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

### 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: $$0$$

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