Properties

Label 2952.a.283392.1
Conductor 2952
Discriminant -283392
Mordell-Weil group \(\Z \times \Z/{2}\Z\)
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

$y^2 + x^3y = -2x^4 - x^3 + 3x^2 + 4x + 4$ (homogenize, simplify)
$y^2 + x^3y = -2x^4z^2 - x^3z^3 + 3x^2z^4 + 4xz^5 + 4z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 8x^4 - 4x^3 + 12x^2 + 16x + 16$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([4, 4, 3, -1, -2]), R([0, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![4, 4, 3, -1, -2], R![0, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([16, 16, 12, -4, -8, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(2952\) \(=\) \( 2^{3} \cdot 3^{2} \cdot 41 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-283392\) \(=\) \( - 2^{8} \cdot 3^{3} \cdot 41 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(-2208\) \(=\)  \( - 2^{5} \cdot 3 \cdot 23 \)
\( I_4 \)  \(=\) \(571392\) \(=\)  \( 2^{11} \cdot 3^{2} \cdot 31 \)
\( I_6 \)  \(=\) \(-583446528\) \(=\)  \( - 2^{12} \cdot 3^{2} \cdot 7^{2} \cdot 17 \cdot 19 \)
\( I_{10} \)  \(=\) \(-1160773632\) \(=\)  \( - 2^{20} \cdot 3^{3} \cdot 41 \)
\( J_2 \)  \(=\) \(-276\) \(=\)  \( - 2^{2} \cdot 3 \cdot 23 \)
\( J_4 \)  \(=\) \(-2778\) \(=\)  \( - 2 \cdot 3 \cdot 463 \)
\( J_6 \)  \(=\) \(507940\) \(=\)  \( 2^{2} \cdot 5 \cdot 109 \cdot 233 \)
\( J_8 \)  \(=\) \(-36977181\) \(=\)  \( - 3 \cdot 149 \cdot 82723 \)
\( J_{10} \)  \(=\) \(-283392\) \(=\)  \( - 2^{8} \cdot 3^{3} \cdot 41 \)
\( g_1 \)  \(=\) \(231708348/41\)
\( g_2 \)  \(=\) \(-16899963/82\)
\( g_3 \)  \(=\) \(-67175065/492\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((-1 : -1 : 1)\) \((0 : -2 : 1)\) \((0 : 2 : 1)\) \((-1 : 2 : 1)\)
\((-2 : 4 : 1)\) \((2 : -4 : 1)\) \((5 : -61 : 2)\) \((5 : -64 : 2)\)

magma: [C![-2,4,1],C![-1,-1,1],C![-1,2,1],C![0,-2,1],C![0,2,1],C![1,-1,0],C![1,0,0],C![2,-4,1],C![5,-64,2],C![5,-61,2]];
 

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - xz - 4z^2\) \(=\) \(0,\) \(y\) \(=\) \(-3xz^2 - 2z^3\) \(0.009020\) \(\infty\)
\((-2 : 4 : 1) + (2 : -4 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - 2z) (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2\) \(0\) \(2\)

2-torsion field: 4.2.1968.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.009020 \)
Real period: \( 18.18041 \)
Tamagawa product: \( 8 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.328006 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(3\) \(8\) \(4\) \(1 + 2 T + 2 T^{2}\)
\(3\) \(2\) \(3\) \(2\) \(1 + 2 T + 3 T^{2}\)
\(41\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 8 T + 41 T^{2} )\)

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 \(\Q\).