Properties

Label 10005.b.450225.1
Conductor 10005
Discriminant 450225
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

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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 1, 3, 0, -2], R![1, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 1, 3, 0, -2]), R([1, 1, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([9, 6, 13, 2, -6, 0, 1]))
 

Invariants

Conductor: \( N \)  =  \(10005\) = \( 3 \cdot 5 \cdot 23 \cdot 29 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(450225\) = \( 3^{3} \cdot 5^{2} \cdot 23 \cdot 29 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-888\) =  \( - 2^{3} \cdot 3 \cdot 37 \)
\( I_4 \)  = \(435108\) =  \( 2^{2} \cdot 3 \cdot 101 \cdot 359 \)
\( I_6 \)  = \(-170979288\) =  \( - 2^{3} \cdot 3 \cdot 137 \cdot 149 \cdot 349 \)
\( I_{10} \)  = \(1844121600\) =  \( 2^{12} \cdot 3^{3} \cdot 5^{2} \cdot 23 \cdot 29 \)
\( J_2 \)  = \(-111\) =  \( - 3 \cdot 37 \)
\( J_4 \)  = \(-4019\) =  \( - 4019 \)
\( J_6 \)  = \(153925\) =  \( 5^{2} \cdot 47 \cdot 131 \)
\( J_8 \)  = \(-8309509\) =  \( - 13 \cdot 23 \cdot 27791 \)
\( J_{10} \)  = \(450225\) =  \( 3^{3} \cdot 5^{2} \cdot 23 \cdot 29 \)
\( g_1 \)  = \(-624095613/16675\)
\( g_2 \)  = \(203574407/16675\)
\( g_3 \)  = \(8428933/2001\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 1 : 1)\) \((-1 : -1 : 1)\) \((1 : 1 : 1)\) \((0 : -2 : 1)\)
\((-1 : 2 : 1)\) \((1 : -4 : 1)\) \((-2 : 4 : 1)\) \((-2 : 5 : 1)\) \((3 : -5 : 1)\) \((-3 : 8 : 2)\)
\((-3 : 23 : 2)\) \((3 : -26 : 1)\)

magma: [C![-3,8,2],C![-3,23,2],C![-2,4,1],C![-2,5,1],C![-1,-1,1],C![-1,2,1],C![0,-2,1],C![0,1,1],C![1,-4,1],C![1,-1,0],C![1,0,0],C![1,1,1],C![3,-26,1],C![3,-5,1]];
 

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
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 + z^3\) \(0.061340\) \(\infty\)
\((-2 : 4 : 1) + (1 : 1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 + 2z^3\) \(0.075791\) \(\infty\)

2-torsion field: 6.2.8012006001.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.003683 \)
Real period: \( 16.51737 \)
Tamagawa product: \( 6 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.365022 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(3\) \(3\) \(1\) \(3\) \(( 1 - T )( 1 + 3 T + 3 T^{2} )\)
\(5\) \(2\) \(1\) \(2\) \(( 1 + T )( 1 + 2 T + 5 T^{2} )\)
\(23\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 6 T + 23 T^{2} )\)
\(29\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 3 T + 29 T^{2} )\)

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 \(\Q\).