# Properties

 Label 720.a.6480.1 Conductor $720$ Discriminant $-6480$ Mordell-Weil group $$\Z/{2}\Z \oplus \Z/{4}\Z$$ Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands: Magma / SageMath

## Simplified equation

 $y^2 + (x^3 + x)y = 2x^4 + 7x^2 + 5$ (homogenize, simplify) $y^2 + (x^3 + xz^2)y = 2x^4z^2 + 7x^2z^4 + 5z^6$ (dehomogenize, simplify) $y^2 = x^6 + 10x^4 + 29x^2 + 20$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([5, 0, 7, 0, 2]), R([0, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![5, 0, 7, 0, 2], R![0, 1, 0, 1]);

sage: X = HyperellipticCurve(R([20, 0, 29, 0, 10, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$720$$ $$=$$ $$2^{4} \cdot 3^{2} \cdot 5$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-6480$$ $$=$$ $$- 2^{4} \cdot 3^{4} \cdot 5$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$2360$$ $$=$$ $$2^{3} \cdot 5 \cdot 59$$ $$I_4$$ $$=$$ $$11992$$ $$=$$ $$2^{3} \cdot 1499$$ $$I_6$$ $$=$$ $$9047820$$ $$=$$ $$2^{2} \cdot 3 \cdot 5 \cdot 150797$$ $$I_{10}$$ $$=$$ $$25920$$ $$=$$ $$2^{6} \cdot 3^{4} \cdot 5$$ $$J_2$$ $$=$$ $$1180$$ $$=$$ $$2^{2} \cdot 5 \cdot 59$$ $$J_4$$ $$=$$ $$56018$$ $$=$$ $$2 \cdot 37 \cdot 757$$ $$J_6$$ $$=$$ $$3453120$$ $$=$$ $$2^{6} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 109$$ $$J_8$$ $$=$$ $$234166319$$ $$=$$ $$3299 \cdot 70981$$ $$J_{10}$$ $$=$$ $$6480$$ $$=$$ $$2^{4} \cdot 3^{4} \cdot 5$$ $$g_1$$ $$=$$ $$28596971960000/81$$ $$g_2$$ $$=$$ $$1150492082200/81$$ $$g_3$$ $$=$$ $$6677950400/9$$

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: $$\Z/{2}\Z \oplus \Z/{4}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
$$D_0 - 2 \cdot(1 : -1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$4$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
$$D_0 - 2 \cdot(1 : -1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$4$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2 + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + xz^2$$ $$0$$ $$2$$
$$D_0 - 2 \cdot(1 : -1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + xz^2$$ $$0$$ $$4$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$9.444268$$ Tamagawa product: $$2$$ Torsion order: $$8$$ Leading coefficient: $$0.295133$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$4$$ $$4$$ $$1$$ $$1 + T + 2 T^{2}$$
$$3$$ $$2$$ $$4$$ $$2$$ $$( 1 - T )( 1 + T )$$
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 2 T + 5 T^{2} )$$

## 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?
$$2$$ 2.180.7 yes
$$3$$ 3.90.1 no

## Sato-Tate group

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

## 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 48.a
Elliptic curve isogeny class 15.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 $$\Z \times \Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.

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);