a])
        sage: L = E.period_lattice(emb)
        sage: L.normalised_basis(64)
        (1.90726488608927255 - 1.34047785962440202*I,
        -1.90726488608927255 - 1.34047785962440202*I)

        sage: # needs sage.rings.number_field
        sage: emb = K.embeddings(ComplexField())[0]
        sage: L = E.period_lattice(emb)
        sage: w1, w2 = L.normalised_basis(); w1, w2
        (-1.37588604166076 - 2.58560946624443*I,
         -2.10339907847356 + 0.428378776460622*I)
        sage: L.is_real()
        False
        sage: tau = w1/w2; tau
        0.387694505032876 + 1.30821088214407*I

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> x = polygen(ZZ, 'x')
        >>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
        >>> emb = K.embeddings(RealField())[Integer(0)]
        >>> E = EllipticCurve([Integer(0),Integer(1),Integer(0),a,a])
        >>> L = E.period_lattice(emb)
        >>> L.normalised_basis(Integer(64))
        (1.90726488608927255 - 1.34047785962440202*I,
        -1.90726488608927255 - 1.34047785962440202*I)

        >>> # needs sage.rings.number_field
        >>> emb = K.embeddings(ComplexField())[Integer(0)]
        >>> L = E.period_lattice(emb)
        >>> w1, w2 = L.normalised_basis(); w1, w2
        (-1.37588604166076 - 2.58560946624443*I,
         -2.10339907847356 + 0.428378776460622*I)
        >>> L.is_real()
        False
        >>> tau = w1/w2; tau
        0.387694505032876 + 1.30821088214407*I

  omega(prec=None, bsd_normalise=False)

    Return the real or complex volume of this period lattice.

    INPUT:

    - `prec` – integer or `None` (default); real precision in bits (default real precision if `None`)

    - `bsd_normalise` – boolean (default: `False`); flag to use BSD normalisation in the complex case

    OUTPUT:

    (real) For real lattices, this is the real period times the number of connected components. For non-real lattices it is the complex area, or double the area if `bsd_normalise` is `True`.

    Note

    If the curve is given by a global minimal Weierstrass equation, then with `bsd_normalise` = `True`, this gives the correct period in the BSD conjecture: the product of this quantity over all embeddings appears in the BSD formula. In general a correction factor is required to make allowance for the model.

    EXAMPLES:

    Sage

        sage: E = EllipticCurve('37a')
        sage: E.period_lattice().omega()
        5.98691729246392

    Python

        >>> from sage.all import *
        >>> E = EllipticCurve('37a')
        >>> E.period_lattice().omega()
        5.98691729246392

    This is not a minimal model:

    Sage

        sage: E = EllipticCurve([0, -432*6^2])
        sage: E.period_lattice().omega()
        0.486109385710056

    [Python]

        >>> from sage.all import *
        >>> E = EllipticCurve([Integer(0), -Integer(432)*Integer(6)**Integer(2)])
        >>> E.period_lattice().omega()
        0.486109385710056

    If you were to plug the above omega into the BSD conjecture, you would get an incorrect value, out by a factor of 2. The following works though:

    Sage

        sage: F = E.minimal_model()
        sage: F.period_lattice().omega()
        0.972218771420113

    Python

        >>> from sage.all import *
        >>> F = E.minimal_model()
        >>> F.period_lattice().omega()
        0.972218771420113

    Sage

        sage: # needs sage.rings.number_field
        sage: x = polygen(ZZ, 'x')
        sage: K.<a> = NumberField(x^3 - 2)
        sage: emb = K.embeddings(RealField())[0]
        sage: E = EllipticCurve([0,1,0,a,a])
        sage: L = E.period_lattice(emb)
        sage: L.omega(64)
        3.81452977217854509

