Show commands for: Magma / SageMath

## Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-7, 0, 0, 0, 1], R![0, 1, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-7, 0, 0, 0, 1]), R([0, 1, 0, 1]))

$y^2 + (x^3 + x)y = x^4 - 7$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$448$$ = $$2^{6} \cdot 7$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$448$$ = $$2^{6} \cdot 7$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$6624$$ = $$2^{5} \cdot 3^{2} \cdot 23$$ $$I_4$$ = $$1064640$$ = $$2^{6} \cdot 3 \cdot 5 \cdot 1109$$ $$I_6$$ = $$2717927424$$ = $$2^{11} \cdot 3^{2} \cdot 147457$$ $$I_{10}$$ = $$1835008$$ = $$2^{18} \cdot 7$$ $$J_2$$ = $$828$$ = $$2^{2} \cdot 3^{2} \cdot 23$$ $$J_4$$ = $$17476$$ = $$2^{2} \cdot 17 \cdot 257$$ $$J_6$$ = $$-853888$$ = $$-1 \cdot 2^{7} \cdot 7 \cdot 953$$ $$J_8$$ = $$-253107460$$ = $$-1 \cdot 2^{2} \cdot 5 \cdot 43 \cdot 294311$$ $$J_{10}$$ = $$448$$ = $$2^{6} \cdot 7$$ $$g_1$$ = $$6080953884912/7$$ $$g_2$$ = $$155007628668/7$$ $$g_3$$ = $$-1306723104$$
Alternative geometric invariants: G2

## Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$ $$V_4$$ (GAP id : [4,2]) magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $$V_4$$ (GAP id : [4,2])

## 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,-1,0],C![1,0,0]];

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

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

Number of rational Weierstrass points: $$0$$

## Invariants of the Jacobian:

Analytic rank: $$0$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$1$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Regulator: 1.0 Real period: 7.7927889653116448250938531878 Tamagawa numbers: 1 (p = 2), 1 (p = 7) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

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

### Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $N(G_{1,3})$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{U}(1)\times\mathrm{SU}(2)$$

### Decomposition

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 32.a1
Elliptic curve 14.a4

### Endomorphisms

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 $$K = \Q (a) \simeq$$ $$\Q(\sqrt{-1})$$ with defining polynomial $$x^{2} + 1$$

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

Endomorphism ring over $$\overline{\Q}$$:
 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ an order of index $$8$$ in $$\Z \times \Z [\sqrt{-1}]$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q(\sqrt{-1})$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\R \times \C$$