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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].

## Simplified equation

 $y^2 + (x + 1)y = x^6 - 2x^5 + 2x^3 - x^2$ (homogenize, simplify) $y^2 + (xz^2 + z^3)y = x^6 - 2x^5z + 2x^3z^3 - x^2z^4$ (dehomogenize, simplify) $y^2 = 4x^6 - 8x^5 + 8x^3 - 3x^2 + 2x + 1$ (minimize, homogenize)

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

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

sage: X = HyperellipticCurve(R([1, 2, -3, 8, 0, -8, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

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

### G2 invariants

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

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

All points: $$(1 : -1 : 0),\, (1 : 1 : 0),\, (0 : 0 : 1),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (1 : 0 : 1),\, (1 : -2 : 1)$$

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]];

Number of rational Weierstrass points: $$1$$

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

This curve is locally solvable everywhere.

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);

## Mordell-Weil group of the Jacobian

Group structure: $$\Z/{29}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$29$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$12.90036$$ Tamagawa product: $$29$$ Torsion order: $$29$$ Leading coefficient: $$0.444840$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$14$$ $$4$$ $$29$$ $$1 - T$$
$$61$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 4 T + 61 T^{2} )$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{USp}(4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{USp}(4)$$

## Decomposition of the Jacobian

Simple over $$\overline{\Q}$$

## Endomorphisms of the Jacobian

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].