Properties

Label 100352.d.100352.1
Conductor 100352
Discriminant 100352
Mordell-Weil group \(\Z \times \Z/{2}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

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Minimal equation

Minimal equation

Simplified equation

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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, 4, 3, -4, -1, 1], R![]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, 4, 3, -4, -1, 1]), R([]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([-2, 4, 3, -4, -1, 1]))
 

Invariants

Conductor: \( N \)  =  \(100352\) = \( 2^{11} \cdot 7^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(100352\) = \( 2^{11} \cdot 7^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(4864\) =  \( 2^{8} \cdot 19 \)
\( I_4 \)  = \(351232\) =  \( 2^{10} \cdot 7^{3} \)
\( I_6 \)  = \(614154240\) =  \( 2^{14} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 17 \)
\( I_{10} \)  = \(411041792\) =  \( 2^{23} \cdot 7^{2} \)
\( J_2 \)  = \(608\) =  \( 2^{5} \cdot 19 \)
\( J_4 \)  = \(11744\) =  \( 2^{5} \cdot 367 \)
\( J_6 \)  = \(71936\) =  \( 2^{8} \cdot 281 \)
\( J_8 \)  = \(-23546112\) =  \( - 2^{8} \cdot 3 \cdot 23 \cdot 31 \cdot 43 \)
\( J_{10} \)  = \(100352\) =  \( 2^{11} \cdot 7^{2} \)
\( g_1 \)  = \(40568406016/49\)
\( g_2 \)  = \(1288833536/49\)
\( g_3 \)  = \(12984448/49\)

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

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

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

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: \(\Z \times \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.695932\) \(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)

2-torsion field: 6.6.1229312.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.695932 \)
Real period: \( 11.58932 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 2.016345 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(11\) \(11\) \(1\) \(1\)
\(7\) \(2\) \(2\) \(1\) \(1 + 7 T^{2}\)

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 \(\Q\).