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> x = polygen(ZZ, 'x')
        >>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
        >>> emb = K.embeddings(RealField())[Integer(0)]
        >>> E = EllipticCurve([Integer(0),Integer(1),Integer(0),a,a])
        >>> L = E.period_lattice(emb)
        >>> L.omega(Integer(64))
        3.81452977217854509

    A complex example (taken from J.E.Cremona and E.Whitley, Periods of cusp forms and elliptic curves over imaginary quadratic fields, Mathematics of Computation 62 No. 205 (1994), 407-429). See Issue #29645 and Issue #29782:

    Sage

        sage: # needs sage.rings.number_field
        sage: K.<i> = QuadraticField(-1)
        sage: E = EllipticCurve([0,1-i,i,-i,0])
        sage: L = E.period_lattice(K.embeddings(CC)[0])
        sage: L.omega()
        8.80694160502647
        sage: L.omega(prec=200)
        8.8069416050264741493250743632295462227858630765392114070032
        sage: L.omega(bsd_normalise=True)
        17.6138832100529

    Python

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)
        >>> E = EllipticCurve([Integer(0),Integer(1)-i,i,-i,Integer(0)])
        >>> L = E.period_lattice(K.embeddings(CC)[Integer(0)])
        >>> L.omega()
        8.80694160502647
        >>> L.omega(prec=Integer(200))
        8.8069416050264741493250743632295462227858630765392114070032
        >>> L.omega(bsd_normalise=True)
        17.6138832100529

  real_period(prec=None, algorithm='sage')

    Return the real period of this period lattice.

    INPUT:

    - `prec` – integer or `None` (default); real precision in bits (default real precision if `None`)

    - `algorithm` – string (default: `'sage'`); choice of implementation (for real embeddings only) between `'sage'` (native Sage implementation) or `'pari'` (use the PARI library: only available for real embeddings)

    Note

    Only defined for real lattices; a `RuntimeError` is raised for non-real lattices.

    EXAMPLES:

    Sage

        sage: E = EllipticCurve('37a')
        sage: E.period_lattice().real_period()
        2.99345864623196

    Python

        >>> from sage.all import *
        >>> E = EllipticCurve('37a')
        >>> E.period_lattice().real_period()
        2.99345864623196

    Sage

        sage: # needs sage.rings.number_field
        sage: x = polygen(ZZ, 'x')
        sage: K.<a> = NumberField(x^3 - 2)
        sage: emb = K.embeddings(RealField())[0]
        sage: E = EllipticCurve([0,1,0,a,a])
        sage: L = E.period_lattice(emb)
        sage: L.real_period(64)
        3.81452977217854509

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> x = polygen(ZZ, 'x')
        >>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
        >>> emb = K.embeddings(RealField())[Integer(0)]
        >>> E = EllipticCurve([Integer(0),Integer(1),Integer(0),a,a])
        >>> L = E.period_lattice(emb)
        >>> L.real_period(Integer(64))
        3.81452977217854509

  reduce(z)

    Reduce a complex number modulo the lattice.

    INPUT:

    - `z` – complex number

    OUTPUT:

    (complex) the reduction of \(z\) modulo the lattice, lying in the fundamental period parallelogram with respect to the lattice basis. For curves defined over the reals (i.e. real embeddings) the output will be real when possible.

    EXAMPLES:

    Sage

        sage: E = EllipticCurve('389a')
        sage: L = E.period_lattice()
        sage: w1, w2 = L.basis(prec=100)
        sage: P = E([-1,1])
        sage: zP = P.elliptic_logarithm(precision=100); zP  # abs tol 1e-28
        0.47934825019021931612953301006 + 0.98586885077582410221120384908*I
        sage: z = zP + 10*w1 - 20*w2; z  # abs tol 1e-28
        25.381473858740770069343110929 - 38.448885180257139986236950114*I
        sage: L.reduce(z)  # abs tol 1e-28
        0.47934825019021931612953301006 + 0.98586885077582410221120384908*I
        sage: L.elliptic_logarithm(2*P)  # abs tol 1e-15
        0.958696500380439
        sage: L.reduce(L.elliptic_logarithm(2*P))  # abs tol 1e-15
        0.958696500380439
        sage: L.reduce(L.elliptic_logarithm(2*P) + 10*w1 - 20*w2)  # abs tol 1e-15
        0.958696500380444

