# Properties

 Label 10080.c.141120.1 Conductor $10080$ Discriminant $141120$ Mordell-Weil group $$\Z/{2}\Z \times \Z/{4}\Z$$ 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: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + x)y = -x^6 + 35x^4 - 560x^2 + 2940$ (homogenize, simplify) $y^2 + (x^3 + xz^2)y = -x^6 + 35x^4z^2 - 560x^2z^4 + 2940z^6$ (dehomogenize, simplify) $y^2 = -3x^6 + 142x^4 - 2239x^2 + 11760$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2940, 0, -560, 0, 35, 0, -1]), R([0, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2940, 0, -560, 0, 35, 0, -1], R![0, 1, 0, 1]);

sage: X = HyperellipticCurve(R([11760, 0, -2239, 0, 142, 0, -3]))

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: $$\Delta$$ $$=$$ $$141120$$ $$=$$ $$2^{6} \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$3388552$$ $$=$$ $$2^{3} \cdot 467 \cdot 907$$ $$I_4$$ $$=$$ $$174712$$ $$=$$ $$2^{3} \cdot 21839$$ $$I_6$$ $$=$$ $$197326050612$$ $$=$$ $$2^{2} \cdot 3 \cdot 227 \cdot 2713 \cdot 26701$$ $$I_{10}$$ $$=$$ $$564480$$ $$=$$ $$2^{8} \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ $$J_2$$ $$=$$ $$1694276$$ $$=$$ $$2^{2} \cdot 467 \cdot 907$$ $$J_4$$ $$=$$ $$119607102722$$ $$=$$ $$2 \cdot 23 \cdot 6211 \cdot 418637$$ $$J_6$$ $$=$$ $$11258185829425920$$ $$=$$ $$2^{8} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 11 \cdot 1813122589$$ $$J_8$$ $$=$$ $$1192153758196342556159$$ $$=$$ $$1619 \cdot 2861 \cdot 902767 \cdot 285096503$$ $$J_{10}$$ $$=$$ $$141120$$ $$=$$ $$2^{6} \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ $$g_1$$ $$=$$ $$218142768611210403574323981584/2205$$ $$g_2$$ $$=$$ $$9089279812657801356650662498/2205$$ $$g_3$$ $$=$$ $$229006686528379459553216$$

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: $$(-4 : 34 : 1),\, (4 : -34 : 1)$$ All points: $$(-4 : 34 : 1),\, (4 : -34 : 1)$$ All points: $$(-4 : 0 : 1),\, (4 : 0 : 1)$$

magma: [C![-4,34,1],C![4,-34,1]]; // minimal model

magma: [C![-4,0,1],C![4,0,1]]; // 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: $$\Z/{2}\Z \times \Z/{4}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-4 : 34 : 1) + (4 : -34 : 1) - D_\infty$$ $$(x - 4z) (x + 4z)$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-17xz^2$$ $$0$$ $$2$$
$$D_0 - D_\infty$$ $$13x^2 - 210z^2$$ $$=$$ $$0,$$ $$13y$$ $$=$$ $$-111xz^2$$ $$0$$ $$4$$
Generator $D_0$ Height Order
$$(-4 : 34 : 1) + (4 : -34 : 1) - D_\infty$$ $$(x - 4z) (x + 4z)$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-17xz^2$$ $$0$$ $$2$$
$$D_0 - D_\infty$$ $$13x^2 - 210z^2$$ $$=$$ $$0,$$ $$13y$$ $$=$$ $$-111xz^2$$ $$0$$ $$4$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$(x - 4z) (x + 4z)$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$x^3 - 33xz^2$$ $$0$$ $$2$$
$$D_0 - D_\infty$$ $$13x^2 - 210z^2$$ $$=$$ $$0,$$ $$13y$$ $$=$$ $$x^3 - 221xz^2$$ $$0$$ $$4$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$4$$ Regulator: $$1$$ Real period: $$5.655296$$ Tamagawa product: $$4$$ Torsion order: $$8$$ Leading coefficient: $$1.413824$$ Analytic order of Ш: $$4$$   (rounded) Order of Ш: square

## Local invariants

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

## 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 210.c
Elliptic curve isogeny class 48.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 $$\Q$$.