Properties

Label 966.a.834624.1
Conductor 966
Discriminant 834624
Mordell-Weil group \(\Z/{2}\Z \times \Z/{12}\Z\)
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

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -1, 1, 1, -1, 1]), R([0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, -1, 1, 1, -1, 1], R![0, 1, 1]);
 
sage: X = HyperellipticCurve(R([4, -4, 5, 6, -3, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  =  \(966\) = \( 2 \cdot 3 \cdot 7 \cdot 23 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(834624\) = \( 2^{6} \cdot 3^{4} \cdot 7 \cdot 23 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-184\) =  \( - 2^{3} \cdot 23 \)
\( I_4 \)  = \(98692\) =  \( 2^{2} \cdot 11 \cdot 2243 \)
\( I_6 \)  = \(4458120\) =  \( 2^{3} \cdot 3 \cdot 5 \cdot 97 \cdot 383 \)
\( I_{10} \)  = \(3418619904\) =  \( 2^{18} \cdot 3^{4} \cdot 7 \cdot 23 \)
\( J_2 \)  = \(-23\) =  \( - 23 \)
\( J_4 \)  = \(-1006\) =  \( - 2 \cdot 503 \)
\( J_6 \)  = \(-14336\) =  \( - 2^{11} \cdot 7 \)
\( J_8 \)  = \(-170577\) =  \( - 3^{2} \cdot 11 \cdot 1723 \)
\( J_{10} \)  = \(834624\) =  \( 2^{6} \cdot 3^{4} \cdot 7 \cdot 23 \)
\( g_1 \)  = \(-279841/36288\)
\( g_2 \)  = \(266087/18144\)
\( g_3 \)  = \(-736/81\)

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: \((1 : 0 : 0),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (2 : 3 : 1),\, (2 : -9 : 1)\)

magma: [C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,0,0],C![2,-9,1],C![2,3,1]];
 

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: \(\Z/{2}\Z \times \Z/{12}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: \(\Q(\sqrt{-7}, \sqrt{-23})\)

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 9.526770 \)
Tamagawa product: \( 24 \)
Torsion order:\( 24 \)
Leading coefficient: \( 0.396948 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(6\) \(1\) \(6\) \(( 1 - T )( 1 + T + 2 T^{2} )\)
\(3\) \(4\) \(1\) \(4\) \(( 1 - T )( 1 + 2 T + 3 T^{2} )\)
\(7\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 4 T + 7 T^{2} )\)
\(23\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 23 T^{2} )\)

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 \(\Q\).