Properties

Label 15957.c.143613.1
Conductor $15957$
Discriminant $-143613$
Mordell-Weil group \(\Z \oplus \Z\)
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + 1)y = -2x^4 + 9x^2 + 9x + 2$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = -2x^4z^2 + 9x^2z^4 + 9xz^5 + 2z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 8x^4 + 2x^3 + 36x^2 + 36x + 9$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 9, 9, 0, -2]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 9, 9, 0, -2], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([9, 36, 36, 2, -8, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(15957\) \(=\) \( 3^{4} \cdot 197 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-143613\) \(=\) \( - 3^{6} \cdot 197 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(412\) \(=\)  \( 2^{2} \cdot 103 \)
\( I_4 \)  \(=\) \(4641\) \(=\)  \( 3 \cdot 7 \cdot 13 \cdot 17 \)
\( I_6 \)  \(=\) \(548591\) \(=\)  \( 548591 \)
\( I_{10} \)  \(=\) \(-75648\) \(=\)  \( - 2^{7} \cdot 3 \cdot 197 \)
\( J_2 \)  \(=\) \(309\) \(=\)  \( 3 \cdot 103 \)
\( J_4 \)  \(=\) \(2238\) \(=\)  \( 2 \cdot 3 \cdot 373 \)
\( J_6 \)  \(=\) \(11956\) \(=\)  \( 2^{2} \cdot 7^{2} \cdot 61 \)
\( J_8 \)  \(=\) \(-328560\) \(=\)  \( - 2^{4} \cdot 3 \cdot 5 \cdot 37^{2} \)
\( J_{10} \)  \(=\) \(-143613\) \(=\)  \( - 3^{6} \cdot 197 \)
\( g_1 \)  \(=\) \(-11592740743/591\)
\( g_2 \)  \(=\) \(-815174342/1773\)
\( g_3 \)  \(=\) \(-126841204/15957\)

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

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((-1 : 0 : 1)\) \((0 : 1 : 1)\) \((0 : -2 : 1)\) \((-2 : 3 : 1)\)
\((3 : -2 : 1)\) \((-2 : 4 : 1)\) \((-3 : 8 : 2)\) \((-3 : 11 : 2)\) \((3 : -26 : 1)\) \((-1 : 50 : 5)\)
\((-1 : -174 : 5)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((-1 : 0 : 1)\) \((0 : 1 : 1)\) \((0 : -2 : 1)\) \((-2 : 3 : 1)\)
\((3 : -2 : 1)\) \((-2 : 4 : 1)\) \((-3 : 8 : 2)\) \((-3 : 11 : 2)\) \((3 : -26 : 1)\) \((-1 : 50 : 5)\)
\((-1 : -174 : 5)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((-1 : 0 : 1)\) \((-2 : -1 : 1)\) \((-2 : 1 : 1)\) \((0 : -3 : 1)\)
\((0 : 3 : 1)\) \((-3 : -3 : 2)\) \((-3 : 3 : 2)\) \((3 : -24 : 1)\) \((3 : 24 : 1)\) \((-1 : -224 : 5)\)
\((-1 : 224 : 5)\)

magma: [C![-3,8,2],C![-3,11,2],C![-2,3,1],C![-2,4,1],C![-1,-174,5],C![-1,0,1],C![-1,50,5],C![0,-2,1],C![0,1,1],C![1,-1,0],C![1,0,0],C![3,-26,1],C![3,-2,1]]; // minimal model
 
magma: [C![-3,-3,2],C![-3,3,2],C![-2,-1,1],C![-2,1,1],C![-1,-224,5],C![-1,0,1],C![-1,224,5],C![0,-3,1],C![0,3,1],C![1,-1,0],C![1,1,0],C![3,-24,1],C![3,24,1]]; // simplified model
 

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: \(\Z \oplus \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.146215\) \(\infty\)
\((0 : -2 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.065797\) \(\infty\)
Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.146215\) \(\infty\)
\((0 : -2 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.065797\) \(\infty\)
Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : 1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.146215\) \(\infty\)
\((0 : -3 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 3z^3\) \(0.065797\) \(\infty\)

2-torsion field: 5.3.255312.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.009511 \)
Real period: \( 19.28012 \)
Tamagawa product: \( 3 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.550119 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(3\) \(4\) \(6\) \(3\) \(1 + 2 T + 3 T^{2}\)
\(197\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 26 T + 197 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.6.1 no

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

magma: HeuristicDecompositionFactors(C);
 

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 \(\Q\).

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);