# Properties

 Label 720.b.116640.1 Conductor $720$ Discriminant $116640$ Mordell-Weil group $$\Z/{2}\Z \oplus \Z/{12}\Z$$ 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

 $y^2 + (x^3 + x)y = -6x^4 + 39x^2 - 90$ (homogenize, simplify) $y^2 + (x^3 + xz^2)y = -6x^4z^2 + 39x^2z^4 - 90z^6$ (dehomogenize, simplify) $y^2 = x^6 - 22x^4 + 157x^2 - 360$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-90, 0, 39, 0, -6]), R([0, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-90, 0, 39, 0, -6], R![0, 1, 0, 1]);

sage: X = HyperellipticCurve(R([-360, 0, 157, 0, -22, 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: $$\Delta$$ $$=$$ $$116640$$ $$=$$ $$2^{5} \cdot 3^{6} \cdot 5$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$35416$$ $$=$$ $$2^{3} \cdot 19 \cdot 233$$ $$I_4$$ $$=$$ $$45688$$ $$=$$ $$2^{3} \cdot 5711$$ $$I_6$$ $$=$$ $$537039964$$ $$=$$ $$2^{2} \cdot 6373 \cdot 21067$$ $$I_{10}$$ $$=$$ $$466560$$ $$=$$ $$2^{7} \cdot 3^{6} \cdot 5$$ $$J_2$$ $$=$$ $$17708$$ $$=$$ $$2^{2} \cdot 19 \cdot 233$$ $$J_4$$ $$=$$ $$13057938$$ $$=$$ $$2 \cdot 3^{2} \cdot 17 \cdot 139 \cdot 307$$ $$J_6$$ $$=$$ $$12831384960$$ $$=$$ $$2^{7} \cdot 3^{4} \cdot 5 \cdot 247519$$ $$J_8$$ $$=$$ $$14177105014959$$ $$=$$ $$3^{4} \cdot 29 \cdot 6035378891$$ $$J_{10}$$ $$=$$ $$116640$$ $$=$$ $$2^{5} \cdot 3^{6} \cdot 5$$ $$g_1$$ $$=$$ $$54412363190235229024/3645$$ $$g_2$$ $$=$$ $$251762275020280012/405$$ $$g_3$$ $$=$$ $$310461362928064/9$$

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: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (-3 : 15 : 1),\, (3 : -15 : 1)$$
All points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (-3 : 15 : 1),\, (3 : -15 : 1)$$
All points: $$(1 : -1 : 0),\, (1 : 1 : 0),\, (-3 : 0 : 1),\, (3 : 0 : 1)$$

magma: [C![-3,15,1],C![1,-1,0],C![1,0,0],C![3,-15,1]]; // minimal model

magma: [C![-3,0,1],C![1,-1,0],C![1,1,0],C![3,0,1]]; // simplified model

Number of rational Weierstrass points: $$2$$

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: $$\Z/{2}\Z \oplus \Z/{12}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - 5z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-3xz^2$$ $$0$$ $$2$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$2x^2 + xz - 13z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-4xz^2 + z^3$$ $$0$$ $$12$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - 5z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-3xz^2$$ $$0$$ $$2$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$2x^2 + xz - 13z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-4xz^2 + z^3$$ $$0$$ $$12$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2 - 5z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - 5xz^2$$ $$0$$ $$2$$
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$2x^2 + xz - 13z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - 7xz^2 + 2z^3$$ $$0$$ $$12$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$14.45705$$ Tamagawa product: $$12$$ Torsion order: $$24$$ Leading coefficient: $$0.301188$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$4$$ $$5$$ $$2$$ $$1 + T$$
$$3$$ $$2$$ $$6$$ $$6$$ $$( 1 - T )( 1 + T )$$
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 2 T + 5 T^{2} )$$

## 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?
$$2$$ 2.180.3 yes
$$3$$ 3.720.4 yes

## 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 30.a
Elliptic curve isogeny class 24.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);