Properties

Label 249362.a.498724.1
Conductor $249362$
Discriminant $-498724$
Mordell-Weil group \(\Z \oplus \Z \oplus \Z\)
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

$y^2 + (x^2 + x + 1)y = 2x^5 - 9x^4 + 9x^3 - 2x$ (homogenize, simplify)
$y^2 + (x^2z + xz^2 + z^3)y = 2x^5z - 9x^4z^2 + 9x^3z^3 - 2xz^5$ (dehomogenize, simplify)
$y^2 = 8x^5 - 35x^4 + 38x^3 + 3x^2 - 6x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -2, 0, 9, -9, 2]), R([1, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -2, 0, 9, -9, 2], R![1, 1, 1]);
 
sage: X = HyperellipticCurve(R([1, -6, 3, 38, -35, 8]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(249362\) \(=\) \( 2 \cdot 41 \cdot 3041 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-498724\) \(=\) \( - 2^{2} \cdot 41 \cdot 3041 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(4212\) \(=\)  \( 2^{2} \cdot 3^{4} \cdot 13 \)
\( I_4 \)  \(=\) \(154569\) \(=\)  \( 3 \cdot 67 \cdot 769 \)
\( I_6 \)  \(=\) \(192076389\) \(=\)  \( 3^{2} \cdot 21341821 \)
\( I_{10} \)  \(=\) \(-63836672\) \(=\)  \( - 2^{9} \cdot 41 \cdot 3041 \)
\( J_2 \)  \(=\) \(1053\) \(=\)  \( 3^{4} \cdot 13 \)
\( J_4 \)  \(=\) \(39760\) \(=\)  \( 2^{4} \cdot 5 \cdot 7 \cdot 71 \)
\( J_6 \)  \(=\) \(1918804\) \(=\)  \( 2^{2} \cdot 479701 \)
\( J_8 \)  \(=\) \(109910753\) \(=\)  \( 67 \cdot 127 \cdot 12917 \)
\( J_{10} \)  \(=\) \(-498724\) \(=\)  \( - 2^{2} \cdot 41 \cdot 3041 \)
\( g_1 \)  \(=\) \(-1294618640600493/498724\)
\( g_2 \)  \(=\) \(-11605704217380/124681\)
\( g_3 \)  \(=\) \(-531896786109/124681\)

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
\((1 : 0 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((1 : -2 : 2)\) \((1 : -3 : 1)\)
\((2 : -3 : 1)\) \((2 : -4 : 1)\) \((3 : -6 : 2)\) \((1 : -12 : 2)\) \((5 : 29 : 1)\) \((3 : -32 : 2)\)
\((5 : -60 : 1)\) \((7 : -2184 : 18)\) \((7 : -6798 : 18)\)
Known points
\((1 : 0 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((1 : -2 : 2)\) \((1 : -3 : 1)\)
\((2 : -3 : 1)\) \((2 : -4 : 1)\) \((3 : -6 : 2)\) \((1 : -12 : 2)\) \((5 : 29 : 1)\) \((3 : -32 : 2)\)
\((5 : -60 : 1)\) \((7 : -2184 : 18)\) \((7 : -6798 : 18)\)
Known points
\((1 : 0 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((2 : -1 : 1)\) \((2 : 1 : 1)\) \((1 : -3 : 1)\)
\((1 : 3 : 1)\) \((1 : -10 : 2)\) \((1 : 10 : 2)\) \((3 : -26 : 2)\) \((3 : 26 : 2)\) \((5 : -89 : 1)\)
\((5 : 89 : 1)\) \((7 : -4614 : 18)\) \((7 : 4614 : 18)\)

magma: [C![0,-1,1],C![0,0,1],C![1,-12,2],C![1,-3,1],C![1,-2,2],C![1,0,0],C![1,0,1],C![2,-4,1],C![2,-3,1],C![3,-32,2],C![3,-6,2],C![5,-60,1],C![5,29,1],C![7,-6798,18],C![7,-2184,18]]; // minimal model
 
magma: [C![0,-1,1],C![0,1,1],C![1,-10,2],C![1,-3,1],C![1,10,2],C![1,0,0],C![1,3,1],C![2,-1,1],C![2,1,1],C![3,-26,2],C![3,26,2],C![5,-89,1],C![5,89,1],C![7,-4614,18],C![7,4614,18]]; // 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: \(\Z \oplus \Z \oplus \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 3xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0.880650\) \(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.293429\) \(\infty\)
\((0 : -1 : 1) + (2 : -3 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0.204690\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 3xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0.880650\) \(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.293429\) \(\infty\)
\((0 : -1 : 1) + (2 : -3 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0.204690\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 3xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^2z - xz^2 + z^3\) \(0.880650\) \(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(x^2z + xz^2 - z^3\) \(0.293429\) \(\infty\)
\((0 : -1 : 1) + (2 : 1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(x^2z - xz^2 - z^3\) \(0.204690\) \(\infty\)

2-torsion field: 5.3.1994896.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.044768 \)
Real period: \( 16.47752 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.475362 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + T + 2 T^{2} )\)
\(41\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 10 T + 41 T^{2} )\)
\(3041\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 46 T + 3041 T^{2} )\)

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);