Properties

Label 1142.a.2284.1
Conductor 1142
Discriminant -2284
Mordell-Weil group \(\Z/{10}\Z\)
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

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Minimal equation

Minimal equation

Simplified equation

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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, -1, 1, -1, -1], R![0, 0, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, -1, 1, -1, -1]), R([0, 0, 1, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([-8, -4, 4, -4, -3, 2, 1]))
 

Invariants

Conductor: \( N \)  =  \(1142\) = \( 2 \cdot 571 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-2284\) = \( - 2^{2} \cdot 571 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(1888\) =  \( 2^{5} \cdot 59 \)
\( I_4 \)  = \(-46016\) =  \( - 2^{6} \cdot 719 \)
\( I_6 \)  = \(-27370048\) =  \( - 2^{6} \cdot 53 \cdot 8069 \)
\( I_{10} \)  = \(-9355264\) =  \( - 2^{14} \cdot 571 \)
\( J_2 \)  = \(236\) =  \( 2^{2} \cdot 59 \)
\( J_4 \)  = \(2800\) =  \( 2^{4} \cdot 5^{2} \cdot 7 \)
\( J_6 \)  = \(46521\) =  \( 3^{3} \cdot 1723 \)
\( J_8 \)  = \(784739\) =  \( 227 \cdot 3457 \)
\( J_{10} \)  = \(-2284\) =  \( - 2^{2} \cdot 571 \)
\( g_1 \)  = \(-183020620544/571\)
\( g_2 \)  = \(-9200979200/571\)
\( g_3 \)  = \(-647758404/571\)

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 0 : 1),\, (-2 : 2 : 1)\)

magma: [C![-2,2,1],C![-1,0,1],C![1,-1,0],C![1,0,0]];
 

Number of rational Weierstrass points: \(2\)

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: \(\Z/{10}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(10\)

2-torsion field: 4.2.2284.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 16.87134 \)
Tamagawa product: \( 2 \)
Torsion order:\( 10 \)
Leading coefficient: \( 0.337426 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(2\) \(1\) \(2\) \(( 1 - T )( 1 + 2 T + 2 T^{2} )\)
\(571\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 28 T + 571 T^{2} )\)

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 \(\Q\).