Properties

Label 10137.a.10137.1
Conductor $10137$
Discriminant $10137$
Mordell-Weil group \(\Z\)
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

$y^2 + (x^3 + x + 1)y = -2x^4 + 4x^2 - x - 4$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -2x^4z^2 + 4x^2z^4 - xz^5 - 4z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 6x^4 + 2x^3 + 17x^2 - 2x - 15$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-4, -1, 4, 0, -2]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-4, -1, 4, 0, -2], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([-15, -2, 17, 2, -6, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(10137\) \(=\) \( 3 \cdot 31 \cdot 109 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(10137\) \(=\) \( 3 \cdot 31 \cdot 109 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(2628\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 73 \)
\( I_4 \)  \(=\) \(3417\) \(=\)  \( 3 \cdot 17 \cdot 67 \)
\( I_6 \)  \(=\) \(2633517\) \(=\)  \( 3^{2} \cdot 53 \cdot 5521 \)
\( I_{10} \)  \(=\) \(1297536\) \(=\)  \( 2^{7} \cdot 3 \cdot 31 \cdot 109 \)
\( J_2 \)  \(=\) \(657\) \(=\)  \( 3^{2} \cdot 73 \)
\( J_4 \)  \(=\) \(17843\) \(=\)  \( 7 \cdot 2549 \)
\( J_6 \)  \(=\) \(645873\) \(=\)  \( 3 \cdot 41 \cdot 59 \cdot 89 \)
\( J_8 \)  \(=\) \(26491478\) \(=\)  \( 2 \cdot 13 \cdot 1018903 \)
\( J_{10} \)  \(=\) \(10137\) \(=\)  \( 3 \cdot 31 \cdot 109 \)
\( g_1 \)  \(=\) \(40804268165019/3379\)
\( g_2 \)  \(=\) \(1686718970433/3379\)
\( g_3 \)  \(=\) \(92930144859/3379\)

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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-2 : 3 : 1),\, (-2 : 6 : 1)\)
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-2 : 3 : 1),\, (-2 : 6 : 1)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (-2 : -3 : 1),\, (-2 : 3 : 1)\)

magma: [C![-2,3,1],C![-2,6,1],C![1,-1,0],C![1,0,0]]; // minimal model
 
magma: [C![-2,-3,1],C![-2,3,1],C![1,-1,0],C![1,1,0]]; // simplified model
 

Number of rational Weierstrass points: \(0\)

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: \(\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 - z^3\) \(0.057473\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 - z^3\) \(0.057473\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 3xz^2 - z^3\) \(0.057473\) \(\infty\)

2-torsion field: 6.2.1041665641353.3

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.057473 \)
Real period: \( 12.96773 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.745306 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(3\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 3 T^{2} )\)
\(31\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 3 T + 31 T^{2} )\)
\(109\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 9 T + 109 T^{2} )\)

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 \(\Q\).