Properties

Label 64829.b.64829.1
Conductor 64829
Discriminant 64829
Mordell-Weil group trivial
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

$y^2 + (x^2 + 1)y = x^5 - 2x^4 + 4x^3 + x^2 - 2x - 1$ (homogenize, simplify)
$y^2 + (x^2z + z^3)y = x^5z - 2x^4z^2 + 4x^3z^3 + x^2z^4 - 2xz^5 - z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 - 7x^4 + 16x^3 + 6x^2 - 8x - 3$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, -2, 1, 4, -2, 1], R![1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, -2, 1, 4, -2, 1]), R([1, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([-3, -8, 6, 16, -7, 4]))
 

Invariants

Conductor: \( N \)  =  \(64829\) = \( 241 \cdot 269 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(64829\) = \( 241 \cdot 269 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(928\) =  \( 2^{5} \cdot 29 \)
\( I_4 \)  = \(-595712\) =  \( - 2^{8} \cdot 13 \cdot 179 \)
\( I_6 \)  = \(-157539136\) =  \( - 2^{6} \cdot 17 \cdot 29 \cdot 4993 \)
\( I_{10} \)  = \(265539584\) =  \( 2^{12} \cdot 241 \cdot 269 \)
\( J_2 \)  = \(116\) =  \( 2^{2} \cdot 29 \)
\( J_4 \)  = \(6766\) =  \( 2 \cdot 17 \cdot 199 \)
\( J_6 \)  = \(77169\) =  \( 3 \cdot 29 \cdot 887 \)
\( J_8 \)  = \(-9206788\) =  \( - 2^{2} \cdot 113 \cdot 20369 \)
\( J_{10} \)  = \(64829\) =  \( 241 \cdot 269 \)
\( g_1 \)  = \(21003416576/64829\)
\( g_2 \)  = \(10561022336/64829\)
\( g_3 \)  = \(1038386064/64829\)

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

All points: \((1 : 0 : 0)\)

magma: [C![1,0,0]];
 

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: trivial

magma: MordellWeilGroupGenus2(Jacobian(C));
 

2-torsion field: 5.1.259316.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 3.186146 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 3.186146 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(241\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 13 T + 241 T^{2} )\)
\(269\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 8 T + 269 T^{2} )\)

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 \(\Q\).