Show commands: Magma / SageMath

## Simplified equation

 $y^2 + (x^3 + x)y = x^4 - 7$ (homogenize, simplify) $y^2 + (x^3 + xz^2)y = x^4z^2 - 7z^6$ (dehomogenize, simplify) $y^2 = x^6 + 6x^4 + x^2 - 28$ (homogenize, minimize)

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

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

sage: X = HyperellipticCurve(R([-28, 0, 1, 0, 6, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

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

### G2 invariants

 $$I_2$$ $$=$$ $$828$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 23$$ $$I_4$$ $$=$$ $$16635$$ $$=$$ $$3 \cdot 5 \cdot 1109$$ $$I_6$$ $$=$$ $$5308452$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 147457$$ $$I_{10}$$ $$=$$ $$56$$ $$=$$ $$2^{3} \cdot 7$$ $$J_2$$ $$=$$ $$828$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 23$$ $$J_4$$ $$=$$ $$17476$$ $$=$$ $$2^{2} \cdot 17 \cdot 257$$ $$J_6$$ $$=$$ $$-853888$$ $$=$$ $$- 2^{7} \cdot 7 \cdot 953$$ $$J_8$$ $$=$$ $$-253107460$$ $$=$$ $$- 2^{2} \cdot 5 \cdot 43 \cdot 294311$$ $$J_{10}$$ $$=$$ $$448$$ $$=$$ $$2^{6} \cdot 7$$ $$g_1$$ $$=$$ $$6080953884912/7$$ $$g_2$$ $$=$$ $$155007628668/7$$ $$g_3$$ $$=$$ $$-1306723104$$

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_2^2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

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

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

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

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

magma: MordellWeilGroupGenus2(Jacobian(C));

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

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$7.792788$$ Tamagawa product: $$1$$ Torsion order: $$6$$ Leading coefficient: $$0.216466$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.

Prime $$\ell$$ mod-$$\ell$$ image Is torsion prime?
$$2$$ 2.45.1 yes
$$3$$ 3.2160.5 yes

## Sato-Tate group

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

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 32.a
Elliptic curve isogeny class 14.a

magma: HeuristicDecompositionFactors(C);

## 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 $$K = \Q (a) \simeq$$ $$\Q(\sqrt{-1})$$ 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 $$\Z \times \Z [\sqrt{-1}]$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q(\sqrt{-1})$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\R \times \C$$

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);