Properties

Label 2457.a.154791.1
Conductor 2457
Discriminant -154791
Mordell-Weil group \(\Z \times \Z/{2}\Z \times \Z/{2}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

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Minimal equation

Minimal equation

Simplified equation

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

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

Invariants

Conductor: \( N \)  \(=\)  \(2457\) \(=\) \( 3^{3} \cdot 7 \cdot 13 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-154791\) \(=\) \( - 3^{5} \cdot 7^{2} \cdot 13 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(-1560\) \(=\)  \( - 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
\( I_4 \)  \(=\) \(254628\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 11 \cdot 643 \)
\( I_6 \)  \(=\) \(-187288920\) \(=\)  \( - 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \cdot 5717 \)
\( I_{10} \)  \(=\) \(-634023936\) \(=\)  \( - 2^{12} \cdot 3^{5} \cdot 7^{2} \cdot 13 \)
\( J_2 \)  \(=\) \(-195\) \(=\)  \( - 3 \cdot 5 \cdot 13 \)
\( J_4 \)  \(=\) \(-1068\) \(=\)  \( - 2^{2} \cdot 3 \cdot 89 \)
\( J_6 \)  \(=\) \(164320\) \(=\)  \( 2^{5} \cdot 5 \cdot 13 \cdot 79 \)
\( J_8 \)  \(=\) \(-8295756\) \(=\)  \( - 2^{2} \cdot 3 \cdot 7 \cdot 61 \cdot 1619 \)
\( J_{10} \)  \(=\) \(-154791\) \(=\)  \( - 3^{5} \cdot 7^{2} \cdot 13 \)
\( g_1 \)  \(=\) \(89253125/49\)
\( g_2 \)  \(=\) \(-7520500/147\)
\( g_3 \)  \(=\) \(-53404000/1323\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 1 : 1)\) \((1 : -1 : 1)\) \((0 : -2 : 1)\) \((-2 : 2 : 1)\)
\((-2 : 5 : 1)\) \((1 : 6 : 2)\) \((-3 : 13 : 1)\) \((1 : -15 : 2)\)

magma: [C![-3,13,1],C![-2,2,1],C![-2,5,1],C![0,-2,1],C![0,1,1],C![1,-15,2],C![1,-1,0],C![1,-1,1],C![1,0,0],C![1,6,2]];
 

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
\((1 : -1 : 1) - (1 : 0 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3\) \(0.033247\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + xz - 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 + z^3\) \(0\) \(2\)
\((-3 : 13 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + 3z)\) \(=\) \(0,\) \(2y\) \(=\) \(-7xz^2 + 5z^3\) \(0\) \(2\)

2-torsion field: \(\Q(\sqrt{-3}, \sqrt{13})\)

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(3\)
Regulator: \( 0.033247 \)
Real period: \( 18.98310 \)
Tamagawa product: \( 8 \)
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
\(3\) \(3\) \(5\) \(4\) \(1 + T\)
\(7\) \(1\) \(2\) \(2\) \(( 1 - T )( 1 + 4 T + 7 T^{2} )\)
\(13\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 6 T + 13 T^{2} )\)

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 \(\Q\).