# Properties

 Label 10115.a.10115.1 Conductor $10115$ Discriminant $10115$ Mordell-Weil group $$\Z \times \Z/{2}\Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

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Show commands for: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + x^2 + x)y = -x^4 + 3x^2 + x - 2$ (homogenize, simplify) $y^2 + (x^3 + x^2z + xz^2)y = -x^4z^2 + 3x^2z^4 + xz^5 - 2z^6$ (dehomogenize, simplify) $y^2 = x^6 + 2x^5 - x^4 + 2x^3 + 13x^2 + 4x - 8$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, 1, 3, 0, -1]), R([0, 1, 1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, 1, 3, 0, -1], R![0, 1, 1, 1]);

sage: X = HyperellipticCurve(R([-8, 4, 13, 2, -1, 2, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$10115$$ $$=$$ $$5 \cdot 7 \cdot 17^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$10115$$ $$=$$ $$5 \cdot 7 \cdot 17^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$1236$$ $$=$$ $$2^{2} \cdot 3 \cdot 103$$ $$I_4$$ $$=$$ $$3825$$ $$=$$ $$3^{2} \cdot 5^{2} \cdot 17$$ $$I_6$$ $$=$$ $$1561977$$ $$=$$ $$3^{3} \cdot 17 \cdot 41 \cdot 83$$ $$I_{10}$$ $$=$$ $$1294720$$ $$=$$ $$2^{7} \cdot 5 \cdot 7 \cdot 17^{2}$$ $$J_2$$ $$=$$ $$309$$ $$=$$ $$3 \cdot 103$$ $$J_4$$ $$=$$ $$3819$$ $$=$$ $$3 \cdot 19 \cdot 67$$ $$J_6$$ $$=$$ $$60281$$ $$=$$ $$13 \cdot 4637$$ $$J_8$$ $$=$$ $$1010517$$ $$=$$ $$3 \cdot 167 \cdot 2017$$ $$J_{10}$$ $$=$$ $$10115$$ $$=$$ $$5 \cdot 7 \cdot 17^{2}$$ $$g_1$$ $$=$$ $$2817036000549/10115$$ $$g_2$$ $$=$$ $$112674359151/10115$$ $$g_3$$ $$=$$ $$5755690161/10115$$

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

 All points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (-2 : 2 : 1),\, (-2 : 4 : 1)$$ All points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (-2 : 2 : 1),\, (-2 : 4 : 1)$$ All points: $$(1 : -1 : 0),\, (1 : 1 : 0),\, (-2 : -2 : 1),\, (-2 : 2 : 1)$$

magma: [C![-2,2,1],C![-2,4,1],C![1,-1,0],C![1,0,0]]; // minimal model

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

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : -1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.269551$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 + xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : -1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.269551$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 + xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : -1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + x^2z + xz^2$$ $$0.269551$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2 + xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + x^2z - xz^2$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$2$$ Regulator: $$0.269551$$ Real period: $$11.75100$$ Tamagawa product: $$1$$ Torsion order: $$2$$ Leading coefficient: $$0.791876$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 5 T^{2} )$$
$$7$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 7 T^{2} )$$
$$17$$ $$2$$ $$2$$ $$1$$ $$1 + 2 T + 17 T^{2}$$

## 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 $$\Q$$.