Properties

Label 100124.a.400496.1
Conductor 100124
Discriminant -400496
Mordell-Weil group \(\Z \times \Z/{2}\Z \times \Z/{2}\Z\)
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

$y^2 + (x^2 + x)y = x^5 - 2x^4 + 2x^2 - 2x$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = x^5z - 2x^4z^2 + 2x^2z^4 - 2xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 - 7x^4 + 2x^3 + 9x^2 - 8x$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -2, 2, 0, -2, 1], R![0, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -2, 2, 0, -2, 1]), R([0, 1, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([0, -8, 9, 2, -7, 4]))
 

Invariants

Conductor: \( N \)  =  \(100124\) = \( 2^{2} \cdot 25031 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-400496\) = \( - 2^{4} \cdot 25031 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-248\) =  \( - 2^{3} \cdot 31 \)
\( I_4 \)  = \(-60956\) =  \( - 2^{2} \cdot 7^{2} \cdot 311 \)
\( I_6 \)  = \(-839960\) =  \( - 2^{3} \cdot 5 \cdot 11 \cdot 23 \cdot 83 \)
\( I_{10} \)  = \(-1640431616\) =  \( - 2^{16} \cdot 25031 \)
\( J_2 \)  = \(-31\) =  \( - 31 \)
\( J_4 \)  = \(675\) =  \( 3^{3} \cdot 5^{2} \)
\( J_6 \)  = \(6857\) =  \( 6857 \)
\( J_8 \)  = \(-167048\) =  \( - 2^{3} \cdot 7 \cdot 19 \cdot 157 \)
\( J_{10} \)  = \(-400496\) =  \( - 2^{4} \cdot 25031 \)
\( g_1 \)  = \(28629151/400496\)
\( g_2 \)  = \(20108925/400496\)
\( g_3 \)  = \(-6589577/400496\)

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

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

Number of rational Weierstrass points: \(3\)

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : -1 : 1) - (1 : 0 : 0)\) \(x + z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.822181\) \(\infty\)
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
\((1 : -1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(2\)

2-torsion field: 3.1.25031.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(3\)
Regulator: \( 0.822181 \)
Real period: \( 11.14761 \)
Tamagawa product: \( 4 \)
Torsion order:\( 4 \)
Leading coefficient: \( 2.291339 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(4\) \(2\) \(4\) \(( 1 - T )( 1 + T )\)
\(25031\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 56 T + 25031 T^{2} )\)

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 \(\Q\).