# Properties

 Label 10130.a.162080.1 Conductor $10130$ Discriminant $162080$ Mordell-Weil group $$\Z$$ 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

# Related objects

Show commands: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + x^2)y = -x^4 + 3x^2 - 5x - 10$ (homogenize, simplify) $y^2 + (x^3 + x^2z)y = -x^4z^2 + 3x^2z^4 - 5xz^5 - 10z^6$ (dehomogenize, simplify) $y^2 = x^6 + 2x^5 - 3x^4 + 12x^2 - 20x - 40$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-10, -5, 3, 0, -1]), R([0, 0, 1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-10, -5, 3, 0, -1], R![0, 0, 1, 1]);

sage: X = HyperellipticCurve(R([-40, -20, 12, 0, -3, 2, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$10130$$ $$=$$ $$2 \cdot 5 \cdot 1013$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$162080$$ $$=$$ $$2^{5} \cdot 5 \cdot 1013$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$2144$$ $$=$$ $$2^{5} \cdot 67$$ $$I_4$$ $$=$$ $$5248$$ $$=$$ $$2^{7} \cdot 41$$ $$I_6$$ $$=$$ $$3753415$$ $$=$$ $$5 \cdot 71 \cdot 97 \cdot 109$$ $$I_{10}$$ $$=$$ $$648320$$ $$=$$ $$2^{7} \cdot 5 \cdot 1013$$ $$J_2$$ $$=$$ $$1072$$ $$=$$ $$2^{4} \cdot 67$$ $$J_4$$ $$=$$ $$47008$$ $$=$$ $$2^{5} \cdot 13 \cdot 113$$ $$J_6$$ $$=$$ $$2695089$$ $$=$$ $$3 \cdot 71 \cdot 12653$$ $$J_8$$ $$=$$ $$169845836$$ $$=$$ $$2^{2} \cdot 37 \cdot 97 \cdot 11831$$ $$J_{10}$$ $$=$$ $$162080$$ $$=$$ $$2^{5} \cdot 5 \cdot 1013$$ $$g_1$$ $$=$$ $$44240899506176/5065$$ $$g_2$$ $$=$$ $$1809698189312/5065$$ $$g_3$$ $$=$$ $$96786036168/5065$$

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

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

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

Number of rational Weierstrass points: $$1$$

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: $$\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-2 : 2 : 1) - (1 : 0 : 0)$$ $$z (x + 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - 6z^3$$ $$0.134122$$ $$\infty$$
Generator $D_0$ Height Order
$$(-2 : 2 : 1) - (1 : 0 : 0)$$ $$z (x + 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - 6z^3$$ $$0.134122$$ $$\infty$$
Generator $D_0$ Height Order
$$(-2 : 0 : 1) - (1 : 1 : 0)$$ $$z (x + 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 + x^2z - 12z^3$$ $$0.134122$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.134122$$ Real period: $$5.992173$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.803683$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$1$$ $$5$$ $$1$$ $$( 1 + T )( 1 + 2 T^{2} )$$
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 3 T + 5 T^{2} )$$
$$1013$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 35 T + 1013 T^{2} )$$

## 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 $$\Q$$.