Properties

Label 1146.a.2292.1
Conductor 1146
Discriminant 2292
Mordell-Weil group \(\Z/{6}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

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

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

Invariants

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

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

G2 invariants

magma: G2Invariants(C);
 

\( I_2 \)  = \(416\) =  \( 2^{5} \cdot 13 \)
\( I_4 \)  = \(17536\) =  \( 2^{7} \cdot 137 \)
\( I_6 \)  = \(3904704\) =  \( 2^{6} \cdot 3^{2} \cdot 6779 \)
\( I_{10} \)  = \(9388032\) =  \( 2^{14} \cdot 3 \cdot 191 \)
\( J_2 \)  = \(52\) =  \( 2^{2} \cdot 13 \)
\( J_4 \)  = \(-70\) =  \( - 2 \cdot 5 \cdot 7 \)
\( J_6 \)  = \(-3815\) =  \( - 5 \cdot 7 \cdot 109 \)
\( J_8 \)  = \(-50820\) =  \( - 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2} \)
\( J_{10} \)  = \(2292\) =  \( 2^{2} \cdot 3 \cdot 191 \)
\( g_1 \)  = \(95051008/573\)
\( g_2 \)  = \(-2460640/573\)
\( g_3 \)  = \(-2578940/573\)

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2$

Rational points

magma: [C![0,0,1],C![1,0,0]];
 

Points: \((0 : 0 : 1),\, (1 : 0 : 0)\)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

Number of rational Weierstrass points: \(2\)

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);
 

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

Generator Height Order
\(x^2 + 2xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(6\)

2-torsion field: 4.0.2292.1

BSD invariants

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
\(2\) \(2\) \(1\) \(2\) \(( 1 - T )( 1 + 2 T^{2} )\)
\(3\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 2 T + 3 T^{2} )\)
\(191\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 191 T^{2} )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

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 \(\Q\).