Hyperelliptic curves of genus 2 in the smooth model¶

Hyperelliptic curves of genus 2 in the smooth model

AUTHORS:

David Kohel (2006): initial version
Sabrina Kunzweiler, Gareth Ma, Giacomo Pope (2024): adapt to smooth model

class sage.schemes.hyperelliptic_curves.hyperelliptic_g2.HyperellipticCurve_g2(defining_polynomial, f, h, genus: Integer, names=['x', 'y'])[source]¶

Bases: HyperellipticCurve_generic

absolute_igusa_invariants_kohel()[source]¶

Return the three absolute Igusa invariants used by Kohel [KohECHIDNA].

EXAMPLES:

sage: R.<x> = QQ[]
sage: f = x^5 - x + 2
sage: HyperellipticCurve(f).igusa_clebsch_invariants()
(-640, -20480, 1310720, 52160364544)
sage: HyperellipticCurve(f(2*x)).igusa_clebsch_invariants()
(-40960, -83886080, 343597383680, 56006764965979488256)

sage: HyperellipticCurve(f, x).igusa_clebsch_invariants()
(-640, 17920, -1966656, 52409511936)
sage: HyperellipticCurve(f(2*x), 2*x).igusa_clebsch_invariants()
(-40960, 73400320, -515547070464, 56274284941110411264)

is_odd_degree()[source]¶

Return True if the curve is an odd degree model.

EXAMPLES:

sage: R.<x> = QQ[]
sage: f = x^5 - x^4 + 3
sage: HyperellipticCurve(f).is_odd_degree()
True

jacobian()[source]¶

Return the Jacobian of the hyperelliptic curve.

Elements of the Jacobian are represented by tuples of the form \((u, v : n)\), where

\((u,v)\) is the Mumford representative of a divisor \(P_1 + ... + P_r\),
\(n\) is a non-negative integer

This tuple represents the equivalence class \([P_1 + ... + P_r + n \cdot \infty_+ + m \cdot \infty_- - D_\infty]\), where \(m = g - \deg(u) - n\), and \(\infty_+\), \(\infty_-\) are the points at infinity of the hyperelliptic curve,

\[D_\infty = \lceil g/2 \rceil \infty_+ + \lfloor g/2 \rfloor \infty_-.\]

Here, \(\infty_- = \infty_+\), if the hyperelliptic curve is ramified.

EXAMPLES:

sage: R.<x> = QQ[]
sage: H = HyperellipticCurve(2*x^5 + 4*x^4 + x^3 - x, x^3 + x + 1)
sage: J = H.jacobian(); J
Jacobian of Hyperelliptic Curve over Rational Field defined by y^2 + (x^3 + x + 1)*y = 2*x^5 + 4*x^4 + x^3 - x

The points \(P = (0, 0)\) and \(Q = (-1, -1)\) are on \(H\). We construct the element \(D_1 = [P - Q] = [P + (-Q) - D_\infty]\) on the Jacobian:

sage: P = H.point([0, 0])
sage: Q = H.point([-1, -1])
sage: D1 = J(P,Q); D1
(x^2 + x, -2*x : 0)

Elements of the Jacobian can also be constructed by directly providing the Mumford representation:

sage: D1 == J(x^2 + x, -2*x, 0)
True

We can also embed single points into the Jacobian. Below we construct \(D_2 = [P - P_0]\), where \(P_0\) is the distinguished point of \(H\) (by default one of the points at infinity):

sage: D2 = J(P); D2
(x, 0 : 0)
sage: P0 = H.distinguished_point(); P0
(1 : 0 : 0)
sage: D2 == J(P, P0)
True

We may add elements, or multiply by integers:

sage: 2*D1
(x, -1 : 1)
sage: D1 + D2
(x^2 + x, -1 : 0)
sage: -D2
(x, -1 : 1)

Note that the neutral element is given by \([D_\infty - D_\infty]\), in particular \(n = 1\):

sage: J.zero()
(1, 0 : 1)

There are two more elements of the Jacobian that are only supported at infinity: \([\infty_+ - \infty_-]\) and \([\infty_- - \infty_+]\):

sage: [P_plus, P_minus] = H.points_at_infinity()
sage: P_plus == P0
True
sage: J(P_plus,P_minus)
(1, 0 : 2)
sage: J(P_minus, P_plus)
(1, 0 : 0)

class sage.schemes.hyperelliptic_curves.hyperelliptic_g2.HyperellipticCurve_g2_finite_field(projective_model, f, h, genus: Integer, names=['x', 'y'])[source]¶: Bases: HyperellipticCurve_g2, HyperellipticCurve_finite_field

class sage.schemes.hyperelliptic_curves.hyperelliptic_g2.HyperellipticCurve_g2_padic_field(projective_model, f, h, genus: Integer, names=['x', 'y'])[source]¶: Bases: HyperellipticCurve_g2, HyperellipticCurve_padic_field

class sage.schemes.hyperelliptic_curves.hyperelliptic_g2.HyperellipticCurve_g2_rational_field(projective_model, f, h, genus: Integer, names=['x', 'y'])[source]¶: Bases: HyperellipticCurve_g2, HyperellipticCurve_rational_field