Properties

Label 2457.a.95823.1
Conductor $2457$
Discriminant $-95823$
Mordell-Weil group \(\Z \times \Z/{2}\Z \times \Z/{2}\Z\)
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

$y^2 + (x^3 + 1)y = x^5 - x^4 - 3x^3 + 6x^2 - 6x + 2$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = x^5z - x^4z^2 - 3x^3z^3 + 6x^2z^4 - 6xz^5 + 2z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 - 4x^4 - 10x^3 + 24x^2 - 24x + 9$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, -6, 6, -3, -1, 1]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, -6, 6, -3, -1, 1], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([9, -24, 24, -10, -4, 4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(2457\) \(=\) \( 3^{3} \cdot 7 \cdot 13 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-95823\) \(=\) \( - 3^{4} \cdot 7 \cdot 13^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1932\) \(=\)  \( 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
\( I_4 \)  \(=\) \(57897\) \(=\)  \( 3^{2} \cdot 7 \cdot 919 \)
\( I_6 \)  \(=\) \(45198315\) \(=\)  \( 3^{2} \cdot 5 \cdot 73 \cdot 13759 \)
\( I_{10} \)  \(=\) \(12265344\) \(=\)  \( 2^{7} \cdot 3^{4} \cdot 7 \cdot 13^{2} \)
\( J_2 \)  \(=\) \(483\) \(=\)  \( 3 \cdot 7 \cdot 23 \)
\( J_4 \)  \(=\) \(7308\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 7 \cdot 29 \)
\( J_6 \)  \(=\) \(-43264\) \(=\)  \( - 2^{8} \cdot 13^{2} \)
\( J_8 \)  \(=\) \(-18575844\) \(=\)  \( - 2^{2} \cdot 3 \cdot 7 \cdot 19 \cdot 103 \cdot 113 \)
\( J_{10} \)  \(=\) \(95823\) \(=\)  \( 3^{4} \cdot 7 \cdot 13^{2} \)
\( g_1 \)  \(=\) \(46360978629/169\)
\( g_2 \)  \(=\) \(1452301788/169\)
\( g_3 \)  \(=\) \(-947968/9\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 1 : 1)\) \((1 : -1 : 1)\) \((0 : -2 : 1)\) \((-3 : 13 : 1)\)
\((4 : -13 : 3)\) \((4 : -78 : 3)\)

magma: [C![-3,13,1],C![0,-2,1],C![0,1,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![4,-78,3],C![4,-13,3]];
 

Number of rational Weierstrass points: \(2\)

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: \(\Z \times \Z/{2}\Z \times \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : -2 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - 2z^3\) \(0.066494\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + 3xz - 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(-6xz^2 + 4z^3\) \(0\) \(2\)
\((-3 : 13 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + 3z)\) \(=\) \(0,\) \(2y\) \(=\) \(-7xz^2 + 5z^3\) \(0\) \(2\)

2-torsion field: \(\Q(\sqrt{-3}, \sqrt{-7})\)

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(3\)
Regulator: \( 0.066494 \)
Real period: \( 18.98310 \)
Tamagawa product: \( 4 \)
Torsion order:\( 4 \)
Leading coefficient: \( 0.315566 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(3\) \(3\) \(4\) \(2\) \(1 + T\)
\(7\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 4 T + 7 T^{2} )\)
\(13\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + 6 T + 13 T^{2} )\)

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 \(\Q\).