Properties

Label 610423.a.610423.1
Conductor 610423
Discriminant -610423
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x + 1)y = x^5 - 9x^3 + 7x^2 + 13x - 14$ (homogenize, simplify)
$y^2 + (xz^2 + z^3)y = x^5z - 9x^3z^3 + 7x^2z^4 + 13xz^5 - 14z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 - 36x^3 + 29x^2 + 54x - 55$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-14, 13, 7, -9, 0, 1], R![1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-14, 13, 7, -9, 0, 1]), R([1, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([-55, 54, 29, -36, 0, 4]))
 

Invariants

Conductor: \( N \)  =  \(610423\) = \( 11 \cdot 211 \cdot 263 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-610423\) = \( - 11 \cdot 211 \cdot 263 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(16416\) =  \( 2^{5} \cdot 3^{3} \cdot 19 \)
\( I_4 \)  = \(142464\) =  \( 2^{7} \cdot 3 \cdot 7 \cdot 53 \)
\( I_6 \)  = \(802171584\) =  \( 2^{6} \cdot 3^{2} \cdot 769 \cdot 1811 \)
\( I_{10} \)  = \(-2500292608\) =  \( - 2^{12} \cdot 11 \cdot 211 \cdot 263 \)
\( J_2 \)  = \(2052\) =  \( 2^{2} \cdot 3^{3} \cdot 19 \)
\( J_4 \)  = \(173962\) =  \( 2 \cdot 86981 \)
\( J_6 \)  = \(19454065\) =  \( 5 \cdot 89 \cdot 43717 \)
\( J_8 \)  = \(2414240984\) =  \( 2^{3} \cdot 301780123 \)
\( J_{10} \)  = \(-610423\) =  \( - 11 \cdot 211 \cdot 263 \)
\( g_1 \)  = \(-36382017816364032/610423\)
\( g_2 \)  = \(-1503095107936896/610423\)
\( g_3 \)  = \(-81915309311760/610423\)

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: \((1 : 0 : 0),\, (2 : 0 : 1),\, (2 : -3 : 1)\)

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

Number of rational Weierstrass points: \(1\)

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
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + xz - 4z^2\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 + 2z^3\) \(0.687204\) \(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0.648883\) \(\infty\)

2-torsion field: 5.3.2441692.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.397075 \)
Real period: \( 11.34796 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 4.505994 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(11\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 2 T + 11 T^{2} )\)
\(211\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - T + 211 T^{2} )\)
\(263\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 18 T + 263 T^{2} )\)

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 \(\Q\).