# Properties

 Label 336.a.172032.1 Conductor $336$ Discriminant $-172032$ Mordell-Weil group $$\Z/{2}\Z$$ 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

 $y^2 + (x^3 + x)y = -x^6 + 15x^4 - 75x^2 - 56$ (homogenize, simplify) $y^2 + (x^3 + xz^2)y = -x^6 + 15x^4z^2 - 75x^2z^4 - 56z^6$ (dehomogenize, simplify) $y^2 = -3x^6 + 62x^4 - 299x^2 - 224$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-56, 0, -75, 0, 15, 0, -1]), R([0, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-56, 0, -75, 0, 15, 0, -1], R![0, 1, 0, 1]);

sage: X = HyperellipticCurve(R([-224, 0, -299, 0, 62, 0, -3]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$336$$ $$=$$ $$2^{4} \cdot 3 \cdot 7$$ magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(336,2),R![1, 1]>*])); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-172032$$ $$=$$ $$- 2^{13} \cdot 3 \cdot 7$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$16916$$ $$=$$ $$2^{2} \cdot 4229$$ $$I_4$$ $$=$$ $$151117825$$ $$=$$ $$5^{2} \cdot 6044713$$ $$I_6$$ $$=$$ $$232872423961$$ $$=$$ $$397 \cdot 586580413$$ $$I_{10}$$ $$=$$ $$-21504$$ $$=$$ $$- 2^{10} \cdot 3 \cdot 7$$ $$J_2$$ $$=$$ $$16916$$ $$=$$ $$2^{2} \cdot 4229$$ $$J_4$$ $$=$$ $$-88822256$$ $$=$$ $$- 2^{4} \cdot 103 \cdot 53897$$ $$J_6$$ $$=$$ $$277597802496$$ $$=$$ $$2^{12} \cdot 3 \cdot 7 \cdot 3227281$$ $$J_8$$ $$=$$ $$-798387183476800$$ $$=$$ $$- 2^{6} \cdot 5^{2} \cdot 107 \cdot 991 \cdot 4705829$$ $$J_{10}$$ $$=$$ $$-172032$$ $$=$$ $$- 2^{13} \cdot 3 \cdot 7$$ $$g_1$$ $$=$$ $$-1352659309173012149/168$$ $$g_2$$ $$=$$ $$419870026410625699/168$$ $$g_3$$ $$=$$ $$-461744933079368$$

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

This curve has no rational points.
This curve has no rational points.
This curve has no rational points.

magma: []; // minimal model

magma: []; // simplified model

Number of rational Weierstrass points: $$0$$

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

This curve is locally solvable except over $\Q_{2}$.

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

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$3x^2 - 32z^2$$ $$=$$ $$0,$$ $$6y$$ $$=$$ $$-35xz^2$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$3x^2 - 32z^2$$ $$=$$ $$0,$$ $$6y$$ $$=$$ $$-35xz^2$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$3x^2 - 32z^2$$ $$=$$ $$0,$$ $$6y$$ $$=$$ $$x^3 - 69xz^2$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$0.356066$$ Tamagawa product: $$1$$ Torsion order: $$2$$ Leading coefficient: $$0.178033$$ Analytic order of Ш: $$2$$   (rounded) Order of Ш: twice a square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$4$$ $$13$$ $$1$$ $$1 + T$$
$$3$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 2 T + 3 T^{2} )$$
$$7$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 7 T^{2} )$$

## 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?
$$2$$ 2.45.1 yes
$$3$$ 3.720.5 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 14.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);