Properties

Label 1462.a.11696.1
Conductor 1462
Discriminant -11696
Mordell-Weil group \(\Z/{10}\Z\)
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

$y^2 + (x^3 + x)y = 2x^5 - 27x^3 - 38x^2 + 94x + 148$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = 2x^5z - 27x^3z^3 - 38x^2z^4 + 94xz^5 + 148z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 8x^5 + 2x^4 - 108x^3 - 151x^2 + 376x + 592$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([148, 94, -38, -27, 0, 2]), R([0, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![148, 94, -38, -27, 0, 2], R![0, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([592, 376, -151, -108, 2, 8, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1462\) \(=\) \( 2 \cdot 17 \cdot 43 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-11696\) \(=\) \( - 2^{4} \cdot 17 \cdot 43 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(53056\) \(=\)  \( 2^{6} \cdot 829 \)
\( I_4 \)  \(=\) \(-8307776\) \(=\)  \( - 2^{6} \cdot 271 \cdot 479 \)
\( I_6 \)  \(=\) \(-139339433536\) \(=\)  \( - 2^{6} \cdot 331 \cdot 6577579 \)
\( I_{10} \)  \(=\) \(-47906816\) \(=\)  \( - 2^{16} \cdot 17 \cdot 43 \)
\( J_2 \)  \(=\) \(6632\) \(=\)  \( 2^{3} \cdot 829 \)
\( J_4 \)  \(=\) \(1919182\) \(=\)  \( 2 \cdot 61 \cdot 15731 \)
\( J_6 \)  \(=\) \(757711065\) \(=\)  \( 3 \cdot 5 \cdot 4639 \cdot 10889 \)
\( J_8 \)  \(=\) \(335470058489\) \(=\)  \( 419 \cdot 800644531 \)
\( J_{10} \)  \(=\) \(-11696\) \(=\)  \( - 2^{4} \cdot 17 \cdot 43 \)
\( g_1 \)  \(=\) \(-801867487713585152/731\)
\( g_2 \)  \(=\) \(-34988855092435136/731\)
\( g_3 \)  \(=\) \(-2082920440086660/731\)

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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-4 : 34 : 1)\)

magma: [C![-4,34,1],C![1,-1,0],C![1,0,0]];
 

Number of rational Weierstrass points: \(1\)

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: \(\Z/{10}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + 2xz - 12z^2\) \(=\) \(0,\) \(y\) \(=\) \(-8xz^2 + 10z^3\) \(0\) \(10\)

2-torsion field: 6.2.145346192.3

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 9.232590 \)
Tamagawa product: \( 4 \)
Torsion order:\( 10 \)
Leading coefficient: \( 0.369303 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(4\) \(4\) \(( 1 - T )( 1 + 2 T + 2 T^{2} )\)
\(17\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 7 T + 17 T^{2} )\)
\(43\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 4 T + 43 T^{2} )\)

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 \(\Q\).