Properties

Label 24704.a.790528.1
Conductor 24704
Discriminant -790528
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

Related objects

Learn more about

Show commands for: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

$y^2 + x^3y = 6x^3 + 14x^2 + 12x + 4$ (homogenize, simplify)
$y^2 + x^3y = 6x^3z^3 + 14x^2z^4 + 12xz^5 + 4z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 24x^3 + 56x^2 + 48x + 16$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![4, 12, 14, 6], R![0, 0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([4, 12, 14, 6]), R([0, 0, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([16, 48, 56, 24, 0, 0, 1]))
 

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

\( I_2 \)  = \(-384\) =  \( - 2^{7} \cdot 3 \)
\( I_4 \)  = \(218112\) =  \( 2^{10} \cdot 3 \cdot 71 \)
\( I_6 \)  = \(-42369024\) =  \( - 2^{15} \cdot 3 \cdot 431 \)
\( I_{10} \)  = \(-3238002688\) =  \( - 2^{24} \cdot 193 \)
\( J_2 \)  = \(-48\) =  \( - 2^{4} \cdot 3 \)
\( J_4 \)  = \(-2176\) =  \( - 2^{7} \cdot 17 \)
\( J_6 \)  = \(43008\) =  \( 2^{11} \cdot 3 \cdot 7 \)
\( J_8 \)  = \(-1699840\) =  \( - 2^{12} \cdot 5 \cdot 83 \)
\( J_{10} \)  = \(-790528\) =  \( - 2^{12} \cdot 193 \)
\( g_1 \)  = \(62208/193\)
\( g_2 \)  = \(-58752/193\)
\( g_3 \)  = \(-24192/193\)

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

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((-1 : 0 : 1)\) \((-1 : 1 : 1)\) \((0 : -2 : 1)\) \((0 : 2 : 1)\)
\((-2 : 2 : 1)\) \((-2 : 6 : 1)\) \((-3 : 11 : 2)\) \((-3 : 16 : 2)\) \((-5 : 44 : 3)\) \((-5 : 81 : 3)\)
\((8 : 2058 : 7)\) \((8 : -2570 : 7)\)

magma: [C![-5,44,3],C![-5,81,3],C![-3,11,2],C![-3,16,2],C![-2,2,1],C![-2,6,1],C![-1,0,1],C![-1,1,1],C![0,-2,1],C![0,2,1],C![1,-1,0],C![1,0,0],C![8,-2570,7],C![8,2058,7]];
 

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: \(\Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.144322\) \(\infty\)
\((-2 : 6 : 1) - (1 : 0 : 0)\) \(z (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - 2z^3\) \(0.051963\) \(\infty\)

2-torsion field: 6.0.49408.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.007471 \)
Real period: \( 15.42238 \)
Tamagawa product: \( 8 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.921871 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(12\) \(7\) \(8\) \(1\)
\(193\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 17 T + 193 T^{2} )\)

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 \(\Q\).