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Properties

Label 24704.b.790528.1
Conductor $24704$
Discriminant $-790528$
Mordell-Weil group \(\Z\)
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

$y^2 + x^3y = x^5 + 3x^4 + 4x^3 + 6x^2 + 4x + 4$ (homogenize, simplify)
$y^2 + x^3y = x^5z + 3x^4z^2 + 4x^3z^3 + 6x^2z^4 + 4xz^5 + 4z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 + 12x^4 + 16x^3 + 24x^2 + 16x + 16$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([4, 4, 6, 4, 3, 1]), R([0, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![4, 4, 6, 4, 3, 1], R![0, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([16, 16, 24, 16, 12, 4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(24704\) \(=\) \( 2^{7} \cdot 193 \)
magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(24704,2),R![1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-790528\) \(=\) \( - 2^{12} \cdot 193 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(272\) \(=\)  \( 2^{4} \cdot 17 \)
\( I_4 \)  \(=\) \(340\) \(=\)  \( 2^{2} \cdot 5 \cdot 17 \)
\( I_6 \)  \(=\) \(25304\) \(=\)  \( 2^{3} \cdot 3163 \)
\( I_{10} \)  \(=\) \(3088\) \(=\)  \( 2^{4} \cdot 193 \)
\( J_2 \)  \(=\) \(544\) \(=\)  \( 2^{5} \cdot 17 \)
\( J_4 \)  \(=\) \(11424\) \(=\)  \( 2^{5} \cdot 3 \cdot 7 \cdot 17 \)
\( J_6 \)  \(=\) \(329728\) \(=\)  \( 2^{11} \cdot 7 \cdot 23 \)
\( J_8 \)  \(=\) \(12216064\) \(=\)  \( 2^{8} \cdot 7 \cdot 17 \cdot 401 \)
\( J_{10} \)  \(=\) \(790528\) \(=\)  \( 2^{12} \cdot 193 \)
\( g_1 \)  \(=\) \(11631468544/193\)
\( g_2 \)  \(=\) \(449008896/193\)
\( g_3 \)  \(=\) \(23822848/193\)

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

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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (0 : -2 : 1),\, (0 : 2 : 1)\)
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (0 : -2 : 1),\, (0 : 2 : 1)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (0 : -4 : 1),\, (0 : 4 : 1)\)

magma: [C![0,-2,1],C![0,2,1],C![1,-1,0],C![1,0,0]]; // minimal model
 
magma: [C![0,-4,1],C![0,4,1],C![1,-1,0],C![1,1,0]]; // simplified model
 

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: \(\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.020871\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.020871\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 + xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 4z^3\) \(0.020871\) \(\infty\)

2-torsion field: 6.0.49408.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.020871 \)
Real period: \( 8.575631 \)
Tamagawa product: \( 8 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.431915 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(7\) \(12\) \(8\) \(1\)
\(193\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 15 T + 193 T^{2} )\)

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?
\(2\) 2.10.1 no

Sato-Tate group

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

Decomposition of the Jacobian

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

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 \(\Q\).

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

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

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