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Properties

Label 11664.e.314928.1
Conductor $11664$
Discriminant $-314928$
Mordell-Weil group \(\Z \oplus \Z/{2}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + x^2y = x^5 - 5x^3 - 7x^2 - 4x - 1$ (homogenize, simplify)
$y^2 + x^2zy = x^5z - 5x^3z^3 - 7x^2z^4 - 4xz^5 - z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 + x^4 - 20x^3 - 28x^2 - 16x - 4$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, -4, -7, -5, 0, 1]), R([0, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, -4, -7, -5, 0, 1], R![0, 0, 1]);
 
Copy content sage:X = HyperellipticCurve(R([-4, -16, -28, -20, 1, 4]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(11664\) \(=\) \( 2^{4} \cdot 3^{6} \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-314928\) \(=\) \( - 2^{4} \cdot 3^{9} \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(24\) \(=\)  \( 2^{3} \cdot 3 \)
\( I_4 \)  \(=\) \(-72\) \(=\)  \( - 2^{3} \cdot 3^{2} \)
\( I_6 \)  \(=\) \(-7236\) \(=\)  \( - 2^{2} \cdot 3^{3} \cdot 67 \)
\( I_{10} \)  \(=\) \(-5184\) \(=\)  \( - 2^{6} \cdot 3^{4} \)
\( J_2 \)  \(=\) \(36\) \(=\)  \( 2^{2} \cdot 3^{2} \)
\( J_4 \)  \(=\) \(162\) \(=\)  \( 2 \cdot 3^{4} \)
\( J_6 \)  \(=\) \(20736\) \(=\)  \( 2^{8} \cdot 3^{4} \)
\( J_8 \)  \(=\) \(180063\) \(=\)  \( 3^{6} \cdot 13 \cdot 19 \)
\( J_{10} \)  \(=\) \(-314928\) \(=\)  \( - 2^{4} \cdot 3^{9} \)
\( g_1 \)  \(=\) \(-192\)
\( g_2 \)  \(=\) \(-24\)
\( g_3 \)  \(=\) \(-256/3\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

Copy content sage:C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
Copy content magma:IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
Copy content magma:AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2$
Copy content magma:AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

All points: \((1 : 0 : 0),\, (-1 : 0 : 1),\, (-1 : -1 : 1)\)
All points: \((1 : 0 : 0),\, (-1 : 0 : 1),\, (-1 : -1 : 1)\)
All points: \((1 : 0 : 0),\, (-1 : -1 : 1),\, (-1 : 1 : 1)\)

Copy content magma:[C![-1,-1,1],C![-1,0,1],C![1,0,0]]; // minimal model
 
Copy content magma:[C![-1,-1,1],C![-1,1,1],C![1,0,0]]; // simplified model
 

Number of rational Weierstrass points: \(1\)

Copy content magma:#Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

This curve is locally solvable everywhere.

Copy content 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: \(\Z \oplus \Z/{2}\Z\)

Copy content magma:MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : -1 : 1) - (1 : 0 : 0)\) \(x + z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.213510\) \(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 2xz - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\((-1 : -1 : 1) - (1 : 0 : 0)\) \(x + z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.213510\) \(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 2xz - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\((-1 : -1 : 1) - (1 : 0 : 0)\) \(x + z\) \(=\) \(0,\) \(y\) \(=\) \(x^2z - 2z^3\) \(0.213510\) \(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 2xz - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^2z - 2xz^2 - 2z^3\) \(0\) \(2\)

2-torsion field: 6.0.314928.2

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.213510 \)
Real period: \( 7.338972 \)
Tamagawa product: \( 2 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.783474 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa Root number* L-factor Cluster picture Tame reduction?
\(2\) \(4\) \(4\) \(1\) \(1^*\) \(1 + T + 2 T^{2}\) no
\(3\) \(6\) \(9\) \(2\) \(-1^*\) \(1\) no

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.60.1 yes

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Copy content magma:HeuristicDecompositionFactors(C);
 

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 \(\Q\).

Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

Copy content magma:HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

Copy content magma:HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);