# Properties

 Label 932261.a.932261.1 Conductor $932261$ Discriminant $-932261$ Mordell-Weil group $$\Z \times \Z \times \Z$$ 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 for: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + 1)y = -2x^4 + 19x^3 - 71x^2 + 98x - 46$ (homogenize, simplify) $y^2 + (x^3 + z^3)y = -2x^4z^2 + 19x^3z^3 - 71x^2z^4 + 98xz^5 - 46z^6$ (dehomogenize, simplify) $y^2 = x^6 - 8x^4 + 78x^3 - 284x^2 + 392x - 183$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-46, 98, -71, 19, -2]), R([1, 0, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-46, 98, -71, 19, -2], R![1, 0, 0, 1]);

sage: X = HyperellipticCurve(R([-183, 392, -284, 78, -8, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$932261$$ $$=$$ $$11 \cdot 84751$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-932261$$ $$=$$ $$- 11 \cdot 84751$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$22036$$ $$=$$ $$2^{2} \cdot 7 \cdot 787$$ $$I_4$$ $$=$$ $$-129335$$ $$=$$ $$- 5 \cdot 25867$$ $$I_6$$ $$=$$ $$-943515931$$ $$=$$ $$-943515931$$ $$I_{10}$$ $$=$$ $$-119329408$$ $$=$$ $$- 2^{7} \cdot 11 \cdot 84751$$ $$J_2$$ $$=$$ $$5509$$ $$=$$ $$7 \cdot 787$$ $$J_4$$ $$=$$ $$1269934$$ $$=$$ $$2 \cdot 17 \cdot 41 \cdot 911$$ $$J_6$$ $$=$$ $$391878820$$ $$=$$ $$2^{2} \cdot 5 \cdot 41 \cdot 53 \cdot 71 \cdot 127$$ $$J_8$$ $$=$$ $$136532013756$$ $$=$$ $$2^{2} \cdot 3 \cdot 41 \cdot 277504093$$ $$J_{10}$$ $$=$$ $$-932261$$ $$=$$ $$- 11 \cdot 84751$$ $$g_1$$ $$=$$ $$-5074156546952986549/932261$$ $$g_2$$ $$=$$ $$-212324186037072886/932261$$ $$g_3$$ $$=$$ $$-11893162050364420/932261$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(2 : -2 : 1)$$ $$(2 : -7 : 1)$$ $$(3 : -16 : 2)$$ $$(3 : -19 : 2)$$
$$(4 : -45 : 3)$$ $$(4 : -46 : 3)$$ $$(-122 : 2762 : 3)$$ $$(-122 : 1813059 : 3)$$
Known points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(2 : -2 : 1)$$ $$(2 : -7 : 1)$$ $$(3 : -16 : 2)$$ $$(3 : -19 : 2)$$
$$(4 : -45 : 3)$$ $$(4 : -46 : 3)$$ $$(-122 : 2762 : 3)$$ $$(-122 : 1813059 : 3)$$
Known points
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(3 : -3 : 2)$$ $$(3 : 3 : 2)$$ $$(4 : -1 : 3)$$ $$(4 : 1 : 3)$$
$$(2 : -5 : 1)$$ $$(2 : 5 : 1)$$ $$(-122 : -1810297 : 3)$$ $$(-122 : 1810297 : 3)$$

magma: [C![-122,2762,3],C![-122,1813059,3],C![1,-1,0],C![1,0,0],C![2,-7,1],C![2,-2,1],C![3,-19,2],C![3,-16,2],C![4,-46,3],C![4,-45,3]]; // minimal model

magma: [C![-122,-1810297,3],C![-122,1810297,3],C![1,-1,0],C![1,1,0],C![2,-5,1],C![2,5,1],C![3,-3,2],C![3,3,2],C![4,-1,3],C![4,1,3]]; // 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: $$\Z \times \Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(2 : -7 : 1) + (4 : -45 : 3) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$(x - 2z) (3x - 4z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-8xz^2 + 9z^3$$ $$1.317139$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$2x^2 + 10xz - 17z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-51xz^2 + 65z^3$$ $$1.645093$$ $$\infty$$
$$(4 : -46 : 3) - (1 : 0 : 0)$$ $$z (3x - 4z)$$ $$=$$ $$0,$$ $$3y$$ $$=$$ $$-3x^3 + 2z^3$$ $$0.205701$$ $$\infty$$
Generator $D_0$ Height Order
$$(2 : -7 : 1) + (4 : -45 : 3) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$(x - 2z) (3x - 4z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-8xz^2 + 9z^3$$ $$1.317139$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$2x^2 + 10xz - 17z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-51xz^2 + 65z^3$$ $$1.645093$$ $$\infty$$
$$(4 : -46 : 3) - (1 : 0 : 0)$$ $$z (3x - 4z)$$ $$=$$ $$0,$$ $$3y$$ $$=$$ $$-3x^3 + 2z^3$$ $$0.205701$$ $$\infty$$
Generator $D_0$ Height Order
$$(2 : -5 : 1) + (4 : 1 : 3) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$(x - 2z) (3x - 4z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - 16xz^2 + 19z^3$$ $$1.317139$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$2x^2 + 10xz - 17z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$x^3 - 102xz^2 + 131z^3$$ $$1.645093$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$z (3x - 4z)$$ $$=$$ $$0,$$ $$3y$$ $$=$$ $$-5x^3 + 5z^3$$ $$0.205701$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$3$$   (upper bound) Mordell-Weil rank: $$3$$ 2-Selmer rank: $$3$$ Regulator: $$0.384040$$ Real period: $$6.904498$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$2.651607$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$11$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 4 T + 11 T^{2} )$$
$$84751$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 395 T + 84751 T^{2} )$$

## 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 $$\Q$$.