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

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Minimal equation

Minimal equation

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);
 

Igusa-Clebsch invariants

Igusa invariants

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
\((2 : -5 : 1) + (4 : 1 : 3) - (1 : -1 : 0) - (1 : 1 : 0)\) \((x - 2z) (3x - 4z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 16xz^2 + 19z^3\) \(1.317139\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(2x^2 + 10xz - 17z^2\) \(=\) \(0,\) \(2y\) \(=\) \(x^3 - 102xz^2 + 131z^3\) \(1.645093\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(z (3x - 4z)\) \(=\) \(0,\) \(3y\) \(=\) \(-5x^3 + 5z^3\) \(0.205701\) \(\infty\)

2-torsion field: 6.4.14916176.1

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 \(\Q\).