Show commands: SageMath / Magma

Simplified equation

 $y^2 = x^5 - x$ (homogenize, simplify) $y^2 = x^5z - xz^5$ (dehomogenize, simplify) $y^2 = x^5 - x$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 0, 0, 0, 1]), R([]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 0, 0, 0, 1], R![]);

sage: X = HyperellipticCurve(R([0, -1, 0, 0, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

Invariants

 Conductor: $$N$$ $$=$$ $$4096$$ $$=$$ $$2^{12}$$ magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(4096,2),R!>*])); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-65536$$ $$=$$ $$- 2^{16}$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ $$=$$ $$20$$ $$=$$ $$2^{2} \cdot 5$$ $$I_4$$ $$=$$ $$-20$$ $$=$$ $$- 2^{2} \cdot 5$$ $$I_6$$ $$=$$ $$-40$$ $$=$$ $$- 2^{3} \cdot 5$$ $$I_{10}$$ $$=$$ $$8$$ $$=$$ $$2^{3}$$ $$J_2$$ $$=$$ $$80$$ $$=$$ $$2^{4} \cdot 5$$ $$J_4$$ $$=$$ $$480$$ $$=$$ $$2^{5} \cdot 3 \cdot 5$$ $$J_6$$ $$=$$ $$-1280$$ $$=$$ $$- 2^{8} \cdot 5$$ $$J_8$$ $$=$$ $$-83200$$ $$=$$ $$- 2^{8} \cdot 5^{2} \cdot 13$$ $$J_{10}$$ $$=$$ $$65536$$ $$=$$ $$2^{16}$$ $$g_1$$ $$=$$ $$50000$$ $$g_2$$ $$=$$ $$3750$$ $$g_3$$ $$=$$ $$-125$$

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_4$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$GL(2,3)$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

Rational points

All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : 0 : 1),\, (1 : 0 : 1)$$
All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : 0 : 1),\, (1 : 0 : 1)$$
All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : 0 : 1),\, (1 : 0 : 1)$$

magma: [C![-1,0,1],C![0,0,1],C![1,0,0],C![1,0,1]]; // minimal model

magma: [C![-1,0,1],C![0,0,1],C![1,0,0],C![1,0,1]]; // simplified model

Number of rational Weierstrass points: $$4$$

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

This curve is locally solvable everywhere.

magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);

Mordell-Weil group of the Jacobian

Group structure: $$\Z/{2}\Z \times \Z/{2}\Z \times \Z/{2}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : 0 : 1) - (1 : 0 : 0)$$ $$x + z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$(-1 : 0 : 1) - (1 : 0 : 0)$$ $$x + z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$(-1 : 0 : 1) - (1 : 0 : 0)$$ $$x + z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$

BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$3$$ Regulator: $$1$$ Real period: $$12.68998$$ Tamagawa product: $$4$$ Torsion order: $$8$$ Leading coefficient: $$0.793124$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$12$$ $$16$$ $$4$$ $$1$$

Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $J(C_2)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{U}(1)$$

Decomposition of the Jacobian

Splits over the number field $$\Q (b) \simeq$$ $$\Q(\sqrt{2})$$ with defining polynomial:
$$x^{2} - 2$$

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
Elliptic curve isogeny class 2.2.8.1-64.1-a

Endomorphisms of the Jacobian

Of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z [\sqrt{-1}]$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-1})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\C$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\zeta_{8})$$ with defining polynomial $$x^{4} + 1$$

Not of $$\GL_2$$-type over $$\overline{\Q}$$

Endomorphism ring over $$\overline{\Q}$$:

 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ a maximal order of $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q(\sqrt{-2})$$$$)$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\C)$$

Remainder of the endomorphism lattice by field

Over subfield $$F \simeq$$ $$\Q(\sqrt{-2})$$ with generator $$-a^{3} - a$$ with minimal polynomial $$x^{2} + 2$$:

 $$\End (J_{F})$$ $$\simeq$$ the maximal order of $$\End (J_{F}) \otimes \Q$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\zeta_{8})$$ (CM) $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\C \times \C$$
Sato Tate group: $C_2$
Not of $$\GL_2$$-type, simple

Over subfield $$F \simeq$$ $$\Q(\sqrt{2})$$ with generator $$a^{3} - a$$ with minimal polynomial $$x^{2} - 2$$:

 $$\End (J_{F})$$ $$\simeq$$ a non-Eichler order of index $$4$$ in a maximal order of $$\End (J_{F}) \otimes \Q$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$
Sato Tate group: $C_{2,1}$
Not of $$\GL_2$$-type, not simple

Over subfield $$F \simeq$$ $$\Q(\sqrt{-1})$$ with generator $$-a^{2}$$ with minimal polynomial $$x^{2} + 1$$:

 $$\End (J_{F})$$ $$\simeq$$ a maximal order of $$\End (J_{F}) \otimes \Q$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ the quaternion algebra over $$\Q$$ of discriminant 2 $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\H$$
Sato Tate group: $J(C_1)$
Not of $$\GL_2$$-type, simple