Properties

Label 1142.b.9136.1
Conductor $1142$
Discriminant $-9136$
Mordell-Weil group \(\Z/{12}\Z\)
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

Learn more

Show commands: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x + 1)y = -x^5 + 3x^4 - 6x^2 + x + 3$ (homogenize, simplify)
$y^2 + (xz^2 + z^3)y = -x^5z + 3x^4z^2 - 6x^2z^4 + xz^5 + 3z^6$ (dehomogenize, simplify)
$y^2 = -4x^5 + 12x^4 - 23x^2 + 6x + 13$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([3, 1, -6, 0, 3, -1]), R([1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![3, 1, -6, 0, 3, -1], R![1, 1]);
 
sage: X = HyperellipticCurve(R([13, 6, -23, 0, 12, -4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1142\) \(=\) \( 2 \cdot 571 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-9136\) \(=\) \( - 2^{4} \cdot 571 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(864\) \(=\)  \( 2^{5} \cdot 3^{3} \)
\( I_4 \)  \(=\) \(-4488\) \(=\)  \( - 2^{3} \cdot 3 \cdot 11 \cdot 17 \)
\( I_6 \)  \(=\) \(-1442025\) \(=\)  \( - 3^{2} \cdot 5^{2} \cdot 13 \cdot 17 \cdot 29 \)
\( I_{10} \)  \(=\) \(-36544\) \(=\)  \( - 2^{6} \cdot 571 \)
\( J_2 \)  \(=\) \(432\) \(=\)  \( 2^{4} \cdot 3^{3} \)
\( J_4 \)  \(=\) \(8524\) \(=\)  \( 2^{2} \cdot 2131 \)
\( J_6 \)  \(=\) \(257089\) \(=\)  \( 7 \cdot 19 \cdot 1933 \)
\( J_8 \)  \(=\) \(9600968\) \(=\)  \( 2^{3} \cdot 13 \cdot 92317 \)
\( J_{10} \)  \(=\) \(-9136\) \(=\)  \( - 2^{4} \cdot 571 \)
\( g_1 \)  \(=\) \(-940369969152/571\)
\( g_2 \)  \(=\) \(-42951140352/571\)
\( g_3 \)  \(=\) \(-2998686096/571\)

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: \((1 : 0 : 0),\, (-1 : 0 : 1),\, (1 : 0 : 1),\, (1 : -2 : 1)\)
All points: \((1 : 0 : 0),\, (-1 : 0 : 1),\, (1 : 0 : 1),\, (1 : -2 : 1)\)
All points: \((1 : 0 : 0),\, (-1 : 0 : 1),\, (1 : -2 : 1),\, (1 : 2 : 1)\)

magma: [C![-1,0,1],C![1,-2,1],C![1,0,0],C![1,0,1]]; // minimal model
 
magma: [C![-1,0,1],C![1,-2,1],C![1,0,0],C![1,2,1]]; // 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: \(\Z/{12}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -2 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0\) \(12\)
Generator $D_0$ Height Order
\((1 : -2 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0\) \(12\)
Generator $D_0$ Height Order
\((1 : -2 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - 3z^3\) \(0\) \(12\)

2-torsion field: 4.2.2284.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 17.75728 \)
Tamagawa product: \( 4 \)
Torsion order:\( 12 \)
Leading coefficient: \( 0.493257 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(4\) \(4\) \(( 1 - T )( 1 + 2 T^{2} )\)
\(571\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 4 T + 571 T^{2} )\)

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 \(\Q\).