Properties

Label 62233.a.62233.1
Conductor 62233
Discriminant -62233
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x + 1)y = -3x^4 + x^3 + 5x^2 - 5x$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -3x^4z^2 + x^3z^3 + 5x^2z^4 - 5xz^5$ (dehomogenize, simplify)
$y^2 = x^6 - 10x^4 + 6x^3 + 21x^2 - 18x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -5, 5, 1, -3]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -5, 5, 1, -3], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, -18, 21, 6, -10, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(62233\) \(=\) \( 62233 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-62233\) \(=\) \( -62233 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(3336\) \(=\)  \( 2^{3} \cdot 3 \cdot 139 \)
\( I_4 \)  \(=\) \(257892\) \(=\)  \( 2^{2} \cdot 3 \cdot 21491 \)
\( I_6 \)  \(=\) \(244275816\) \(=\)  \( 2^{3} \cdot 3 \cdot 29 \cdot 350971 \)
\( I_{10} \)  \(=\) \(-254906368\) \(=\)  \( - 2^{12} \cdot 62233 \)
\( J_2 \)  \(=\) \(417\) \(=\)  \( 3 \cdot 139 \)
\( J_4 \)  \(=\) \(4559\) \(=\)  \( 47 \cdot 97 \)
\( J_6 \)  \(=\) \(54933\) \(=\)  \( 3 \cdot 18311 \)
\( J_8 \)  \(=\) \(530645\) \(=\)  \( 5 \cdot 106129 \)
\( J_{10} \)  \(=\) \(-62233\) \(=\)  \( -62233 \)
\( g_1 \)  \(=\) \(-12608989261857/62233\)
\( g_2 \)  \(=\) \(-330580899567/62233\)
\( g_3 \)  \(=\) \(-9552244437/62233\)

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

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : -1 : 1)\) \((-1 : -2 : 1)\)
\((1 : -2 : 1)\) \((-1 : 3 : 1)\) \((2 : -5 : 1)\) \((2 : -6 : 1)\) \((-3 : 14 : 1)\) \((-3 : 15 : 1)\)
\((-4 : -27 : 3)\) \((5 : -53 : 2)\) \((-4 : 100 : 3)\) \((5 : -100 : 2)\) \((11 : -891 : 6)\) \((11 : -1052 : 6)\)
\((73 : -251641 : 52)\) \((73 : -475376 : 52)\)

magma: [C![-4,-27,3],C![-4,100,3],C![-3,14,1],C![-3,15,1],C![-1,-2,1],C![-1,3,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![2,-6,1],C![2,-5,1],C![5,-100,2],C![5,-53,2],C![11,-1052,6],C![11,-891,6],C![73,-475376,52],C![73,-251641,52]];
 

Number of rational Weierstrass points: \(0\)

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
\((0 : 0 : 1) + (2 : -6 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((-x + 2z) x\) \(=\) \(0,\) \(y\) \(=\) \(-3xz^2\) \(0.562559\) \(\infty\)
\((1 : -2 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.285327\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.212996\) \(\infty\)

2-torsion field: 6.4.3982912.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.026759 \)
Real period: \( 21.72187 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.581268 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(62233\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 146 T + 62233 T^{2} )\)

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 \(\Q\).