Properties

Label 62411.b.62411.1
Conductor 62411
Discriminant -62411
Mordell-Weil group \(\Z \times \Z \times \Z\)
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

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

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

Invariants

Conductor: \( N \)  =  \(62411\) = \( 139 \cdot 449 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-62411\) = \( - 139 \cdot 449 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-640\) =  \( - 2^{7} \cdot 5 \)
\( I_4 \)  = \(-20480\) =  \( - 2^{12} \cdot 5 \)
\( I_6 \)  = \(1310720\) =  \( 2^{18} \cdot 5 \)
\( I_{10} \)  = \(-255635456\) =  \( - 2^{12} \cdot 139 \cdot 449 \)
\( J_2 \)  = \(-80\) =  \( - 2^{4} \cdot 5 \)
\( J_4 \)  = \(480\) =  \( 2^{5} \cdot 3 \cdot 5 \)
\( J_6 \)  = \(1280\) =  \( 2^{8} \cdot 5 \)
\( J_8 \)  = \(-83200\) =  \( - 2^{8} \cdot 5^{2} \cdot 13 \)
\( J_{10} \)  = \(-62411\) =  \( - 139 \cdot 449 \)
\( g_1 \)  = \(3276800000/62411\)
\( g_2 \)  = \(245760000/62411\)
\( g_3 \)  = \(-8192000/62411\)

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
\((1 : 0 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((-1 : -1 : 1)\)
\((1 : -1 : 1)\) \((2 : 5 : 1)\) \((2 : -6 : 1)\) \((3 : 15 : 1)\) \((3 : -16 : 1)\) \((1 : -30 : 4)\)
\((1 : -34 : 4)\) \((-15 : 740 : 16)\) \((-15 : -4836 : 16)\) \((30 : 4929 : 1)\) \((30 : -4930 : 1)\)

magma: [C![-15,-4836,16],C![-15,740,16],C![-1,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-34,4],C![1,-30,4],C![1,-1,1],C![1,0,0],C![1,0,1],C![2,-6,1],C![2,5,1],C![3,-16,1],C![3,15,1],C![30,-4930,1],C![30,4929,1]];
 

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: \(\Z \times \Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.602986\) \(\infty\)
\((-1 : -1 : 1) + (0 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.783665\) \(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.155861\) \(\infty\)

2-torsion field: 5.3.998576.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.052702 \)
Real period: \( 14.17064 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.746826 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(139\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 19 T + 139 T^{2} )\)
\(449\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 5 T + 449 T^{2} )\)

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 \(\Q\).