Properties

Label 116384.a.232768.1
Conductor 116384
Discriminant 232768
Mordell-Weil group \(\Z \times \Z\)
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

$y^2 + y = 2x^5 + 2x^4 - 3x^3$ (homogenize, simplify)
$y^2 + z^3y = 2x^5z + 2x^4z^2 - 3x^3z^3$ (dehomogenize, simplify)
$y^2 = 8x^5 + 8x^4 - 12x^3 + 1$ (minimize, homogenize)

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

Invariants

Conductor: \( N \)  \(=\)  \(116384\) \(=\) \( 2^{5} \cdot 3637 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(232768\) \(=\) \( 2^{6} \cdot 3637 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(108\) \(=\)  \( 2^{2} \cdot 3^{3} \)
\( I_4 \)  \(=\) \(2544\) \(=\)  \( 2^{4} \cdot 3 \cdot 53 \)
\( I_6 \)  \(=\) \(17208\) \(=\)  \( 2^{3} \cdot 3^{2} \cdot 239 \)
\( I_{10} \)  \(=\) \(29096\) \(=\)  \( 2^{3} \cdot 3637 \)
\( J_2 \)  \(=\) \(108\) \(=\)  \( 2^{2} \cdot 3^{3} \)
\( J_4 \)  \(=\) \(-1210\) \(=\)  \( - 2 \cdot 5 \cdot 11^{2} \)
\( J_6 \)  \(=\) \(38500\) \(=\)  \( 2^{2} \cdot 5^{3} \cdot 7 \cdot 11 \)
\( J_8 \)  \(=\) \(673475\) \(=\)  \( 5^{2} \cdot 11 \cdot 31 \cdot 79 \)
\( J_{10} \)  \(=\) \(232768\) \(=\)  \( 2^{6} \cdot 3637 \)
\( g_1 \)  \(=\) \(229582512/3637\)
\( g_2 \)  \(=\) \(-23816430/3637\)
\( g_3 \)  \(=\) \(7016625/3637\)

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

Known points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -2 : 2),\, (1 : -6 : 2),\, (2 : 8 : 1),\, (2 : -9 : 1)\)

magma: [C![0,-1,1],C![0,0,1],C![1,-6,2],C![1,-2,2],C![1,0,0],C![2,-9,1],C![2,8,1]];
 

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: \(\Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0.411792\) \(\infty\)
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.144670\) \(\infty\)

2-torsion field: 5.1.58192.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.057517 \)
Real period: \( 17.96507 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 2.066615 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(5\) \(6\) \(2\) \(1 + 2 T^{2}\)
\(3637\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 22 T + 3637 T^{2} )\)

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 \(\Q\).