Properties

Label 110499.a.331497.1
Conductor $110499$
Discriminant $331497$
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

$y^2 + (x^3 + 1)y = -x^5 + 7x^2 - 8x + 2$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = -x^5z + 7x^2z^4 - 8xz^5 + 2z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 4x^5 + 2x^3 + 28x^2 - 32x + 9$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, -8, 7, 0, 0, -1]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, -8, 7, 0, 0, -1], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([9, -32, 28, 2, 0, -4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(110499\) \(=\) \( 3 \cdot 36833 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(331497\) \(=\) \( 3^{2} \cdot 36833 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1492\) \(=\)  \( 2^{2} \cdot 373 \)
\( I_4 \)  \(=\) \(222409\) \(=\)  \( 11 \cdot 20219 \)
\( I_6 \)  \(=\) \(72914981\) \(=\)  \( 72914981 \)
\( I_{10} \)  \(=\) \(42431616\) \(=\)  \( 2^{7} \cdot 3^{2} \cdot 36833 \)
\( J_2 \)  \(=\) \(373\) \(=\)  \( 373 \)
\( J_4 \)  \(=\) \(-3470\) \(=\)  \( - 2 \cdot 5 \cdot 347 \)
\( J_6 \)  \(=\) \(67588\) \(=\)  \( 2^{2} \cdot 61 \cdot 277 \)
\( J_8 \)  \(=\) \(3292356\) \(=\)  \( 2^{2} \cdot 3 \cdot 17 \cdot 16139 \)
\( J_{10} \)  \(=\) \(331497\) \(=\)  \( 3^{2} \cdot 36833 \)
\( g_1 \)  \(=\) \(7220115733093/331497\)
\( g_2 \)  \(=\) \(-180076055990/331497\)
\( g_3 \)  \(=\) \(9403450852/331497\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 1 : 1)\) \((1 : 0 : 1)\) \((0 : -2 : 1)\) \((1 : -2 : 1)\)
\((1 : -3 : 2)\) \((2 : -3 : 1)\) \((-2 : -6 : 1)\) \((1 : -6 : 2)\) \((2 : -6 : 1)\) \((-2 : 13 : 1)\)
\((-4 : -15 : 1)\) \((-4 : 78 : 1)\) \((23 : -344 : 24)\) \((11 : -516 : 3)\) \((11 : -842 : 3)\) \((9 : -1183 : 14)\)
\((9 : -2290 : 14)\) \((23 : -25647 : 24)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 1 : 1)\) \((1 : 0 : 1)\) \((0 : -2 : 1)\) \((1 : -2 : 1)\)
\((1 : -3 : 2)\) \((2 : -3 : 1)\) \((-2 : -6 : 1)\) \((1 : -6 : 2)\) \((2 : -6 : 1)\) \((-2 : 13 : 1)\)
\((-4 : -15 : 1)\) \((-4 : 78 : 1)\) \((23 : -344 : 24)\) \((11 : -516 : 3)\) \((11 : -842 : 3)\) \((9 : -1183 : 14)\)
\((9 : -2290 : 14)\) \((23 : -25647 : 24)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((1 : -2 : 1)\) \((1 : 2 : 1)\) \((0 : -3 : 1)\) \((0 : 3 : 1)\)
\((1 : -3 : 2)\) \((1 : 3 : 2)\) \((2 : -3 : 1)\) \((2 : 3 : 1)\) \((-2 : -19 : 1)\) \((-2 : 19 : 1)\)
\((-4 : -93 : 1)\) \((-4 : 93 : 1)\) \((11 : -326 : 3)\) \((11 : 326 : 3)\) \((9 : -1107 : 14)\) \((9 : 1107 : 14)\)
\((23 : -25303 : 24)\) \((23 : 25303 : 24)\)

magma: [C![-4,-15,1],C![-4,78,1],C![-2,-6,1],C![-2,13,1],C![0,-2,1],C![0,1,1],C![1,-6,2],C![1,-3,2],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![2,-6,1],C![2,-3,1],C![9,-2290,14],C![9,-1183,14],C![11,-842,3],C![11,-516,3],C![23,-25647,24],C![23,-344,24]]; // minimal model
 
magma: [C![-4,-93,1],C![-4,93,1],C![-2,-19,1],C![-2,19,1],C![0,-3,1],C![0,3,1],C![1,-3,2],C![1,3,2],C![1,-2,1],C![1,-1,0],C![1,1,0],C![1,2,1],C![2,-3,1],C![2,3,1],C![9,-1107,14],C![9,1107,14],C![11,-326,3],C![11,326,3],C![23,-25303,24],C![23,25303,24]]; // 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: \(\Z \oplus \Z \oplus \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -2 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.391929\) \(\infty\)
\((1 : -2 : 1) + (2 : -6 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - 2z) (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-4xz^2 + 2z^3\) \(0.454238\) \(\infty\)
\((1 : -3 : 2) - (1 : 0 : 0)\) \(z (2x - z)\) \(=\) \(0,\) \(4y\) \(=\) \(-4x^3 - z^3\) \(0.169029\) \(\infty\)
Generator $D_0$ Height Order
\((1 : -2 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.391929\) \(\infty\)
\((1 : -2 : 1) + (2 : -6 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - 2z) (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-4xz^2 + 2z^3\) \(0.454238\) \(\infty\)
\((1 : -3 : 2) - (1 : 0 : 0)\) \(z (2x - z)\) \(=\) \(0,\) \(4y\) \(=\) \(-4x^3 - z^3\) \(0.169029\) \(\infty\)
Generator $D_0$ Height Order
\((1 : -2 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 3z^3\) \(0.391929\) \(\infty\)
\((1 : -2 : 1) + (2 : -3 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \((x - 2z) (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 8xz^2 + 5z^3\) \(0.454238\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(z (2x - z)\) \(=\) \(0,\) \(4y\) \(=\) \(-7x^3 - z^3\) \(0.169029\) \(\infty\)

2-torsion field: 6.2.2357312.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.027992 \)
Real period: \( 18.92576 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.059548 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(3\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + T + 3 T^{2} )\)
\(36833\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 278 T + 36833 T^{2} )\)

Galois representations

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

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);