Properties

Label 1116.a.214272.1
Conductor 1116
Discriminant -214272
Mordell-Weil group \(\Z/{39}\Z\)
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + 1)y = x^4 + 2x^3 + x^2 - x$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = x^4z^2 + 2x^3z^3 + x^2z^4 - xz^5$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^4 + 10x^3 + 4x^2 - 4x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 1, 2, 1]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 1, 2, 1], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, -4, 4, 10, 4, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1116\) \(=\) \( 2^{2} \cdot 3^{2} \cdot 31 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-214272\) \(=\) \( - 2^{8} \cdot 3^{3} \cdot 31 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(104\) \(=\)  \( 2^{3} \cdot 13 \)
\( I_4 \)  \(=\) \(88804\) \(=\)  \( 2^{2} \cdot 149^{2} \)
\( I_6 \)  \(=\) \(1906280\) \(=\)  \( 2^{3} \cdot 5 \cdot 47657 \)
\( I_{10} \)  \(=\) \(-877658112\) \(=\)  \( - 2^{20} \cdot 3^{3} \cdot 31 \)
\( J_2 \)  \(=\) \(13\) \(=\)  \( 13 \)
\( J_4 \)  \(=\) \(-918\) \(=\)  \( - 2 \cdot 3^{3} \cdot 17 \)
\( J_6 \)  \(=\) \(36\) \(=\)  \( 2^{2} \cdot 3^{2} \)
\( J_8 \)  \(=\) \(-210564\) \(=\)  \( - 2^{2} \cdot 3^{2} \cdot 5849 \)
\( J_{10} \)  \(=\) \(-214272\) \(=\)  \( - 2^{8} \cdot 3^{3} \cdot 31 \)
\( g_1 \)  \(=\) \(-371293/214272\)
\( g_2 \)  \(=\) \(37349/3968\)
\( g_3 \)  \(=\) \(-169/5952\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\)
\((1 : 1 : 1)\) \((1 : -3 : 1)\)

magma: [C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,1,1]];
 

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: \(\Z/{39}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 1 : 1) + (1 : -3 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 - z^3\) \(0\) \(39\)

2-torsion field: 6.0.53568.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 16.98409 \)
Tamagawa product: \( 39 \)
Torsion order:\( 39 \)
Leading coefficient: \( 0.435489 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(2\) \(8\) \(13\) \(( 1 - T )( 1 + T )\)
\(3\) \(2\) \(3\) \(3\) \(1 + T + T^{2}\)
\(31\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 7 T + 31 T^{2} )\)

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 \(\Q\).

Additional information

This example of a curve of genus 2 whose Jacobian has a rational 39-torsion point was discovered by Noam Elkies (see this page).