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Properties

Label 4356.b.470448.1
Conductor $4356$
Discriminant $470448$
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + x^3y = -x^4 - 6x^3 + 22x^2 - 24x + 9$ (homogenize, simplify)
$y^2 + x^3y = -x^4z^2 - 6x^3z^3 + 22x^2z^4 - 24xz^5 + 9z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 4x^4 - 24x^3 + 88x^2 - 96x + 36$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([9, -24, 22, -6, -1]), R([0, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![9, -24, 22, -6, -1], R![0, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([36, -96, 88, -24, -4, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(4356\) \(=\) \( 2^{2} \cdot 3^{2} \cdot 11^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(470448\) \(=\) \( 2^{4} \cdot 3^{5} \cdot 11^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(56\) \(=\)  \( 2^{3} \cdot 7 \)
\( I_4 \)  \(=\) \(4741\) \(=\)  \( 11 \cdot 431 \)
\( I_6 \)  \(=\) \(-21505\) \(=\)  \( - 5 \cdot 11 \cdot 17 \cdot 23 \)
\( I_{10} \)  \(=\) \(58806\) \(=\)  \( 2 \cdot 3^{5} \cdot 11^{2} \)
\( J_2 \)  \(=\) \(56\) \(=\)  \( 2^{3} \cdot 7 \)
\( J_4 \)  \(=\) \(-3030\) \(=\)  \( - 2 \cdot 3 \cdot 5 \cdot 101 \)
\( J_6 \)  \(=\) \(68688\) \(=\)  \( 2^{4} \cdot 3^{4} \cdot 53 \)
\( J_8 \)  \(=\) \(-1333593\) \(=\)  \( - 3^{2} \cdot 71 \cdot 2087 \)
\( J_{10} \)  \(=\) \(470448\) \(=\)  \( 2^{4} \cdot 3^{5} \cdot 11^{2} \)
\( g_1 \)  \(=\) \(34420736/29403\)
\( g_2 \)  \(=\) \(-11085760/9801\)
\( g_3 \)  \(=\) \(166208/363\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((1 : 0 : 1)\) \((1 : -1 : 1)\) \((0 : -3 : 1)\) \((0 : 3 : 1)\)
\((2 : -3 : 1)\) \((2 : -5 : 1)\) \((5 : -8 : 1)\) \((3 : -12 : 2)\) \((3 : -15 : 2)\) \((5 : -117 : 1)\)
All points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((1 : 0 : 1)\) \((1 : -1 : 1)\) \((0 : -3 : 1)\) \((0 : 3 : 1)\)
\((2 : -3 : 1)\) \((2 : -5 : 1)\) \((5 : -8 : 1)\) \((3 : -12 : 2)\) \((3 : -15 : 2)\) \((5 : -117 : 1)\)
All points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((1 : -1 : 1)\) \((1 : 1 : 1)\) \((2 : -2 : 1)\) \((2 : 2 : 1)\)
\((3 : -3 : 2)\) \((3 : 3 : 2)\) \((0 : -6 : 1)\) \((0 : 6 : 1)\) \((5 : -109 : 1)\) \((5 : 109 : 1)\)

magma: [C![0,-3,1],C![0,3,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![1,0,1],C![2,-5,1],C![2,-3,1],C![3,-15,2],C![3,-12,2],C![5,-117,1],C![5,-8,1]]; // minimal model
 
magma: [C![0,-6,1],C![0,6,1],C![1,-1,0],C![1,-1,1],C![1,1,0],C![1,1,1],C![2,-2,1],C![2,2,1],C![3,-3,2],C![3,3,2],C![5,-109,1],C![5,109,1]]; // 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
\((0 : -3 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(2xz^2 - 3z^3\) \(0.001196\) \(\infty\)
Generator $D_0$ Height Order
\((0 : -3 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(2xz^2 - 3z^3\) \(0.001196\) \(\infty\)
Generator $D_0$ Height Order
\((0 : -6 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + 4xz^2 - 6z^3\) \(0.001196\) \(\infty\)

2-torsion field: 6.2.836352.2

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.001196 \)
Real period: \( 17.38106 \)
Tamagawa product: \( 21 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.436802 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(4\) \(3\) \(1 + 2 T + 2 T^{2}\)
\(3\) \(2\) \(5\) \(7\) \(( 1 - T )( 1 + T )\)
\(11\) \(2\) \(2\) \(1\) \(1 + 4 T + 11 T^{2}\)

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?
\(2\) 2.20.3 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);