# 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

# Related objects

Show commands for: Magma / SageMath

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); $\Delta$ = $-214272$ = $-1 \cdot 2^{8} \cdot 3^{3} \cdot 31$

### 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: 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,-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:

$0$

### 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 $\Q$