Show commands for: Magma / SageMath

## Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -9, -13, -4, 1], R![]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -9, -13, -4, 1]), R([]))

$y^2 = x^5 - 4x^4 - 13x^3 - 9x^2 - x$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$784$$ = $$2^{4} \cdot 7^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$614656$$ = $$2^{8} \cdot 7^{4}$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$6368$$ = $$2^{5} \cdot 199$$ $$I_4$$ = $$2308096$$ = $$2^{11} \cdot 7^{2} \cdot 23$$ $$I_6$$ = $$3735904256$$ = $$2^{13} \cdot 7^{2} \cdot 41 \cdot 227$$ $$I_{10}$$ = $$2517630976$$ = $$2^{20} \cdot 7^{4}$$ $$J_2$$ = $$796$$ = $$2^{2} \cdot 199$$ $$J_4$$ = $$2358$$ = $$2 \cdot 3^{2} \cdot 131$$ $$J_6$$ = $$-2348$$ = $$-1 \cdot 2^{2} \cdot 587$$ $$J_8$$ = $$-1857293$$ = $$-1 \cdot 1857293$$ $$J_{10}$$ = $$614656$$ = $$2^{8} \cdot 7^{4}$$ $$g_1$$ = $$1248318403996/2401$$ $$g_2$$ = $$9291226221/4802$$ $$g_3$$ = $$-23245787/9604$$
Alternative geometric invariants: G2

## Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$ $$C_6$$ (GAP id : [6,2]) magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $$D_6$$ (GAP id : [12,4])

## Rational points

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);

This curve is locally solvable everywhere.

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

All rational points: (-1 : 0 : 1), (0 : 0 : 1), (1 : 0 : 0)

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

Number of rational Weierstrass points: $$3$$

## Invariants of the Jacobian:

Analytic rank: $$0$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$2$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: 3 (p = 2), 3 (p = 7) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{2}\Z \times \Z/{6}\Z$$

### Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $E_3$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)$$

### Decomposition

Splits over the number field $$\Q (b) \simeq$$ $$\Q(\zeta_{7})^+$$ with defining polynomial:
$$x^{3} - x^{2} - 2 x + 1$$

Decomposes up to isogeny as the square of the elliptic curve:
Elliptic curve 3.3.49.1-64.1-a3

### Endomorphisms

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

Endomorphism ring over $$\Q$$:
 $$\End (J_{})$$ $$\simeq$$ $$\Z [\frac{1 + \sqrt{-3}}{2}]$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-3})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\C$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\zeta_{7})^+$$ with defining polynomial $$x^{3} - x^{2} - 2 x + 1$$

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

Endomorphism ring over $$\overline{\Q}$$:
 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ an Eichler order of index $$3$$ in a maximal order of $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$