Properties

Label 100130.a.200260.1
Conductor 100130
Discriminant -200260
Mordell-Weil group \(\Z \times \Z \times \Z/{2}\Z\)
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

Learn more about

Show commands for: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + x + 1)y = x^6 + 5x^5 + 6x^4 + x^2 + x$ (homogenize, simplify)
$y^2 + (x^2z + xz^2 + z^3)y = x^6 + 5x^5z + 6x^4z^2 + x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = 4x^6 + 20x^5 + 25x^4 + 2x^3 + 7x^2 + 6x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 1, 0, 6, 5, 1]), R([1, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 1, 0, 6, 5, 1], R![1, 1, 1]);
 
sage: X = HyperellipticCurve(R([1, 6, 7, 2, 25, 20, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(100130\) \(=\) \( 2 \cdot 5 \cdot 17 \cdot 19 \cdot 31 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-200260\) \(=\) \( - 2^{2} \cdot 5 \cdot 17 \cdot 19 \cdot 31 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1064\) \(=\)  \( 2^{3} \cdot 7 \cdot 19 \)
\( I_4 \)  \(=\) \(872740\) \(=\)  \( 2^{2} \cdot 5 \cdot 11 \cdot 3967 \)
\( I_6 \)  \(=\) \(-8682968\) \(=\)  \( - 2^{3} \cdot 7 \cdot 47 \cdot 3299 \)
\( I_{10} \)  \(=\) \(-820264960\) \(=\)  \( - 2^{14} \cdot 5 \cdot 17 \cdot 19 \cdot 31 \)
\( J_2 \)  \(=\) \(133\) \(=\)  \( 7 \cdot 19 \)
\( J_4 \)  \(=\) \(-8354\) \(=\)  \( - 2 \cdot 4177 \)
\( J_6 \)  \(=\) \(356384\) \(=\)  \( 2^{5} \cdot 7 \cdot 37 \cdot 43 \)
\( J_8 \)  \(=\) \(-5597561\) \(=\)  \( -5597561 \)
\( J_{10} \)  \(=\) \(-200260\) \(=\)  \( - 2^{2} \cdot 5 \cdot 17 \cdot 19 \cdot 31 \)
\( g_1 \)  \(=\) \(-2190305047/10540\)
\( g_2 \)  \(=\) \(517208671/5270\)
\( g_3 \)  \(=\) \(-82948376/2635\)

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
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : 1 : 1)\) \((-1 : -2 : 1)\)
\((7 : 602 : 3)\) \((7 : -839 : 3)\)

magma: [C![-1,-2,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,1,0],C![7,-839,3],C![7,602,3]];
 

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: \(\Z \times \Z \times \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(2x^2 + 5xz + z^2\) \(=\) \(0,\) \(4y\) \(=\) \(5xz^2 - z^3\) \(0.429826\) \(\infty\)
\((-1 : -2 : 1) - (1 : 1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - 3z^3\) \(0.300725\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 + 3xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0\) \(2\)

2-torsion field: splitting field of \(x^{6} - 20 x^{4} - 26 x^{3} - 255 x^{2} - 1730 x - 1706\) with Galois group $S_4\times C_2$

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(3\)
Regulator: \( 0.128156 \)
Real period: \( 18.88632 \)
Tamagawa product: \( 2 \)
Torsion order:\( 2 \)
Leading coefficient: \( 1.210198 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + T + 2 T^{2} )\)
\(5\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 5 T^{2} )\)
\(17\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 6 T + 17 T^{2} )\)
\(19\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 6 T + 19 T^{2} )\)
\(31\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 6 T + 31 T^{2} )\)

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 \(\Q\).