# Properties

 Label 1170.a.10530.1 Conductor $1170$ Discriminant $-10530$ Mordell-Weil group $$\Z/{2}\Z \times \Z/{6}\Z$$ 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: SageMath / Magma

## Simplified equation

 $y^2 + (x^2 + x)y = 15x^6 + 28x^5 + 62x^4 + 59x^3 + 62x^2 + 28x + 15$ (homogenize, simplify) $y^2 + (x^2z + xz^2)y = 15x^6 + 28x^5z + 62x^4z^2 + 59x^3z^3 + 62x^2z^4 + 28xz^5 + 15z^6$ (dehomogenize, simplify) $y^2 = 60x^6 + 112x^5 + 249x^4 + 238x^3 + 249x^2 + 112x + 60$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([15, 28, 62, 59, 62, 28, 15]), R([0, 1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![15, 28, 62, 59, 62, 28, 15], R![0, 1, 1]);

sage: X = HyperellipticCurve(R([60, 112, 249, 238, 249, 112, 60]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$1170$$ $$=$$ $$2 \cdot 3^{2} \cdot 5 \cdot 13$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-10530$$ $$=$$ $$- 2 \cdot 3^{4} \cdot 5 \cdot 13$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$507196$$ $$=$$ $$2^{2} \cdot 23 \cdot 37 \cdot 149$$ $$I_4$$ $$=$$ $$192673$$ $$=$$ $$13 \cdot 14821$$ $$I_6$$ $$=$$ $$32552199279$$ $$=$$ $$3^{2} \cdot 3616911031$$ $$I_{10}$$ $$=$$ $$1347840$$ $$=$$ $$2^{8} \cdot 3^{4} \cdot 5 \cdot 13$$ $$J_2$$ $$=$$ $$126799$$ $$=$$ $$23 \cdot 37 \cdot 149$$ $$J_4$$ $$=$$ $$669908072$$ $$=$$ $$2^{3} \cdot 991 \cdot 84499$$ $$J_6$$ $$=$$ $$4718980180980$$ $$=$$ $$2^{2} \cdot 3^{4} \cdot 5 \cdot 13 \cdot 224073133$$ $$J_8$$ $$=$$ $$37396285759331459$$ $$=$$ $$71 \cdot 11505961 \cdot 45776989$$ $$J_{10}$$ $$=$$ $$10530$$ $$=$$ $$2 \cdot 3^{4} \cdot 5 \cdot 13$$ $$g_1$$ $$=$$ $$32777750301275239538233999/10530$$ $$g_2$$ $$=$$ $$682861614668954802420364/5265$$ $$g_3$$ $$=$$ $$7205289570406928666$$

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 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: $$\Z/{2}\Z \times \Z/{6}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$10x^2 + 7xz + 10z^2$$ $$=$$ $$0,$$ $$20y$$ $$=$$ $$-3xz^2 + 10z^3$$ $$0$$ $$2$$
$$D_0 - D_\infty$$ $$17x^2 + 11xz + 17z^2$$ $$=$$ $$0,$$ $$289y$$ $$=$$ $$-65xz^2 + 136z^3$$ $$0$$ $$6$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$10x^2 + 7xz + 10z^2$$ $$=$$ $$0,$$ $$20y$$ $$=$$ $$-3xz^2 + 10z^3$$ $$0$$ $$2$$
$$D_0 - D_\infty$$ $$17x^2 + 11xz + 17z^2$$ $$=$$ $$0,$$ $$289y$$ $$=$$ $$-65xz^2 + 136z^3$$ $$0$$ $$6$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$10x^2 + 7xz + 10z^2$$ $$=$$ $$0,$$ $$20y$$ $$=$$ $$x^2z - 5xz^2 + 20z^3$$ $$0$$ $$2$$
$$D_0 - D_\infty$$ $$17x^2 + 11xz + 17z^2$$ $$=$$ $$0,$$ $$289y$$ $$=$$ $$x^2z - 129xz^2 + 272z^3$$ $$0$$ $$6$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$4$$ Regulator: $$1$$ Real period: $$5.542029$$ Tamagawa product: $$3$$ Torsion order: $$12$$ Leading coefficient: $$0.461835$$ Analytic order of Ш: $$4$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - T + 2 T^{2} )$$
$$3$$ $$2$$ $$4$$ $$3$$ $$( 1 - T )( 1 + T )$$
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 2 T + 5 T^{2} )$$
$$13$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 2 T + 13 T^{2} )$$

## 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 39.a

## 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 $$\Q$$.