# Properties

 Label 10040.c.502000.1 Conductor $10040$ Discriminant $-502000$ 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

 $y^2 + (x^2 + 1)y = x^5 - 3x^4 + 3x^3 - 2x$ (homogenize, simplify) $y^2 + (x^2z + z^3)y = x^5z - 3x^4z^2 + 3x^3z^3 - 2xz^5$ (dehomogenize, simplify) $y^2 = 4x^5 - 11x^4 + 12x^3 + 2x^2 - 8x + 1$ (minimize, homogenize)

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

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

sage: X = HyperellipticCurve(R([1, -8, 2, 12, -11, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$10040$$ $$=$$ $$2^{3} \cdot 5 \cdot 251$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-502000$$ $$=$$ $$- 2^{4} \cdot 5^{3} \cdot 251$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$16$$ $$=$$ $$2^{4}$$ $$I_4$$ $$=$$ $$-4076$$ $$=$$ $$- 2^{2} \cdot 1019$$ $$I_6$$ $$=$$ $$38588$$ $$=$$ $$2^{2} \cdot 11 \cdot 877$$ $$I_{10}$$ $$=$$ $$2008000$$ $$=$$ $$2^{6} \cdot 5^{3} \cdot 251$$ $$J_2$$ $$=$$ $$8$$ $$=$$ $$2^{3}$$ $$J_4$$ $$=$$ $$682$$ $$=$$ $$2 \cdot 11 \cdot 31$$ $$J_6$$ $$=$$ $$-5796$$ $$=$$ $$- 2^{2} \cdot 3^{2} \cdot 7 \cdot 23$$ $$J_8$$ $$=$$ $$-127873$$ $$=$$ $$-127873$$ $$J_{10}$$ $$=$$ $$502000$$ $$=$$ $$2^{4} \cdot 5^{3} \cdot 251$$ $$g_1$$ $$=$$ $$2048/31375$$ $$g_2$$ $$=$$ $$21824/31375$$ $$g_3$$ $$=$$ $$-23184/31375$$

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: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -1 : 1),\, (3 : 5 : 1),\, (3 : -15 : 1)$$ All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -1 : 1),\, (3 : 5 : 1),\, (3 : -15 : 1)$$ All points: $$(1 : 0 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : 0 : 1),\, (3 : -20 : 1),\, (3 : 20 : 1)$$

magma: [C![0,-1,1],C![0,0,1],C![1,-1,1],C![1,0,0],C![3,-15,1],C![3,5,1]]; // minimal model

magma: [C![0,-1,1],C![0,1,1],C![1,0,1],C![1,0,0],C![3,-20,1],C![3,20,1]]; // simplified model

Number of rational Weierstrass points: $$2$$

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
$$(0 : 0 : 1) + (3 : -15 : 1) - 2 \cdot(1 : 0 : 0)$$ $$(-x + 3z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-5xz^2$$ $$0.045171$$ $$\infty$$
$$(1 : -1 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$(0 : 0 : 1) + (3 : -15 : 1) - 2 \cdot(1 : 0 : 0)$$ $$(-x + 3z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-5xz^2$$ $$0.045171$$ $$\infty$$
$$(1 : -1 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$(0 : 1 : 1) + (3 : -20 : 1) - 2 \cdot(1 : 0 : 0)$$ $$(-x + 3z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^2z - 10xz^2 + z^3$$ $$0.045171$$ $$\infty$$
$$(1 : 0 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^2z - z^3$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$2$$ Regulator: $$0.045171$$ Real period: $$11.06047$$ Tamagawa product: $$6$$ Torsion order: $$2$$ Leading coefficient: $$0.749431$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$3$$ $$4$$ $$2$$ $$1 + T + 2 T^{2}$$
$$5$$ $$1$$ $$3$$ $$3$$ $$( 1 - T )( 1 + 4 T + 5 T^{2} )$$
$$251$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 6 T + 251 T^{2} )$$

## 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 $$\Q$$.