Show commands for: SageMath / Magma

## Simplified equation

 $y^2 + x^3y = 2$ (homogenize, simplify) $y^2 + x^3y = 2z^6$ (dehomogenize, simplify) $y^2 = x^6 + 8$ (minimize, homogenize)

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

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

sage: X = HyperellipticCurve(R([8, 0, 0, 0, 0, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$20736$$ $$=$$ $$2^{8} \cdot 3^{4}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-373248$$ $$=$$ $$- 2^{9} \cdot 3^{6}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$-1920$$ $$=$$ $$- 2^{7} \cdot 3 \cdot 5$$ $$I_4$$ $$=$$ $$103680$$ $$=$$ $$2^{8} \cdot 3^{4} \cdot 5$$ $$I_6$$ $$=$$ $$-61378560$$ $$=$$ $$- 2^{12} \cdot 3^{4} \cdot 5 \cdot 37$$ $$I_{10}$$ $$=$$ $$-1528823808$$ $$=$$ $$- 2^{21} \cdot 3^{6}$$ $$J_2$$ $$=$$ $$-240$$ $$=$$ $$- 2^{4} \cdot 3 \cdot 5$$ $$J_4$$ $$=$$ $$1320$$ $$=$$ $$2^{3} \cdot 3 \cdot 5 \cdot 11$$ $$J_6$$ $$=$$ $$2560$$ $$=$$ $$2^{9} \cdot 5$$ $$J_8$$ $$=$$ $$-589200$$ $$=$$ $$- 2^{4} \cdot 3 \cdot 5^{2} \cdot 491$$ $$J_{10}$$ $$=$$ $$-373248$$ $$=$$ $$- 2^{9} \cdot 3^{6}$$ $$g_1$$ $$=$$ $$6400000/3$$ $$g_2$$ $$=$$ $$440000/9$$ $$g_3$$ $$=$$ $$-32000/81$$

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

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

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2^2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_3:D_4$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

All points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : -1 : 1),\, (1 : 1 : 1),\, (-1 : 2 : 1),\, (1 : -2 : 1)$$

magma: [C![-1,-1,1],C![-1,2,1],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,1,1]];

Number of rational Weierstrass points: $$0$$

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: $$\Z \times \Z/{6}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : -1 : 1) - (1 : -1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.326617$$ $$\infty$$
$$(-1 : 2 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$(x - z) (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2$$ $$0$$ $$6$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$2$$ Regulator: $$0.326617$$ Real period: $$12.51227$$ Tamagawa product: $$12$$ Torsion order: $$6$$ Leading coefficient: $$1.362242$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $D_{2,1}$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{U}(1)$$

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 576.e4
Elliptic curve 36.a4

## Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ an order of index $$2$$ in $$\Z \times \Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ with defining polynomial $$x^{4} + 2 x^{2} + 4$$

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

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

 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ a non-Eichler order of index $$16$$ in a maximal order of $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q(\sqrt{-3})$$$$)$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\C)$$

### Remainder of the endomorphism lattice by field

Over subfield $$F \simeq$$ $$\Q(\sqrt{2})$$ with generator $$\frac{1}{2} a^{3}$$ 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 $$F \simeq$$ $$\Q(\sqrt{-6})$$ with generator $$-\frac{1}{2} a^{3} - 2 a$$ with minimal polynomial $$x^{2} + 6$$:

 $$\End (J_{F})$$ $$\simeq$$ a non-Eichler order of index $$12$$ 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 $$F \simeq$$ $$\Q(\sqrt{-3})$$ with generator $$-\frac{1}{2} a^{2}$$ with minimal polynomial $$x^{2} - x + 1$$:

 $$\End (J_{F})$$ $$\simeq$$ an order of index $$4$$ in $$\Z [\frac{1 + \sqrt{-3}}{2}] \times \Z [\frac{1 + \sqrt{-3}}{2}]$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-3})$$ $$\times$$ $$\Q(\sqrt{-3})$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\C \times \C$$
Sato Tate group: $C_2$
Not of $$\GL_2$$-type, not simple