Properties

Label 249.a.249.1
Conductor 249
Discriminant 249
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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The Jacobian $A$ of this curve is the first abelian surface of paramodular type (meaning $\End(A)=\Z$) that appears in the table of Brumer and Kramer [MR:3165645].

Minimal equation

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

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

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 249 \)  =  \( 3 \cdot 83 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(249\)  =  \( 3 \cdot 83 \)

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);
 

\( I_2 \)  =  \(-216\)  =  \( -1 \cdot 2^{3} \cdot 3^{3} \)
\( I_4 \)  =  \(228\)  =  \( 2^{2} \cdot 3 \cdot 19 \)
\( I_6 \)  =  \(-18072\)  =  \( -1 \cdot 2^{3} \cdot 3^{2} \cdot 251 \)
\( I_{10} \)  =  \(1019904\)  =  \( 2^{12} \cdot 3 \cdot 83 \)
\( J_2 \)  =  \(-27\)  =  \( -1 \cdot 3^{3} \)
\( J_4 \)  =  \(28\)  =  \( 2^{2} \cdot 7 \)
\( J_6 \)  =  \(-32\)  =  \( -1 \cdot 2^{5} \)
\( J_8 \)  =  \(20\)  =  \( 2^{2} \cdot 5 \)
\( J_{10} \)  =  \(249\)  =  \( 3 \cdot 83 \)
\( g_1 \)  =  \(-4782969/83\)
\( g_2 \)  =  \(-183708/83\)
\( g_3 \)  =  \(-7776/83\)
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; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);
 

This curve is locally solvable everywhere.

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

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

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

Number of rational Weierstrass points: \(1\)

Invariants of the Jacobian:

Analytic rank: \(0\)

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

2-Selmer rank: \(1\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 1.0

Real period: 25.783703374249836805826663230

Tamagawa numbers: 1 (p = 3), 1 (p = 83)

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

Torsion: \(\Z/{14}\Z\)

2-torsion field: 6.0.186003.1

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 \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).