Properties

Label 100240.a.400960.1
Conductor 100240
Discriminant 400960
Mordell-Weil group \(\Z\)
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

$y^2 + x^3y = x^5 - x^4 - 5x^3 + 6x^2 + 10x - 15$ (homogenize, simplify)
$y^2 + x^3y = x^5z - x^4z^2 - 5x^3z^3 + 6x^2z^4 + 10xz^5 - 15z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 - 4x^4 - 20x^3 + 24x^2 + 40x - 60$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-15, 10, 6, -5, -1, 1]), R([0, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-15, 10, 6, -5, -1, 1], R![0, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([-60, 40, 24, -20, -4, 4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(100240\) \(=\) \( 2^{4} \cdot 5 \cdot 7 \cdot 179 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(400960\) \(=\) \( 2^{6} \cdot 5 \cdot 7 \cdot 179 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(24736\) \(=\)  \( 2^{5} \cdot 773 \)
\( I_4 \)  \(=\) \(202816\) \(=\)  \( 2^{6} \cdot 3169 \)
\( I_6 \)  \(=\) \(1664700928\) \(=\)  \( 2^{9} \cdot 11 \cdot 17 \cdot 17387 \)
\( I_{10} \)  \(=\) \(1642332160\) \(=\)  \( 2^{18} \cdot 5 \cdot 7 \cdot 179 \)
\( J_2 \)  \(=\) \(3092\) \(=\)  \( 2^{2} \cdot 773 \)
\( J_4 \)  \(=\) \(396240\) \(=\)  \( 2^{4} \cdot 3 \cdot 5 \cdot 13 \cdot 127 \)
\( J_6 \)  \(=\) \(67352576\) \(=\)  \( 2^{11} \cdot 32887 \)
\( J_8 \)  \(=\) \(12812006848\) \(=\)  \( 2^{6} \cdot 23 \cdot 103 \cdot 84503 \)
\( J_{10} \)  \(=\) \(400960\) \(=\)  \( 2^{6} \cdot 5 \cdot 7 \cdot 179 \)
\( g_1 \)  \(=\) \(4415881923441488/6265\)
\( g_2 \)  \(=\) \(36603852142416/1253\)
\( g_3 \)  \(=\) \(10061279346176/6265\)

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

All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-3 : 12 : 1),\, (-3 : 15 : 1)\)

magma: [C![-3,12,1],C![-3,15,1],C![1,-1,0],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: \(\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + xz - 4z^2\) \(=\) \(0,\) \(y\) \(=\) \(-3xz^2 + z^3\) \(0.061812\) \(\infty\)

2-torsion field: 6.2.983610638500.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.061812 \)
Real period: \( 6.806168 \)
Tamagawa product: \( 5 \)
Torsion order:\( 1 \)
Leading coefficient: \( 2.103531 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(4\) \(6\) \(5\) \(1\)
\(5\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + T + 5 T^{2} )\)
\(7\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 2 T + 7 T^{2} )\)
\(179\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 24 T + 179 T^{2} )\)

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 \(\Q\).