Properties

Label 1116.a.214272.1
Conductor 1116
Discriminant -214272
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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

Minimal equation

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

$y^2 + (x^3 + 1)y = x^4 + 2x^3 + x^2 - x$

Invariants

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

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];

G2 invariants

magma: G2Invariants(C);

\( I_2 \)  =  \(104\)  =  \( 2^{3} \cdot 13 \)
\( I_4 \)  =  \(88804\)  =  \( 2^{2} \cdot 149^{2} \)
\( I_6 \)  =  \(1906280\)  =  \( 2^{3} \cdot 5 \cdot 47657 \)
\( I_{10} \)  =  \(-877658112\)  =  \( -1 \cdot 2^{20} \cdot 3^{3} \cdot 31 \)
\( J_2 \)  =  \(13\)  =  \( 13 \)
\( J_4 \)  =  \(-918\)  =  \( -1 \cdot 2 \cdot 3^{3} \cdot 17 \)
\( J_6 \)  =  \(36\)  =  \( 2^{2} \cdot 3^{2} \)
\( J_8 \)  =  \(-210564\)  =  \( -1 \cdot 2^{2} \cdot 3^{2} \cdot 5849 \)
\( J_{10} \)  =  \(-214272\)  =  \( -1 \cdot 2^{8} \cdot 3^{3} \cdot 31 \)
\( g_1 \)  =  \(-371293/214272\)
\( g_2 \)  =  \(37349/3968\)
\( g_3 \)  =  \(-169/5952\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
\(\mathrm{Aut}(X)\)\(\simeq\) \(C_2 \) (GAP id : [2,1])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(C_2 \) (GAP id : [2,1])

Rational points

magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsLocallyEverywhere(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);

This curve is locally solvable everywhere.

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]];

All rational points:

(-1 : -1 : 1), (-1 : 1 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -3 : 1), (1 : -1 : 0), (1 : 0 : 0), (1 : 1 : 1)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

Number of rational Weierstrass points:

\(0\)

Invariants of the Jacobian:

Analytic rank:

\(0\)

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);

2-Selmer rank:

\(0\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*:

square

Tamagawa numbers:

13 (p = 2), 3 (p = 3), 1 (p = 31)

magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion:

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

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition

Simple over \(\overline{\Q}\)

Endomorphisms

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 endomorphisms of the Jacobian are defined over \(\Q\)