# Properties

 Label 10110.a.50550.2 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

 $y^2 + (x^2 + x)y = 39x^6 - 45x^5 - 62x^4 + 22x^3 + 69x^2 + 50x - 75$ (homogenize, simplify) $y^2 + (x^2z + xz^2)y = 39x^6 - 45x^5z - 62x^4z^2 + 22x^3z^3 + 69x^2z^4 + 50xz^5 - 75z^6$ (dehomogenize, simplify) $y^2 = 156x^6 - 180x^5 - 247x^4 + 90x^3 + 277x^2 + 200x - 300$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-75, 50, 69, 22, -62, -45, 39]), R([0, 1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-75, 50, 69, 22, -62, -45, 39], R![0, 1, 1]);

sage: X = HyperellipticCurve(R([-300, 200, 277, 90, -247, -180, 156]))

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

 $$I_2$$ $$=$$ $$5467652$$ $$=$$ $$2^{2} \cdot 23 \cdot 103 \cdot 577$$ $$I_4$$ $$=$$ $$-306863$$ $$=$$ $$- 47 \cdot 6529$$ $$I_6$$ $$=$$ $$-559265218983$$ $$=$$ $$- 3^{3} \cdot 20713526629$$ $$I_{10}$$ $$=$$ $$-6470400$$ $$=$$ $$- 2^{8} \cdot 3 \cdot 5^{2} \cdot 337$$ $$J_2$$ $$=$$ $$1366913$$ $$=$$ $$23 \cdot 103 \cdot 577$$ $$J_4$$ $$=$$ $$77852144018$$ $$=$$ $$2 \cdot 11 \cdot 3538733819$$ $$J_6$$ $$=$$ $$5912063482701400$$ $$=$$ $$2^{3} \cdot 5^{2} \cdot 337 \cdot 87716075411$$ $$J_8$$ $$=$$ $$505080025782601398469$$ $$=$$ $$131 \cdot 653 \cdot 10891 \cdot 542135640313$$ $$J_{10}$$ $$=$$ $$-50550$$ $$=$$ $$- 2 \cdot 3 \cdot 5^{2} \cdot 337$$ $$g_1$$ $$=$$ $$-4772043231067501633914862225793/50550$$ $$g_2$$ $$=$$ $$-99417583641640068080475078473/25275$$ $$g_3$$ $$=$$ $$-655572807749456176591436/3$$

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: $$(15 : -2730 : 13)$$ Known points: $$(15 : -2730 : 13)$$ Known points: $$(15 : 0 : 13)$$

magma: [C![15,-2730,13]]; // minimal model

magma: [C![15,0,13]]; // 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: $$\Z \times \Z/{2}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$2x^2 + 2xz - 5z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$41xz^2 - 50z^3$$ $$0.303673$$ $$\infty$$
$$D_0 - D_\infty$$ $$3x^2 - 4z^2$$ $$=$$ $$0,$$ $$6y$$ $$=$$ $$-3xz^2 - 4z^3$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$2x^2 + 2xz - 5z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$41xz^2 - 50z^3$$ $$0.303673$$ $$\infty$$
$$D_0 - D_\infty$$ $$3x^2 - 4z^2$$ $$=$$ $$0,$$ $$6y$$ $$=$$ $$-3xz^2 - 4z^3$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$2x^2 + 2xz - 5z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$x^2z + 83xz^2 - 100z^3$$ $$0.303673$$ $$\infty$$
$$D_0 - D_\infty$$ $$3x^2 - 4z^2$$ $$=$$ $$0,$$ $$6y$$ $$=$$ $$x^2z - 5xz^2 - 8z^3$$ $$0$$ $$2$$

## 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 $$\Q$$.