Properties

Label 10102.a.323264.1
Conductor $10102$
Discriminant $-323264$
Mordell-Weil group \(\Z \times \Z/{2}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

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Show commands: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x + 1)y = -x^5 + 3x^4 + x^3 - 3x^2$ (homogenize, simplify)
$y^2 + (xz^2 + z^3)y = -x^5z + 3x^4z^2 + x^3z^3 - 3x^2z^4$ (dehomogenize, simplify)
$y^2 = -4x^5 + 12x^4 + 4x^3 - 11x^2 + 2x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -3, 1, 3, -1]), R([1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -3, 1, 3, -1], R![1, 1]);
 
sage: X = HyperellipticCurve(R([1, 2, -11, 4, 12, -4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(10102\) \(=\) \( 2 \cdot 5051 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-323264\) \(=\) \( - 2^{6} \cdot 5051 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(472\) \(=\)  \( 2^{3} \cdot 59 \)
\( I_4 \)  \(=\) \(8644\) \(=\)  \( 2^{2} \cdot 2161 \)
\( I_6 \)  \(=\) \(1089623\) \(=\)  \( 367 \cdot 2969 \)
\( I_{10} \)  \(=\) \(-1293056\) \(=\)  \( - 2^{8} \cdot 5051 \)
\( J_2 \)  \(=\) \(236\) \(=\)  \( 2^{2} \cdot 59 \)
\( J_4 \)  \(=\) \(880\) \(=\)  \( 2^{4} \cdot 5 \cdot 11 \)
\( J_6 \)  \(=\) \(3801\) \(=\)  \( 3 \cdot 7 \cdot 181 \)
\( J_8 \)  \(=\) \(30659\) \(=\)  \( 23 \cdot 31 \cdot 43 \)
\( J_{10} \)  \(=\) \(-323264\) \(=\)  \( - 2^{6} \cdot 5051 \)
\( g_1 \)  \(=\) \(-11438788784/5051\)
\( g_2 \)  \(=\) \(-180733520/5051\)
\( g_3 \)  \(=\) \(-13231281/20204\)

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
\((1 : 0 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((1 : -2 : 1)\)
\((3 : 0 : 1)\) \((3 : -4 : 1)\)
All points
\((1 : 0 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((1 : -2 : 1)\)
\((3 : 0 : 1)\) \((3 : -4 : 1)\)
All points
\((1 : 0 : 0)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((1 : -2 : 1)\) \((1 : 2 : 1)\)
\((3 : -4 : 1)\) \((3 : 4 : 1)\)

magma: [C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,0,0],C![1,0,1],C![3,-4,1],C![3,0,1]]; // minimal model
 
magma: [C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,-2,1],C![1,0,0],C![1,2,1],C![3,-4,1],C![3,4,1]]; // simplified model
 

Number of rational Weierstrass points: \(2\)

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
\((1 : -2 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.025482\) \(\infty\)
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(x + z\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\((1 : -2 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.025482\) \(\infty\)
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(x + z\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\((1 : -2 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - 3z^3\) \(0.025482\) \(\infty\)
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(x + z\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 + z^3\) \(0\) \(2\)

2-torsion field: 4.2.20204.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.025482 \)
Real period: \( 17.57152 \)
Tamagawa product: \( 6 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.671646 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(6\) \(6\) \(( 1 - T )( 1 + 2 T + 2 T^{2} )\)
\(5051\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 72 T + 5051 T^{2} )\)

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 \(\Q\).