# Properties

 Label 100130.a.200260.1 Conductor 100130 Discriminant -200260 Mordell-Weil group $$\Z \times \Z \times \Z/{2}\Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands for: SageMath / Magma

## Simplified equation

 $y^2 + (x^2 + x + 1)y = x^6 + 5x^5 + 6x^4 + x^2 + x$ (homogenize, simplify) $y^2 + (x^2z + xz^2 + z^3)y = x^6 + 5x^5z + 6x^4z^2 + x^2z^4 + xz^5$ (dehomogenize, simplify) $y^2 = 4x^6 + 20x^5 + 25x^4 + 2x^3 + 7x^2 + 6x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 1, 0, 6, 5, 1]), R([1, 1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 1, 0, 6, 5, 1], R![1, 1, 1]);

sage: X = HyperellipticCurve(R([1, 6, 7, 2, 25, 20, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$100130$$ $$=$$ $$2 \cdot 5 \cdot 17 \cdot 19 \cdot 31$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-200260$$ $$=$$ $$- 2^{2} \cdot 5 \cdot 17 \cdot 19 \cdot 31$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$1064$$ $$=$$ $$2^{3} \cdot 7 \cdot 19$$ $$I_4$$ $$=$$ $$872740$$ $$=$$ $$2^{2} \cdot 5 \cdot 11 \cdot 3967$$ $$I_6$$ $$=$$ $$-8682968$$ $$=$$ $$- 2^{3} \cdot 7 \cdot 47 \cdot 3299$$ $$I_{10}$$ $$=$$ $$-820264960$$ $$=$$ $$- 2^{14} \cdot 5 \cdot 17 \cdot 19 \cdot 31$$ $$J_2$$ $$=$$ $$133$$ $$=$$ $$7 \cdot 19$$ $$J_4$$ $$=$$ $$-8354$$ $$=$$ $$- 2 \cdot 4177$$ $$J_6$$ $$=$$ $$356384$$ $$=$$ $$2^{5} \cdot 7 \cdot 37 \cdot 43$$ $$J_8$$ $$=$$ $$-5597561$$ $$=$$ $$-5597561$$ $$J_{10}$$ $$=$$ $$-200260$$ $$=$$ $$- 2^{2} \cdot 5 \cdot 17 \cdot 19 \cdot 31$$ $$g_1$$ $$=$$ $$-2190305047/10540$$ $$g_2$$ $$=$$ $$517208671/5270$$ $$g_3$$ $$=$$ $$-82948376/2635$$

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

Known points
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : 1 : 1)$$ $$(-1 : -2 : 1)$$
$$(7 : 602 : 3)$$ $$(7 : -839 : 3)$$

magma: [C![-1,-2,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,1,0],C![7,-839,3],C![7,602,3]];

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

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$2x^2 + 5xz + z^2$$ $$=$$ $$0,$$ $$4y$$ $$=$$ $$5xz^2 - z^3$$ $$0.429826$$ $$\infty$$
$$(-1 : -2 : 1) - (1 : 1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - 3z^3$$ $$0.300725$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2 + 3xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2$$ $$0$$ $$2$$

2-torsion field: splitting field of $$x^{6} - 20 x^{4} - 26 x^{3} - 255 x^{2} - 1730 x - 1706$$ with Galois group $S_4\times C_2$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$3$$ Regulator: $$0.128156$$ Real period: $$18.88632$$ Tamagawa product: $$2$$ Torsion order: $$2$$ Leading coefficient: $$1.210198$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$1$$ $$2$$ $$2$$ $$( 1 + T )( 1 + T + 2 T^{2} )$$
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 5 T^{2} )$$
$$17$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 6 T + 17 T^{2} )$$
$$19$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 6 T + 19 T^{2} )$$
$$31$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 6 T + 31 T^{2} )$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{USp}(4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{USp}(4)$$

## Decomposition of the Jacobian

Simple over $$\overline{\Q}$$

## Endomorphisms of the Jacobian

Not of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.