Properties

Label 5501.a.5501.1
Conductor 5501
Discriminant 5501
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

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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 1, -3, -2], R![1, 0, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 1, -3, -2]), R([1, 0, 1, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([9, 4, -10, -6, 1, 2, 1]))
 

Invariants

Conductor: \( N \)  =  \(5501\) = \( 5501 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(5501\) = \( 5501 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-1464\) =  \( - 2^{3} \cdot 3 \cdot 61 \)
\( I_4 \)  = \(88164\) =  \( 2^{2} \cdot 3^{2} \cdot 31 \cdot 79 \)
\( I_6 \)  = \(-50339736\) =  \( - 2^{3} \cdot 3^{2} \cdot 743 \cdot 941 \)
\( I_{10} \)  = \(22532096\) =  \( 2^{12} \cdot 5501 \)
\( J_2 \)  = \(-183\) =  \( - 3 \cdot 61 \)
\( J_4 \)  = \(477\) =  \( 3^{2} \cdot 53 \)
\( J_6 \)  = \(26525\) =  \( 5^{2} \cdot 1061 \)
\( J_8 \)  = \(-1270401\) =  \( - 3 \cdot 11 \cdot 137 \cdot 281 \)
\( J_{10} \)  = \(5501\) =  \( 5501 \)
\( g_1 \)  = \(-205236901143/5501\)
\( g_2 \)  = \(-2923288299/5501\)
\( g_3 \)  = \(888295725/5501\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((-1 : 0 : 1)\) \((0 : 1 : 1)\) \((-1 : -1 : 1)\) \((1 : -1 : 1)\)
\((0 : -2 : 1)\) \((-2 : -1 : 1)\) \((1 : -2 : 1)\) \((-2 : 4 : 1)\) \((-13 : -128 : 12)\) \((-13 : -1431 : 12)\)

magma: [C![-13,-1431,12],C![-13,-128,12],C![-2,-1,1],C![-2,4,1],C![-1,-1,1],C![-1,0,1],C![0,-2,1],C![0,1,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0]];
 

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: \(\Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : -1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.176934\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0.057232\) \(\infty\)

2-torsion field: 6.2.352064.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.010011 \)
Real period: \( 26.16477 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.261938 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(5501\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 9 T + 5501 T^{2} )\)

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 \(\Q\).