Weierstrass $\wp$ function for elliptic curves.¶

The Weierstrass $\wp$ function associated to an elliptic curve over a field $k$ is a Laurent series of the form

$\wp(z) = \frac{1}{z^2} + c_2 \cdot z^2 + c_4 \cdot z^4 + \cdots.$

If the field is contained in $\mathbb{C}$ , then this is the series expansion of the map from $\mathbb{C}$ to $E(\mathbb{C})$ whose kernel is the period lattice of $E$ .

Over other fields, like finite fields, this still makes sense as a formal power series with coefficients in $k$ - at least its first $p-2$ coefficients where $p$ is the characteristic of $k$ . It can be defined via the formal group as $x+c$ in the variable $z=\log_E(t)$ for a constant $c$ such that the constant term $c_0$ in $\wp(z)$ is zero.

EXAMPLE:

sage: E = EllipticCurve([0,1])
sage: E.weierstrass_p()
z^-2 - 1/7*z^4 + 1/637*z^10 - 1/84721*z^16 + O(z^20)

REFERENCES:

[BMSS] Boston, Morain, Salvy, Schost, “Fast Algorithms for Isogenies.”

AUTHORS:

Dan Shumov 04/09 - original implementation
Chris Wuthrich 11/09 - major restructuring

sage.schemes.elliptic_curves.ell_wp.compute_wp_fast(k, A, B, m)¶

Computes the Weierstrass function of an elliptic curve defined by short Weierstrass model: $y^2 = x^3 + Ax + B$ . It does this with as fast as polynomial of degree $m$ can be multiplied together in the base ring, i.e. $O(M(n))$ in the notation of [BMSS].

Let $p$ be the characteristic of the underlying field: Then we must have either $p=0$ , or $p > m + 3$ .

INPUT:

k - the base field of the curve

A - and

B - as the coeffients of the short Weierstrass model $y^2 = x^3 +Ax +B$ , and

m - the precision to which the function is computed to.

OUTPUT:

the Weierstrass $\wp$ function as a Laurent series to precision $m$ .

ALGORITHM:

This function uses the algorithm described in section 3.3 of [BMSS].

EXAMPLES:

sage: from sage.schemes.elliptic_curves.ell_wp import compute_wp_fast
sage: compute_wp_fast(QQ, 1, 8, 7)
z^-2 - 1/5*z^2 - 8/7*z^4 + 1/75*z^6 + O(z^7)

sage: k = GF(37)
sage: compute_wp_fast(k, k(1), k(8), 5)
z^-2 + 22*z^2 + 20*z^4 + O(z^5)

sage.schemes.elliptic_curves.ell_wp.compute_wp_pari(E, prec)¶

Computes the Weierstrass $\wp$ -function via calling the corresponding function in pari.

EXAMPLES:

sage: E = EllipticCurve([0,1])
sage: E.weierstrass_p(algorithm='pari')
z^-2 - 1/7*z^4 + 1/637*z^10 - 1/84721*z^16 + O(z^20)

sage: E = EllipticCurve(GF(101),[5,4])
sage: E.weierstrass_p(prec=30, algorithm='pari')
z^-2 + 100*z^2 + 86*z^4 + 34*z^6 + 50*z^8 + 82*z^10 + 45*z^12 + 70*z^14 + 33*z^16 + 87*z^18 + 33*z^20 + 36*z^22 + 45*z^24 + 40*z^26 + 12*z^28 + O(z^30)

sage: from sage.schemes.elliptic_curves.ell_wp import compute_wp_pari
sage: compute_wp_pari(E, prec= 20)
z^-2 + 100*z^2 + 86*z^4 + 34*z^6 + 50*z^8 + 82*z^10 + 45*z^12 + 70*z^14 + 33*z^16 + 87*z^18 + O(z^20)

sage.schemes.elliptic_curves.ell_wp.compute_wp_quadratic(k, A, B, prec)¶

Computes the truncated Weierstrass function of an elliptic curve defined by short Weierstrass model: $y^2 = x^3 + Ax + B$ . Uses an algorithm that is of complexity $O(prec^2)$ .

Let p be the characteristic of the underlying field: Then we must have either p=0, or p > prec + 3.

INPUT:

k - the field of definition of the curve

A - and

B - the coefficients of the elliptic curve

prec - the precision to which we compute the series.

OUTPUT: A Laurent series aproximating the Weierstrass $\wp$ -function to precision prec.

