# 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

# Related objects

Show commands for: Magma / SageMath

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); $\Delta$ = $277$ = $277$

### 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: 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![-1,0,1],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:

$0$

### 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 $\Q$