    Python

        >>> from sage.all import *
        >>> E = EllipticCurve('389a')
        >>> L = E.period_lattice()
        >>> w1, w2 = L.basis(prec=Integer(100))
        >>> P = E([-Integer(1),Integer(1)])
        >>> zP = P.elliptic_logarithm(precision=Integer(100)); zP  # abs tol 1e-28
        0.47934825019021931612953301006 + 0.98586885077582410221120384908*I
        >>> z = zP + Integer(10)*w1 - Integer(20)*w2; z  # abs tol 1e-28
        25.381473858740770069343110929 - 38.448885180257139986236950114*I
        >>> L.reduce(z)  # abs tol 1e-28
        0.47934825019021931612953301006 + 0.98586885077582410221120384908*I
        >>> L.elliptic_logarithm(Integer(2)*P)  # abs tol 1e-15
        0.958696500380439
        >>> L.reduce(L.elliptic_logarithm(Integer(2)*P))  # abs tol 1e-15
        0.958696500380439
        >>> L.reduce(L.elliptic_logarithm(Integer(2)*P) + Integer(10)*w1 - Integer(20)*w2)  # abs tol 1e-15
        0.958696500380444

  sigma(z, prec=None, flag=0)

    Return the value of the Weierstrass sigma function for this elliptic curve period lattice.

    INPUT:

    - `z` – a complex number

    - `prec` – (default: `None`) real precision in bits

        (default real precision if `None`)

    - `flag` –

      0: (default) ???;

      1: computes an arbitrary determination of log(sigma(z))

      2, 3: same using the product expansion instead of theta series. ???

    Note

    The reason for the ???’s above, is that the PARI documentation for ellsigma is very vague. Also this is only implemented for curves defined over \(\QQ\).

    Todo

    This function does not use any of the PeriodLattice functions and so should be moved to ell_rational_field.

    EXAMPLES:

    Sage

        sage: EllipticCurve('389a1').period_lattice().sigma(CC(2,1))
        2.60912163570108 - 0.200865080824587*I

    Python

        >>> from sage.all import *
        >>> EllipticCurve('389a1').period_lattice().sigma(CC(Integer(2),Integer(1)))
        2.60912163570108 - 0.200865080824587*I

  tau(prec=None, algorithm='sage')

    Return the upper half-plane parameter in the fundamental region.

    INPUT:

    - `prec` – (default: `None`) precision in bits (default precision if `None`)

    - `algorithm` – string (default: `'sage'`); choice of implementation (for real embeddings only) between ‘sage’ (native Sage implementation) or ‘pari’ (use the PARI library: only available for real embeddings)

    OUTPUT:

    (Complex) \(\tau = \omega_1/\omega_2\) where the lattice has the form \(\ZZ\omega_1 + \ZZ\omega_2\), normalised so that \(\tau = \omega_1/\omega_2\) is in the fundamental region of the upper half-plane.

    EXAMPLES:

    Sage

        sage: E = EllipticCurve('37a')
        sage: L = E.period_lattice()
        sage: L.tau()
        1.22112736076463*I

    Python

        >>> from sage.all import *
        >>> E = EllipticCurve('37a')
        >>> L = E.period_lattice()
        >>> L.tau()
        1.22112736076463*I

