# Properties

 Label 10110.a.50550.1 Conductor $10110$ Discriminant $50550$ 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

# Related objects

Show commands: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + 1)y = 2x^5 + 3x^4 + 2x^3 - x - 1$ (homogenize, simplify) $y^2 + (x^3 + z^3)y = 2x^5z + 3x^4z^2 + 2x^3z^3 - xz^5 - z^6$ (dehomogenize, simplify) $y^2 = x^6 + 8x^5 + 12x^4 + 10x^3 - 4x - 3$ (minimize, homogenize)

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

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

sage: X = HyperellipticCurve(R([-3, -4, 0, 10, 12, 8, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$10110$$ $$=$$ $$2 \cdot 3 \cdot 5 \cdot 337$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$50550$$ $$=$$ $$2 \cdot 3 \cdot 5^{2} \cdot 337$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$20$$ $$=$$ $$2^{2} \cdot 5$$ $$I_4$$ $$=$$ $$16177$$ $$=$$ $$7 \cdot 2311$$ $$I_6$$ $$=$$ $$1232025$$ $$=$$ $$3 \cdot 5^{2} \cdot 16427$$ $$I_{10}$$ $$=$$ $$6470400$$ $$=$$ $$2^{8} \cdot 3 \cdot 5^{2} \cdot 337$$ $$J_2$$ $$=$$ $$5$$ $$=$$ $$5$$ $$J_4$$ $$=$$ $$-673$$ $$=$$ $$-673$$ $$J_6$$ $$=$$ $$-16175$$ $$=$$ $$- 5^{2} \cdot 647$$ $$J_8$$ $$=$$ $$-133451$$ $$=$$ $$-133451$$ $$J_{10}$$ $$=$$ $$50550$$ $$=$$ $$2 \cdot 3 \cdot 5^{2} \cdot 337$$ $$g_1$$ $$=$$ $$125/2022$$ $$g_2$$ $$=$$ $$-3365/2022$$ $$g_3$$ $$=$$ $$-16175/2022$$

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

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

magma: [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
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$2x^2 + 2xz + z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$xz^2 - z^3$$ $$0.303673$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 + xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$2x^2 + 2xz + z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$xz^2 - z^3$$ $$0.303673$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 + xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$2x^2 + 2xz + z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$x^3 + 2xz^2 - z^3$$ $$0.303673$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2 + xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - z^3$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$2$$ Regulator: $$0.303673$$ Real period: $$6.345414$$ Tamagawa product: $$2$$ Torsion order: $$2$$ Leading coefficient: $$0.963467$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - T + 2 T^{2} )$$
$$3$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - T + 3 T^{2} )$$
$$5$$ $$1$$ $$2$$ $$2$$ $$( 1 + T )( 1 + 2 T + 5 T^{2} )$$
$$337$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 14 T + 337 T^{2} )$$

## 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 $$\Q$$.