Properties

Label 1136.a.290816.1
Conductor 1136
Discriminant 290816
Mordell-Weil group \(\Z/{14}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x^2)y = -5x^4 - 9x^3 + 25x^2 + 40x - 24$ (homogenize, simplify)
$y^2 + (x^3 + x^2z)y = -5x^4z^2 - 9x^3z^3 + 25x^2z^4 + 40xz^5 - 24z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^5 - 19x^4 - 36x^3 + 100x^2 + 160x - 96$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-24, 40, 25, -9, -5]), R([0, 0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-24, 40, 25, -9, -5], R![0, 0, 1, 1]);
 
sage: X = HyperellipticCurve(R([-96, 160, 100, -36, -19, 2, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1136\) \(=\) \( 2^{4} \cdot 71 \)
magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(1136,2),R![1, -1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(290816\) \(=\) \( 2^{12} \cdot 71 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(74016\) \(=\)  \( 2^{5} \cdot 3^{2} \cdot 257 \)
\( I_4 \)  \(=\) \(1101888\) \(=\)  \( 2^{6} \cdot 3^{2} \cdot 1913 \)
\( I_6 \)  \(=\) \(27096003072\) \(=\)  \( 2^{9} \cdot 3^{2} \cdot 5880209 \)
\( I_{10} \)  \(=\) \(1191182336\) \(=\)  \( 2^{24} \cdot 71 \)
\( J_2 \)  \(=\) \(9252\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 257 \)
\( J_4 \)  \(=\) \(3555168\) \(=\)  \( 2^{5} \cdot 3 \cdot 29 \cdot 1277 \)
\( J_6 \)  \(=\) \(1815712832\) \(=\)  \( 2^{6} \cdot 577 \cdot 49169 \)
\( J_8 \)  \(=\) \(1039938903360\) \(=\)  \( 2^{6} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 17194757 \)
\( J_{10} \)  \(=\) \(290816\) \(=\)  \( 2^{12} \cdot 71 \)
\( g_1 \)  \(=\) \(66203075280122793/284\)
\( g_2 \)  \(=\) \(1374792164318403/142\)
\( g_3 \)  \(=\) \(151781365064097/284\)

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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-3 : 9 : 1)\)

magma: [C![-3,9,1],C![1,-1,0],C![1,0,0]];
 

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: \(\Z/{14}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - 7z^2\) \(=\) \(0,\) \(y\) \(=\) \(-3xz^2 - 2z^3\) \(0\) \(14\)

2-torsion field: 6.6.10323968.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 13.47670 \)
Tamagawa product: \( 7 \)
Torsion order:\( 14 \)
Leading coefficient: \( 0.481310 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(4\) \(12\) \(7\) \(1 - T\)
\(71\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 16 T + 71 T^{2} )\)

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 \(\Q\).