    Sage

        sage: # needs sage.rings.number_field
        sage: x = polygen(ZZ, 'x')
        sage: K.<a> = NumberField(x^3 - 2)
        sage: emb = K.embeddings(RealField())[0]
        sage: E = EllipticCurve([0,1,0,a,a])
        sage: L = E.period_lattice(emb)
        sage: tau = L.tau(); tau
        -0.338718341018919 + 0.940887817679340*I
        sage: tau.abs()
        1.00000000000000
        sage: -0.5 <= tau.real() <= 0.5
        True

        sage: # needs sage.rings.number_field
        sage: emb = K.embeddings(ComplexField())[0]
        sage: L = E.period_lattice(emb)
        sage: tau = L.tau(); tau
        0.387694505032876 + 1.30821088214407*I
        sage: tau.abs()
        1.36444961115933
        sage: -0.5 <= tau.real() <= 0.5
        True

    [Python]

        >>> from sage.all import *
        >>> # needs sage.rings.number_field
        >>> x = polygen(ZZ, 'x')
        >>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
        >>> emb = K.embeddings(RealField())[Integer(0)]
        >>> E = EllipticCurve([Integer(0),Integer(1),Integer(0),a,a])
        >>> L = E.period_lattice(emb)
        >>> tau = L.tau(); tau
        -0.338718341018919 + 0.940887817679340*I
        >>> tau.abs()
        1.00000000000000
        >>> -RealNumber('0.5') <= tau.real() <= RealNumber('0.5')
        True

        >>> # needs sage.rings.number_field
        >>> emb = K.embeddings(ComplexField())[Integer(0)]
        >>> L = E.period_lattice(emb)
        >>> tau = L.tau(); tau
        0.387694505032876 + 1.30821088214407*I
        >>> tau.abs()
        1.36444961115933
        >>> -RealNumber('0.5') <= tau.real() <= RealNumber('0.5')
        True

sage.schemes.elliptic_curves.period_lattice.extended_agm_iteration(a, b, c)

  Internal function for the extended AGM used in elliptic logarithm computation.

  INPUT:

  - `a`, `b`, `c` – three real or complex numbers

  OUTPUT:

  (3-tuple) \((a_0,b_0,c_0)\), the limit of the iteration \((a,b,c) \mapsto ((a+b)/2,\sqrt{ab},(c+\sqrt{c^2+b^2-a^2})/2)\).

  EXAMPLES:

  Sage

      sage: # needs sage.rings.real_mpfr
      sage: from sage.schemes.elliptic_curves.period_lattice import extended_agm_iteration
      sage: extended_agm_iteration(RR(1), RR(2), RR(3))
      (1.45679103104691, 1.45679103104691, 3.21245294970054)
      sage: extended_agm_iteration(CC(1,2), CC(2,3), CC(3,4))
      (1.46242448156430 + 2.47791311676267*I,
       1.46242448156430 + 2.47791311676267*I,
       3.22202144343535 + 4.28383734262540*I)

  Python

      >>> from sage.all import *
      >>> # needs sage.rings.real_mpfr
      >>> from sage.schemes.elliptic_curves.period_lattice import extended_agm_iteration
      >>> extended_agm_iteration(RR(Integer(1)), RR(Integer(2)), RR(Integer(3)))
      (1.45679103104691, 1.45679103104691, 3.21245294970054)
      >>> extended_agm_iteration(CC(Integer(1),Integer(2)), CC(Integer(2),Integer(3)), CC(Integer(3),Integer(4)))
      (1.46242448156430 + 2.47791311676267*I,
       1.46242448156430 + 2.47791311676267*I,
       3.22202144343535 + 4.28383734262540*I)

sage.schemes.elliptic_curves.period_lattice.normalise_periods(w1, w2)

  Normalise the period basis \((w_1,w_2)\) so that \(w_1/w_2\) is in the fundamental region.

