Show commands for: Magma / SageMath

The Jacobian of this curve has a rational 29-torsion point. This is the largest known prime factor occurring in the torsion of the Mordell-Weil group of any abelian surface over $\mathbb{Q}$. This example was discovered by Franck Leprévost in 1995: see [MR:1413580].

## Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -1, 2, 0, -2, 1], R![1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -1, 2, 0, -2, 1]), R([1, 1]))

$y^2 + (x + 1)y = x^6 - 2x^5 + 2x^3 - x^2$

## Invariants

 magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(976,2),R![1, -1]>*])); Factorization($1); $$N$$ = $$976$$ = $$2^{4} \cdot 61$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$999424$$ = $$2^{14} \cdot 61$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-1216$$ = $$-1 \cdot 2^{6} \cdot 19$$ $$I_4$$ = $$64768$$ = $$2^{8} \cdot 11 \cdot 23$$ $$I_6$$ = $$-35181568$$ = $$-1 \cdot 2^{10} \cdot 17 \cdot 43 \cdot 47$$ $$I_{10}$$ = $$4093640704$$ = $$2^{26} \cdot 61$$ $$J_2$$ = $$-152$$ = $$-1 \cdot 2^{3} \cdot 19$$ $$J_4$$ = $$288$$ = $$2^{5} \cdot 3^{2}$$ $$J_6$$ = $$24464$$ = $$2^{4} \cdot 11 \cdot 139$$ $$J_8$$ = $$-950368$$ = $$-1 \cdot 2^{5} \cdot 17 \cdot 1747$$ $$J_{10}$$ = $$999424$$ = $$2^{14} \cdot 61$$ $$g_1$$ = $$-4952198/61$$ $$g_2$$ = $$-61731/61$$ $$g_3$$ = $$551969/976$$
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![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,0,1],C![1,1,0]];

All rational points: (-1 : 0 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -2 : 1), (1 : -1 : 0), (1 : 0 : 1), (1 : 1 : 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: $$0$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: 29 (p = 2), 1 (p = 61) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{29}\Z$$

### 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 $$\Q$$.

The Jacobian of this curve has a rational 29-torsion point. This is the largest known prime factor occurring in the torsion of the Mordell-Weil group of any abelian surface over $\mathbb{Q}$. This example was discovered by Franck Leprévost in 1995: see [MR:1413580].