# Properties

 Label 1142.b.9136.1 Conductor $1142$ Discriminant $-9136$ Mordell-Weil group $$\Z/{12}\Z$$ 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

 $y^2 + (x + 1)y = -x^5 + 3x^4 - 6x^2 + x + 3$ (homogenize, simplify) $y^2 + (xz^2 + z^3)y = -x^5z + 3x^4z^2 - 6x^2z^4 + xz^5 + 3z^6$ (dehomogenize, simplify) $y^2 = -4x^5 + 12x^4 - 23x^2 + 6x + 13$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([3, 1, -6, 0, 3, -1]), R([1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![3, 1, -6, 0, 3, -1], R![1, 1]);

sage: X = HyperellipticCurve(R([13, 6, -23, 0, 12, -4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$1142$$ $$=$$ $$2 \cdot 571$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-9136$$ $$=$$ $$- 2^{4} \cdot 571$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$864$$ $$=$$ $$2^{5} \cdot 3^{3}$$ $$I_4$$ $$=$$ $$-4488$$ $$=$$ $$- 2^{3} \cdot 3 \cdot 11 \cdot 17$$ $$I_6$$ $$=$$ $$-1442025$$ $$=$$ $$- 3^{2} \cdot 5^{2} \cdot 13 \cdot 17 \cdot 29$$ $$I_{10}$$ $$=$$ $$-36544$$ $$=$$ $$- 2^{6} \cdot 571$$ $$J_2$$ $$=$$ $$432$$ $$=$$ $$2^{4} \cdot 3^{3}$$ $$J_4$$ $$=$$ $$8524$$ $$=$$ $$2^{2} \cdot 2131$$ $$J_6$$ $$=$$ $$257089$$ $$=$$ $$7 \cdot 19 \cdot 1933$$ $$J_8$$ $$=$$ $$9600968$$ $$=$$ $$2^{3} \cdot 13 \cdot 92317$$ $$J_{10}$$ $$=$$ $$-9136$$ $$=$$ $$- 2^{4} \cdot 571$$ $$g_1$$ $$=$$ $$-940369969152/571$$ $$g_2$$ $$=$$ $$-42951140352/571$$ $$g_3$$ $$=$$ $$-2998686096/571$$

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

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

magma: [C![-1,0,1],C![1,-2,1],C![1,0,0],C![1,2,1]]; // 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: $$\Z/{12}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(1 : -2 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2z^3$$ $$0$$ $$12$$
Generator $D_0$ Height Order
$$(1 : -2 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2z^3$$ $$0$$ $$12$$
Generator $D_0$ Height Order
$$(1 : -2 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2 - 3z^3$$ $$0$$ $$12$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$17.75728$$ Tamagawa product: $$4$$ Torsion order: $$12$$ Leading coefficient: $$0.493257$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$1$$ $$4$$ $$4$$ $$( 1 - T )( 1 + 2 T^{2} )$$
$$571$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 4 T + 571 T^{2} )$$

## 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 $$\Q$$.