Properties

Label 1146.a.2292.1
Conductor $1146$
Discriminant $2292$
Mordell-Weil group \(\Z/{6}\Z\)
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

Related objects

Downloads

Learn more

Show commands: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

$y^2 + xy = x^5 + 3x^4 + 5x^3 + 4x^2 + 2x$ (homogenize, simplify)
$y^2 + xz^2y = x^5z + 3x^4z^2 + 5x^3z^3 + 4x^2z^4 + 2xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 + 12x^4 + 20x^3 + 17x^2 + 8x$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 2, 4, 5, 3, 1]), R([0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 2, 4, 5, 3, 1], R![0, 1]);
 
sage: X = HyperellipticCurve(R([0, 8, 17, 20, 12, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1146\) \(=\) \( 2 \cdot 3 \cdot 191 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(2292\) \(=\) \( 2^{2} \cdot 3 \cdot 191 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(104\) \(=\)  \( 2^{3} \cdot 13 \)
\( I_4 \)  \(=\) \(1096\) \(=\)  \( 2^{3} \cdot 137 \)
\( I_6 \)  \(=\) \(61011\) \(=\)  \( 3^{2} \cdot 6779 \)
\( I_{10} \)  \(=\) \(9168\) \(=\)  \( 2^{4} \cdot 3 \cdot 191 \)
\( J_2 \)  \(=\) \(52\) \(=\)  \( 2^{2} \cdot 13 \)
\( J_4 \)  \(=\) \(-70\) \(=\)  \( - 2 \cdot 5 \cdot 7 \)
\( J_6 \)  \(=\) \(-3815\) \(=\)  \( - 5 \cdot 7 \cdot 109 \)
\( J_8 \)  \(=\) \(-50820\) \(=\)  \( - 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2} \)
\( J_{10} \)  \(=\) \(2292\) \(=\)  \( 2^{2} \cdot 3 \cdot 191 \)
\( g_1 \)  \(=\) \(95051008/573\)
\( g_2 \)  \(=\) \(-2460640/573\)
\( g_3 \)  \(=\) \(-2578940/573\)

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: \((1 : 0 : 0),\, (0 : 0 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1)\)

magma: [C![0,0,1],C![1,0,0]]; // minimal model
 
magma: [C![0,0,1],C![1,0,0]]; // simplified model
 

Number of rational Weierstrass points: \(2\)

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: \(\Z/{6}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + 2xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(6\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + 2xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(6\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + 2xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(6\)

2-torsion field: 4.0.2292.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 9.256013 \)
Tamagawa product: \( 2 \)
Torsion order:\( 6 \)
Leading coefficient: \( 0.514222 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(2\) \(2\) \(( 1 - T )( 1 + 2 T^{2} )\)
\(3\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 2 T + 3 T^{2} )\)
\(191\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 191 T^{2} )\)

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?
\(2\) 2.30.3 yes
\(3\) 3.80.1 yes

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);