Properties

Label 277.a.277.1
Conductor 277
Discriminant 277
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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The first genus 2 curve with a primitive L-function which has been proven to be modular. This is the first proven instance of the paramodular conjecture.

Minimal equation

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

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

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 277 \)  =  \( 277 \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(277\)  =  \( 277 \)

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);

\( I_2 \)  =  \(-256\)  =  \( -1 \cdot 2^{8} \)
\( I_4 \)  =  \(5632\)  =  \( 2^{9} \cdot 11 \)
\( I_6 \)  =  \(-611328\)  =  \( -1 \cdot 2^{10} \cdot 3 \cdot 199 \)
\( I_{10} \)  =  \(1134592\)  =  \( 2^{12} \cdot 277 \)
\( J_2 \)  =  \(-32\)  =  \( -1 \cdot 2^{5} \)
\( J_4 \)  =  \(-16\)  =  \( -1 \cdot 2^{4} \)
\( J_6 \)  =  \(464\)  =  \( 2^{4} \cdot 29 \)
\( J_8 \)  =  \(-3776\)  =  \( -1 \cdot 2^{6} \cdot 59 \)
\( J_{10} \)  =  \(277\)  =  \( 277 \)
\( g_1 \)  =  \(-33554432/277\)
\( g_2 \)  =  \(524288/277\)
\( g_3 \)  =  \(475136/277\)
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])
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

Number of rational Weierstrass points:

\(1\)
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);

Locally solvable:

yes

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:

1 (p = 277)
magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion:

\(\Z/{15}\Z\)

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 \(\Q\)