Properties

Label 100096.a.100096.1
Conductor 100096
Discriminant -100096
Mordell-Weil group \(\Z/{2}\Z\)
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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Show commands for: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

$y^2 = x^5 - 2x^4 - 3x^3 - 7x^2 - 7x - 2$ (homogenize, simplify)
$y^2 = x^5z - 2x^4z^2 - 3x^3z^3 - 7x^2z^4 - 7xz^5 - 2z^6$ (dehomogenize, simplify)
$y^2 = x^5 - 2x^4 - 3x^3 - 7x^2 - 7x - 2$ (minimize, homogenize)

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

Invariants

Conductor: \( N \)  =  \(100096\) = \( 2^{8} \cdot 17 \cdot 23 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-100096\) = \( - 2^{8} \cdot 17 \cdot 23 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-7200\) =  \( - 2^{5} \cdot 3^{2} \cdot 5^{2} \)
\( I_4 \)  = \(534528\) =  \( 2^{11} \cdot 3^{2} \cdot 29 \)
\( I_6 \)  = \(-1811128320\) =  \( - 2^{13} \cdot 3^{2} \cdot 5 \cdot 17^{3} \)
\( I_{10} \)  = \(-409993216\) =  \( - 2^{20} \cdot 17 \cdot 23 \)
\( J_2 \)  = \(-900\) =  \( - 2^{2} \cdot 3^{2} \cdot 5^{2} \)
\( J_4 \)  = \(28182\) =  \( 2 \cdot 3 \cdot 7 \cdot 11 \cdot 61 \)
\( J_6 \)  = \(64820\) =  \( 2^{2} \cdot 5 \cdot 7 \cdot 463 \)
\( J_8 \)  = \(-213140781\) =  \( - 3^{3} \cdot 7 \cdot 17 \cdot 66337 \)
\( J_{10} \)  = \(-100096\) =  \( - 2^{8} \cdot 17 \cdot 23 \)
\( g_1 \)  = \(2306601562500/391\)
\( g_2 \)  = \(160505296875/782\)
\( g_3 \)  = \(-820378125/1564\)

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

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: 6.2.2598977.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 3.616385 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.904096 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(8\) \(8\) \(1\) \(1\)
\(17\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 6 T + 17 T^{2} )\)
\(23\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 4 T + 23 T^{2} )\)

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 \(\Q\).