Properties

Label 100152.a.600912.1
Conductor 100152
Discriminant -600912
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

Learn more about

Show commands for: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 0, -4, -6, -2, 1], R![1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 4, 0, -16, -24, -8, 4]))
 

$y^2 + y = x^6 - 2x^5 - 6x^4 - 4x^3 + x$ (homogenize, simplify)
$y^2 + z^3y = x^6 - 2x^5z - 6x^4z^2 - 4x^3z^3 + xz^5$ (dehomogenize, simplify)
$y^2 = 4x^6 - 8x^5 - 24x^4 - 16x^3 + 4x + 1$ (minimize, homogenize)

Invariants

\( N \)  =  \(100152\) = \( 2^{3} \cdot 3^{2} \cdot 13 \cdot 107 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
\( \Delta \)  =  \(-600912\) = \( - 2^{4} \cdot 3^{3} \cdot 13 \cdot 107 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

G2 invariants

magma: G2Invariants(C);
 

\( I_2 \)  = \(-704\) =  \( - 2^{6} \cdot 11 \)
\( I_4 \)  = \(-79040\) =  \( - 2^{6} \cdot 5 \cdot 13 \cdot 19 \)
\( I_6 \)  = \(7366144\) =  \( 2^{9} \cdot 14387 \)
\( I_{10} \)  = \(-2461335552\) =  \( - 2^{16} \cdot 3^{3} \cdot 13 \cdot 107 \)
\( J_2 \)  = \(-88\) =  \( - 2^{3} \cdot 11 \)
\( J_4 \)  = \(1146\) =  \( 2 \cdot 3 \cdot 191 \)
\( J_6 \)  = \(5760\) =  \( 2^{7} \cdot 3^{2} \cdot 5 \)
\( J_8 \)  = \(-455049\) =  \( - 3^{2} \cdot 7 \cdot 31 \cdot 233 \)
\( J_{10} \)  = \(-600912\) =  \( - 2^{4} \cdot 3^{3} \cdot 13 \cdot 107 \)
\( g_1 \)  = \(329832448/37557\)
\( g_2 \)  = \(16270144/12519\)
\( g_3 \)  = \(-309760/4173\)

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2$

Rational points

magma: [C![-1,-45,4],C![-1,-19,4],C![-1,-14,3],C![-1,-13,3],C![-1,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,1,0]];
 

Known points
\((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : -1 : 0)\) \((1 : 1 : 0)\) \((-1 : -1 : 1)\)
\((-1 : -13 : 3)\) \((-1 : -14 : 3)\) \((-1 : -19 : 4)\) \((-1 : -45 : 4)\)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

Number of rational Weierstrass points: \(0\)

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);
 

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Group structure: \(\Z \times \Z\)

Generator Height Order
\(2x^2 + 2xz + z^2\) \(=\) \(0,\) \(2y\) \(=\) \(xz^2 - z^3\) \(0.354578\) \(\infty\)
\(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.058359\) \(\infty\)

2-torsion field: 6.4.9614592.1

BSD invariants

Analytic rank: \(2\)   (upper bound)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.020121 \)
Real period: \( 11.02077 \)
Tamagawa product: \( 6 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.330500 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(4\) \(3\) \(2\) \(1 + 2 T + 2 T^{2}\)
\(3\) \(3\) \(2\) \(3\) \(( 1 - T )( 1 + T )\)
\(13\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 5 T + 13 T^{2} )\)
\(107\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 12 T + 107 T^{2} )\)

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 \(\Q\).