ALGORITHM: This function uses the algorithm described in section 3.2 of [BMSS].

REFERENCES: [BMSS] Boston, Morain, Salvy, Schost, “Fast Algorithms for Isogenies.”

EXAMPLES:

sage: E = EllipticCurve([7,0])
sage: E.weierstrass_p(prec=10, algorithm='quadratic')
z^-2 - 7/5*z^2 + 49/75*z^6 + O(z^10)

sage: E = EllipticCurve(GF(103),[1,2])
sage: E.weierstrass_p(algorithm='quadratic')
z^-2 + 41*z^2 + 88*z^4 + 11*z^6 + 57*z^8 + 55*z^10 + 73*z^12 + 11*z^14 + 17*z^16 + 50*z^18 + O(z^20)

sage: from sage.schemes.elliptic_curves.ell_wp import compute_wp_quadratic
sage: compute_wp_quadratic(E.base_ring(), E.a4(), E.a6(), prec=10)
z^-2 + 41*z^2 + 88*z^4 + 11*z^6 + 57*z^8 + O(z^10)

sage.schemes.elliptic_curves.ell_wp.solve_linear_differential_system(a, b, c, alpha)¶

Solves a system of linear differential equations: $af' + bf = c$ and $f'(0) = \alpha$ where $a$ , $b$ , and $c$ are power series in one variable and $\alpha$ is a constant in the coefficient ring.

ALGORITHM:

due to Brent and Kung ‘78.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.ell_wp import solve_linear_differential_system
sage: k = GF(17)
sage: R.<x> = PowerSeriesRing(k)
sage: a = 1+x+O(x^7); b = x+O(x^7); c = 1+x^3+O(x^7); alpha = k(3)
sage: f = solve_linear_differential_system(a,b,c,alpha)
sage: f
3 + x + 15*x^2 + x^3 + 10*x^5 + 3*x^6 + 13*x^7 + O(x^8)
sage: a*f.derivative()+b*f - c
O(x^7)
sage: f(0) == alpha
True

sage.schemes.elliptic_curves.ell_wp.weierstrass_p(E, prec=20, algorithm=None)¶

Computes the Weierstrass $\wp$ -function on an elliptic curve.

INPUT:

E - an elliptic curve
prec - precision
algorithm - string (default:None) an algorithm identifier indicating using the pari, fast or quadratic algorithm. If the algorithm is None, then this function determines the best algorithm to use.

OUTPUT:

a Laurent series in one variable $z$ with coefficients in the base field $k$ of $E$ .

EXAMPLES:

sage: E = EllipticCurve('11a1')
sage: E.weierstrass_p(prec=10)
z^-2 + 31/15*z^2 + 2501/756*z^4 + 961/675*z^6 + 77531/41580*z^8 + O(z^10)
sage: E.weierstrass_p(prec=8)
z^-2 + 31/15*z^2 + 2501/756*z^4 + 961/675*z^6 + O(z^8)
sage: Esh = E.short_weierstrass_model()
sage: Esh.weierstrass_p(prec=8)
z^-2 + 13392/5*z^2 + 1080432/7*z^4 + 59781888/25*z^6 + O(z^8)

sage: E.weierstrass_p(prec=8, algorithm='pari')
z^-2 + 31/15*z^2 + 2501/756*z^4 + 961/675*z^6 + O(z^8)
sage: E.weierstrass_p(prec=8, algorithm='quadratic')
z^-2 + 31/15*z^2 + 2501/756*z^4 + 961/675*z^6 + O(z^8)

sage: k = GF(11)
sage: E = EllipticCurve(k, [1,1])
sage: E.weierstrass_p(prec=6, algorithm='fast')
z^-2 + 2*z^2 + 3*z^4 + O(z^6)
sage: E.weierstrass_p(prec=7, algorithm='fast')
...
ValueError: For computing the Weierstrass p-function via the fast algorithm, the characteristic (11) of the underlying field must be greater than prec + 4 = 11.
sage: E.weierstrass_p(prec=8 ,algorithm='pari')
z^-2 + 2*z^2 + 3*z^4 + 5*z^6 + O(z^8)
sage: E.weierstrass_p(prec=9, algorithm='pari')
...
ValueError: For computing the Weierstrass p-function via pari, the characteristic (11) of the underlying field must be greater than prec + 2 = 11.

Weierstrass $\wp$ function for elliptic curves.¶

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Weierstrass $\wp$ function for elliptic curves.¶