Properties

Label 3622.a.463616.1
Conductor $3622$
Discriminant $-463616$
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

$y^2 + (x^3 + 1)y = -x^5 + x^3 + 3x^2 - 2x - 1$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = -x^5z + x^3z^3 + 3x^2z^4 - 2xz^5 - z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 4x^5 + 6x^3 + 12x^2 - 8x - 3$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, -2, 3, 1, 0, -1]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, -2, 3, 1, 0, -1], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([-3, -8, 12, 6, 0, -4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(3622\) \(=\) \( 2 \cdot 1811 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-463616\) \(=\) \( - 2^{8} \cdot 1811 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1108\) \(=\)  \( 2^{2} \cdot 277 \)
\( I_4 \)  \(=\) \(11569\) \(=\)  \( 23 \cdot 503 \)
\( I_6 \)  \(=\) \(-1775207\) \(=\)  \( - 7 \cdot 253601 \)
\( I_{10} \)  \(=\) \(-59342848\) \(=\)  \( - 2^{15} \cdot 1811 \)
\( J_2 \)  \(=\) \(277\) \(=\)  \( 277 \)
\( J_4 \)  \(=\) \(2715\) \(=\)  \( 3 \cdot 5 \cdot 181 \)
\( J_6 \)  \(=\) \(110945\) \(=\)  \( 5 \cdot 22189 \)
\( J_8 \)  \(=\) \(5840135\) \(=\)  \( 5 \cdot 7 \cdot 166861 \)
\( J_{10} \)  \(=\) \(-463616\) \(=\)  \( - 2^{8} \cdot 1811 \)
\( g_1 \)  \(=\) \(-1630793025157/463616\)
\( g_2 \)  \(=\) \(-57704428095/463616\)
\( g_3 \)  \(=\) \(-8512698905/463616\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((1 : 0 : 1)\) \((-1 : -2 : 1)\) \((-1 : 2 : 1)\) \((1 : -2 : 1)\)
\((3 : -14 : 1)\) \((7 : -156 : 3)\) \((7 : -214 : 3)\)
All points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((1 : 0 : 1)\) \((-1 : -2 : 1)\) \((-1 : 2 : 1)\) \((1 : -2 : 1)\)
\((3 : -14 : 1)\) \((7 : -156 : 3)\) \((7 : -214 : 3)\)
All points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((1 : -2 : 1)\) \((1 : 2 : 1)\) \((3 : 0 : 1)\) \((-1 : -4 : 1)\)
\((-1 : 4 : 1)\) \((7 : -58 : 3)\) \((7 : 58 : 3)\)

magma: [C![-1,-2,1],C![-1,2,1],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![3,-14,1],C![7,-214,3],C![7,-156,3]]; // minimal model
 
magma: [C![-1,-4,1],C![-1,4,1],C![1,-2,1],C![1,-1,0],C![1,1,0],C![1,2,1],C![3,0,1],C![7,-58,3],C![7,58,3]]; // 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: \(\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : -2 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - z^3\) \(0.004244\) \(\infty\)
Generator $D_0$ Height Order
\((-1 : -2 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - z^3\) \(0.004244\) \(\infty\)
Generator $D_0$ Height Order
\((-1 : -4 : 1) + (1 : 2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + 2xz^2 - z^3\) \(0.004244\) \(\infty\)

2-torsion field: 5.3.28976.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.004244 \)
Real period: \( 14.98353 \)
Tamagawa product: \( 8 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.508828 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(8\) \(8\) \(( 1 - T )( 1 + T + 2 T^{2} )\)
\(1811\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 1811 T^{2} )\)

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);