Properties

Label 10044.a.40176.1
Conductor $10044$
Discriminant $-40176$
Mordell-Weil group \(\Z \times \Z/{2}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

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Show commands: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

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

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

Invariants

Conductor: \( N \)  \(=\)  \(10044\) \(=\) \( 2^{2} \cdot 3^{4} \cdot 31 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-40176\) \(=\) \( - 2^{4} \cdot 3^{4} \cdot 31 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(108\) \(=\)  \( 2^{2} \cdot 3^{3} \)
\( I_4 \)  \(=\) \(-6543\) \(=\)  \( - 3^{2} \cdot 727 \)
\( I_6 \)  \(=\) \(-22041\) \(=\)  \( - 3^{2} \cdot 31 \cdot 79 \)
\( I_{10} \)  \(=\) \(5142528\) \(=\)  \( 2^{11} \cdot 3^{4} \cdot 31 \)
\( J_2 \)  \(=\) \(27\) \(=\)  \( 3^{3} \)
\( J_4 \)  \(=\) \(303\) \(=\)  \( 3 \cdot 101 \)
\( J_6 \)  \(=\) \(-1693\) \(=\)  \( -1693 \)
\( J_8 \)  \(=\) \(-34380\) \(=\)  \( - 2^{2} \cdot 3^{2} \cdot 5 \cdot 191 \)
\( J_{10} \)  \(=\) \(40176\) \(=\)  \( 2^{4} \cdot 3^{4} \cdot 31 \)
\( g_1 \)  \(=\) \(177147/496\)
\( g_2 \)  \(=\) \(73629/496\)
\( g_3 \)  \(=\) \(-15237/496\)

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: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1)\)
All points: \((1 : 0 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1)\)

magma: [C![0,-1,1],C![0,0,1],C![1,0,0]]; // minimal model
 
magma: [C![0,-1,1],C![0,1,1],C![1,0,0]]; // simplified model
 

Number of rational Weierstrass points: \(1\)

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.141330\) \(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(2\)
Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.141330\) \(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(2\)
Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(x^2z + xz^2 - z^3\) \(0.141330\) \(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^2z - xz^2 + z^3\) \(0\) \(2\)

2-torsion field: 6.0.2101707.2

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.141330 \)
Real period: \( 13.03603 \)
Tamagawa product: \( 2 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.921192 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(4\) \(2\) \(1 + T^{2}\)
\(3\) \(4\) \(4\) \(1\) \(1 + T\)
\(31\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 8 T + 31 T^{2} )\)

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 \(\Q\).