  INPUT:

  - `w1`, `w2` – two complex numbers with non-real ratio

  OUTPUT:

  (tuple) \(((\omega_1',\omega_2'),[a,b,c,d])\) where \(a,b,c,d\) are integers such that

  - \(ad-bc=\pm1\);

  - \((\omega_1',\omega_2') = (a\omega_1+b\omega_2,c\omega_1+d\omega_2)\);

  - \(\tau=\omega_1'/\omega_2'\) is in the upper half plane;

  - \(|\tau|\ge1\) and \(|\Re(\tau)|\le\frac{1}{2}\).

  EXAMPLES:

  Sage

      sage: # needs sage.rings.real_mpfr sage.symbolic
      sage: from sage.schemes.elliptic_curves.period_lattice import reduce_tau, normalise_periods
      sage: w1 = CC(1.234, 3.456)
      sage: w2 = CC(1.234, 3.456000001)
      sage: w1/w2    # in lower half plane!
      0.999999999743367 - 9.16334785827644e-11*I
      sage: w1w2, abcd = normalise_periods(w1, w2)
      sage: a,b,c,d = abcd
      sage: w1w2 == (a*w1+b*w2, c*w1+d*w2)
      True
      sage: w1w2[0]/w1w2[1]
      1.23400010389203e9*I
      sage: a*d-b*c # note change of orientation
      -1

  Python

      >>> from sage.all import *
      >>> # needs sage.rings.real_mpfr sage.symbolic
      >>> from sage.schemes.elliptic_curves.period_lattice import reduce_tau, normalise_periods
      >>> w1 = CC(RealNumber('1.234'), RealNumber('3.456'))
      >>> w2 = CC(RealNumber('1.234'), RealNumber('3.456000001'))
      >>> w1/w2    # in lower half plane!
      0.999999999743367 - 9.16334785827644e-11*I
      >>> w1w2, abcd = normalise_periods(w1, w2)
      >>> a,b,c,d = abcd
      >>> w1w2 == (a*w1+b*w2, c*w1+d*w2)
      True
      >>> w1w2[Integer(0)]/w1w2[Integer(1)]
      1.23400010389203e9*I
      >>> a*d-b*c # note change of orientation
      -1

sage.schemes.elliptic_curves.period_lattice.reduce_tau(tau)

  Transform a point in the upper half plane to the fundamental region.

  INPUT:

  - `tau` – complex number with positive imaginary part

  OUTPUT:

  (tuple) \((\tau',[a,b,c,d])\) where \(a,b,c,d\) are integers such that

  - \(ad-bc=1\);

  - \(\tau'=(a\tau+b)/(c\tau+d)\);

  - \(|\tau'|\ge1\);

  - \(|\Re(\tau')|\le\frac{1}{2}\).

  EXAMPLES:

  Sage

      sage: # needs sage.rings.real_mpfr sage.symbolic
      sage: from sage.schemes.elliptic_curves.period_lattice import reduce_tau
      sage: reduce_tau(CC(1.23,3.45))
      (0.230000000000000 + 3.45000000000000*I, [1, -1, 0, 1])
      sage: reduce_tau(CC(1.23,0.0345))
      (-0.463960069171512 + 1.35591888067914*I, [-5, 6, 4, -5])
      sage: reduce_tau(CC(1.23,0.0000345))
      (0.130000000001761 + 2.89855072463768*I, [13, -16, 100, -123])

  Python

      >>> from sage.all import *
      >>> # needs sage.rings.real_mpfr sage.symbolic
      >>> from sage.schemes.elliptic_curves.period_lattice import reduce_tau
      >>> reduce_tau(CC(RealNumber('1.23'),RealNumber('3.45')))
      (0.230000000000000 + 3.45000000000000*I, [1, -1, 0, 1])
      >>> reduce_tau(CC(RealNumber('1.23'),RealNumber('0.0345')))
      (-0.463960069171512 + 1.35591888067914*I, [-5, 6, 4, -5])
      >>> reduce_tau(CC(RealNumber('1.23'),RealNumber('0.0000345')))
      (0.130000000001761 + 2.89855072463768*I, [13, -16, 100, -123])