Properties

Label 6336.a.152064.1
Conductor 6336
Discriminant 152064
Mordell-Weil group \(\Z \times \Z/{3}\Z\)
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

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x)y = -x^4 - 2x^3 + 2x + 1$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = -x^4z^2 - 2x^3z^3 + 2xz^5 + z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 2x^4 - 8x^3 + x^2 + 8x + 4$ (minimize, homogenize)

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

Invariants

Conductor: \( N \)  \(=\)  \(6336\) \(=\) \( 2^{6} \cdot 3^{2} \cdot 11 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(152064\) \(=\) \( 2^{9} \cdot 3^{3} \cdot 11 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(-544\) \(=\)  \( - 2^{5} \cdot 17 \)
\( I_4 \)  \(=\) \(87616\) \(=\)  \( 2^{6} \cdot 37^{2} \)
\( I_6 \)  \(=\) \(-17067520\) \(=\)  \( - 2^{9} \cdot 5 \cdot 59 \cdot 113 \)
\( I_{10} \)  \(=\) \(622854144\) \(=\)  \( 2^{21} \cdot 3^{3} \cdot 11 \)
\( J_2 \)  \(=\) \(-68\) \(=\)  \( - 2^{2} \cdot 17 \)
\( J_4 \)  \(=\) \(-720\) \(=\)  \( - 2^{4} \cdot 3^{2} \cdot 5 \)
\( J_6 \)  \(=\) \(11664\) \(=\)  \( 2^{4} \cdot 3^{6} \)
\( J_8 \)  \(=\) \(-327888\) \(=\)  \( - 2^{4} \cdot 3^{4} \cdot 11 \cdot 23 \)
\( J_{10} \)  \(=\) \(152064\) \(=\)  \( 2^{9} \cdot 3^{3} \cdot 11 \)
\( g_1 \)  \(=\) \(-2839714/297\)
\( g_2 \)  \(=\) \(49130/33\)
\( g_3 \)  \(=\) \(7803/22\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((1 : 0 : 1)\)
\((-1 : 2 : 1)\) \((1 : -2 : 1)\) \((-13 : 64 : 3)\) \((-13 : 2250 : 3)\)

magma: [C![-13,64,3],C![-13,2250,3],C![-1,0,1],C![-1,2,1],C![0,-1,1],C![0,1,1],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1]];
 

Number of rational Weierstrass points: \(0\)

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: \(\Z \times \Z/{3}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.027391\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(3\)

2-torsion field: 6.2.608256.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.027391 \)
Real period: \( 16.86600 \)
Tamagawa product: \( 12 \)
Torsion order:\( 3 \)
Leading coefficient: \( 0.615971 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(6\) \(9\) \(4\) \(1 + T\)
\(3\) \(2\) \(3\) \(3\) \(1 + T + T^{2}\)
\(11\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 3 T + 11 T^{2} )\)

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 \(\Q\).