Properties

Label 4428.a.239112.1
Conductor 4428
Discriminant 239112
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

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 0, 0, -1, -2]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 0, 0, -1, -2], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, -2, 1, 2, -2, -8, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(4428\) \(=\) \( 2^{2} \cdot 3^{3} \cdot 41 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(239112\) \(=\) \( 2^{3} \cdot 3^{6} \cdot 41 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(456\) \(=\)  \( 2^{3} \cdot 3 \cdot 19 \)
\( I_4 \)  \(=\) \(-11196\) \(=\)  \( - 2^{2} \cdot 3^{2} \cdot 311 \)
\( I_6 \)  \(=\) \(-3802680\) \(=\)  \( - 2^{3} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 503 \)
\( I_{10} \)  \(=\) \(979402752\) \(=\)  \( 2^{15} \cdot 3^{6} \cdot 41 \)
\( J_2 \)  \(=\) \(57\) \(=\)  \( 3 \cdot 19 \)
\( J_4 \)  \(=\) \(252\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 7 \)
\( J_6 \)  \(=\) \(5184\) \(=\)  \( 2^{6} \cdot 3^{4} \)
\( J_8 \)  \(=\) \(57996\) \(=\)  \( 2^{2} \cdot 3^{4} \cdot 179 \)
\( J_{10} \)  \(=\) \(239112\) \(=\)  \( 2^{3} \cdot 3^{6} \cdot 41 \)
\( g_1 \)  \(=\) \(2476099/984\)
\( g_2 \)  \(=\) \(48013/246\)
\( g_3 \)  \(=\) \(2888/41\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 2 : 1)\)
\((1 : -5 : 2)\) \((1 : -8 : 2)\) \((-5 : 1462 : 17)\) \((-5 : -4805 : 17)\)

magma: [C![-5,-4805,17],C![-5,1462,17],C![-1,-1,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-8,2],C![1,-5,2],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
\((1 : -5 : 2) - (1 : 0 : 0)\) \(z (2x - z)\) \(=\) \(0,\) \(2y\) \(=\) \(-2x^3 - z^3\) \(0.003691\) \(\infty\)

2-torsion field: 6.2.188928.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.003691 \)
Real period: \( 11.60915 \)
Tamagawa product: \( 12 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.514252 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(3\) \(3\) \(1 + T + T^{2}\)
\(3\) \(3\) \(6\) \(4\) \(1 + T\)
\(41\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 3 T + 41 T^{2} )\)

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 \(\Q\).