# Properties

 Label 10080.a.60480.1 Conductor $10080$ Discriminant $60480$ Mordell-Weil group $$\Z/{2}\Z \times \Z/{2}\Z$$ 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: SageMath / Magma

## Simplified equation

 $y^2 + xy = -15x^6 + 58x^4 - 60x^2 + 7$ (homogenize, simplify) $y^2 + xz^2y = -15x^6 + 58x^4z^2 - 60x^2z^4 + 7z^6$ (dehomogenize, simplify) $y^2 = -60x^6 + 232x^4 - 239x^2 + 28$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([7, 0, -60, 0, 58, 0, -15]), R([0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![7, 0, -60, 0, 58, 0, -15], R![0, 1]);

sage: X = HyperellipticCurve(R([28, 0, -239, 0, 232, 0, -60]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$10080$$ $$=$$ $$2^{5} \cdot 3^{2} \cdot 5 \cdot 7$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$60480$$ $$=$$ $$2^{6} \cdot 3^{3} \cdot 5 \cdot 7$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$161296$$ $$=$$ $$2^{4} \cdot 17 \cdot 593$$ $$I_4$$ $$=$$ $$406586887$$ $$=$$ $$7 \cdot 43 \cdot 67 \cdot 20161$$ $$I_6$$ $$=$$ $$19127473723714$$ $$=$$ $$2 \cdot 347 \cdot 27561201331$$ $$I_{10}$$ $$=$$ $$7560$$ $$=$$ $$2^{3} \cdot 3^{3} \cdot 5 \cdot 7$$ $$J_2$$ $$=$$ $$161296$$ $$=$$ $$2^{4} \cdot 17 \cdot 593$$ $$J_4$$ $$=$$ $$812958726$$ $$=$$ $$2 \cdot 3 \cdot 2399 \cdot 56479$$ $$J_6$$ $$=$$ $$4856153621760$$ $$=$$ $$2^{8} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 89 \cdot 151 \cdot 4481$$ $$J_8$$ $$=$$ $$30594066098964471$$ $$=$$ $$3^{2} \cdot 191 \cdot 311 \cdot 57226994119$$ $$J_{10}$$ $$=$$ $$60480$$ $$=$$ $$2^{6} \cdot 3^{3} \cdot 5 \cdot 7$$ $$g_1$$ $$=$$ $$1705838896690345318825984/945$$ $$g_2$$ $$=$$ $$17767980154611986862208/315$$ $$g_3$$ $$=$$ $$6266846885932235776/3$$

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

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$15x^2 - 28z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-xz^2$$ $$0$$ $$2$$
$$D_0 - D_\infty$$ $$2x^2 - 2xz - z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-xz^2$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$15x^2 - 28z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-xz^2$$ $$0$$ $$2$$
$$D_0 - D_\infty$$ $$2x^2 - 2xz - z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-xz^2$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$15x^2 - 28z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-xz^2$$ $$0$$ $$2$$
$$D_0 - D_\infty$$ $$2x^2 - 2xz - z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-xz^2$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$4$$ Regulator: $$1$$ Real period: $$0.556034$$ Tamagawa product: $$6$$ Torsion order: $$4$$ Leading coefficient: $$0.834051$$ Analytic order of Ш: $$4$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$5$$ $$6$$ $$2$$ $$1 + T + 2 T^{2}$$
$$3$$ $$2$$ $$3$$ $$3$$ $$( 1 - T )^{2}$$
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 2 T + 5 T^{2} )$$
$$7$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 4 T + 7 T^{2} )$$

## 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 480.f
Elliptic curve isogeny class 21.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 $$\Q$$.