Properties

Label 100128.a.801024.1
Conductor 100128
Discriminant 801024
Mordell-Weil group trivial
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

$y^2 + (x^2 + 1)y = 4x^5 - x^4 - 13x^3 - 2x^2 + 7x - 2$ (homogenize, simplify)
$y^2 + (x^2z + z^3)y = 4x^5z - x^4z^2 - 13x^3z^3 - 2x^2z^4 + 7xz^5 - 2z^6$ (dehomogenize, simplify)
$y^2 = 16x^5 - 3x^4 - 52x^3 - 6x^2 + 28x - 7$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, 7, -2, -13, -1, 4]), R([1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, 7, -2, -13, -1, 4], R![1, 0, 1]);
 
sage: X = HyperellipticCurve(R([-7, 28, -6, -52, -3, 16]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  =  \(100128\) = \( 2^{5} \cdot 3 \cdot 7 \cdot 149 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(801024\) = \( 2^{8} \cdot 3 \cdot 7 \cdot 149 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(33856\) =  \( 2^{6} \cdot 23^{2} \)
\( I_4 \)  = \(34483648\) =  \( 2^{6} \cdot 269 \cdot 2003 \)
\( I_6 \)  = \(317292855552\) =  \( 2^{8} \cdot 3^{2} \cdot 137713913 \)
\( I_{10} \)  = \(3280994304\) =  \( 2^{20} \cdot 3 \cdot 7 \cdot 149 \)
\( J_2 \)  = \(4232\) =  \( 2^{3} \cdot 23^{2} \)
\( J_4 \)  = \(387038\) =  \( 2 \cdot 431 \cdot 449 \)
\( J_6 \)  = \(46859332\) =  \( 2^{2} \cdot 13 \cdot 901141 \)
\( J_8 \)  = \(12127569895\) =  \( 5 \cdot 7 \cdot 307 \cdot 397 \cdot 2843 \)
\( J_{10} \)  = \(801024\) =  \( 2^{8} \cdot 3 \cdot 7 \cdot 149 \)
\( g_1 \)  = \(5302593435347072/3129\)
\( g_2 \)  = \(114591028813564/3129\)
\( g_3 \)  = \(3278290581553/3129\)

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: \((1 : 0 : 0)\)

magma: [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: trivial

magma: MordellWeilGroupGenus2(Jacobian(C));
 

2-torsion field: 5.1.200256.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 0.725352 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.450704 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(8\) \(5\) \(2\) \(1 + T\)
\(3\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + T + 3 T^{2} )\)
\(7\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 2 T + 7 T^{2} )\)
\(149\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 4 T + 149 T^{2} )\)

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 \(\Q\).