Properties

Label 10056.a.181008.1
Conductor $10056$
Discriminant $181008$
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

$y^2 + xy = 2x^5 - 3x^4 + 4x^3 - 2x^2 + x$ (homogenize, simplify)
$y^2 + xz^2y = 2x^5z - 3x^4z^2 + 4x^3z^3 - 2x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = 8x^5 - 12x^4 + 16x^3 - 7x^2 + 4x$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -2, 4, -3, 2]), R([0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, -2, 4, -3, 2], R![0, 1]);
 
sage: X = HyperellipticCurve(R([0, 4, -7, 16, -12, 8]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(10056\) \(=\) \( 2^{3} \cdot 3 \cdot 419 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(181008\) \(=\) \( 2^{4} \cdot 3^{3} \cdot 419 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(368\) \(=\)  \( 2^{4} \cdot 23 \)
\( I_4 \)  \(=\) \(4132\) \(=\)  \( 2^{2} \cdot 1033 \)
\( I_6 \)  \(=\) \(532068\) \(=\)  \( 2^{2} \cdot 3 \cdot 101 \cdot 439 \)
\( I_{10} \)  \(=\) \(724032\) \(=\)  \( 2^{6} \cdot 3^{3} \cdot 419 \)
\( J_2 \)  \(=\) \(184\) \(=\)  \( 2^{3} \cdot 23 \)
\( J_4 \)  \(=\) \(722\) \(=\)  \( 2 \cdot 19^{2} \)
\( J_6 \)  \(=\) \(-9500\) \(=\)  \( - 2^{2} \cdot 5^{3} \cdot 19 \)
\( J_8 \)  \(=\) \(-567321\) \(=\)  \( - 3 \cdot 19 \cdot 37 \cdot 269 \)
\( J_{10} \)  \(=\) \(181008\) \(=\)  \( 2^{4} \cdot 3^{3} \cdot 419 \)
\( g_1 \)  \(=\) \(13181630464/11313\)
\( g_2 \)  \(=\) \(281106368/11313\)
\( g_3 \)  \(=\) \(-20102000/11313\)

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: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : 1 : 1),\, (1 : -2 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : 1 : 1),\, (1 : -2 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : -3 : 1),\, (1 : 3 : 1)\)

magma: [C![0,0,1],C![1,-2,1],C![1,0,0],C![1,1,1]]; // minimal model
 
magma: [C![0,0,1],C![1,-3,1],C![1,0,0],C![1,3,1]]; // simplified model
 

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -2 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.055780\) \(\infty\)
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\((1 : -2 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.055780\) \(\infty\)
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\((1 : -3 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - 4z^3\) \(0.055780\) \(\infty\)
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0\) \(2\)

2-torsion field: 4.0.1257.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.055780 \)
Real period: \( 8.723347 \)
Tamagawa product: \( 6 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.729895 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(3\) \(4\) \(2\) \(1 + T + 2 T^{2}\)
\(3\) \(1\) \(3\) \(3\) \(( 1 - T )( 1 + 2 T + 3 T^{2} )\)
\(419\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 20 T + 419 T^{2} )\)

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 \(\Q\).