Properties

Label 10005.b.450225.1
Conductor $10005$
Discriminant $450225$
Mordell-Weil group \(\Z \times \Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

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Show commands: SageMath / Magma

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)

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

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

\( I_2 \)  \(=\) \(444\) \(=\)  \( 2^{2} \cdot 3 \cdot 37 \)
\( I_4 \)  \(=\) \(108777\) \(=\)  \( 3 \cdot 101 \cdot 359 \)
\( I_6 \)  \(=\) \(21372411\) \(=\)  \( 3 \cdot 137 \cdot 149 \cdot 349 \)
\( I_{10} \)  \(=\) \(-57628800\) \(=\)  \( - 2^{7} \cdot 3^{3} \cdot 5^{2} \cdot 23 \cdot 29 \)
\( J_2 \)  \(=\) \(111\) \(=\)  \( 3 \cdot 37 \)
\( J_4 \)  \(=\) \(-4019\) \(=\)  \( -4019 \)
\( J_6 \)  \(=\) \(-153925\) \(=\)  \( - 5^{2} \cdot 47 \cdot 131 \)
\( J_8 \)  \(=\) \(-8309509\) \(=\)  \( - 13 \cdot 23 \cdot 27791 \)
\( J_{10} \)  \(=\) \(-450225\) \(=\)  \( - 3^{3} \cdot 5^{2} \cdot 23 \cdot 29 \)
\( g_1 \)  \(=\) \(-624095613/16675\)
\( g_2 \)  \(=\) \(203574407/16675\)
\( g_3 \)  \(=\) \(8428933/2001\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 1 : 1)\) \((-1 : -1 : 1)\) \((1 : 1 : 1)\) \((0 : -2 : 1)\)
\((-1 : 2 : 1)\) \((1 : -4 : 1)\) \((-2 : 4 : 1)\) \((-2 : 5 : 1)\) \((3 : -5 : 1)\) \((-3 : 8 : 2)\)
\((-3 : 23 : 2)\) \((3 : -26 : 1)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 1 : 1)\) \((-1 : -1 : 1)\) \((1 : 1 : 1)\) \((0 : -2 : 1)\)
\((-1 : 2 : 1)\) \((1 : -4 : 1)\) \((-2 : 4 : 1)\) \((-2 : 5 : 1)\) \((3 : -5 : 1)\) \((-3 : 8 : 2)\)
\((-3 : 23 : 2)\) \((3 : -26 : 1)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((-2 : -1 : 1)\) \((-2 : 1 : 1)\) \((0 : -3 : 1)\) \((0 : 3 : 1)\)
\((-1 : -3 : 1)\) \((-1 : 3 : 1)\) \((1 : -5 : 1)\) \((1 : 5 : 1)\) \((-3 : -15 : 2)\) \((-3 : 15 : 2)\)
\((3 : -21 : 1)\) \((3 : 21 : 1)\)

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]]; // minimal model
 
magma: [C![-3,-15,2],C![-3,15,2],C![-2,-1,1],C![-2,1,1],C![-1,-3,1],C![-1,3,1],C![0,-3,1],C![0,3,1],C![1,-5,1],C![1,-1,0],C![1,1,0],C![1,5,1],C![3,-21,1],C![3,21,1]]; // simplified model
 

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
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 + z^3\) \(0.061340\) \(\infty\)
\((-2 : 4 : 1) + (1 : 1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 + 2z^3\) \(0.075791\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 + z^3\) \(0.061340\) \(\infty\)
\((-2 : 4 : 1) + (1 : 1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 + 2z^3\) \(0.075791\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 - 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - xz^2 + 3z^3\) \(0.061340\) \(\infty\)
\((-2 : -1 : 1) + (1 : 5 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \((x - z) (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - xz^2 + 5z^3\) \(0.075791\) \(\infty\)

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 Cluster picture
\(3\) \(1\) \(3\) \(3\) \(( 1 - T )( 1 + 3 T + 3 T^{2} )\)
\(5\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + 2 T + 5 T^{2} )\)
\(23\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 6 T + 23 T^{2} )\)
\(29\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 3 T + 29 T^{2} )\)

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 \(\Q\).