# Properties

 Label 61553.a.61553.1 Conductor 61553 Discriminant -61553 Mordell-Weil group $$\Z \times \Z \times \Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands for: Magma / SageMath

## Simplified equation

 $y^2 + (x^3 + x + 1)y = -x^4 - 3x^3 + x^2 + x$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = -x^4z^2 - 3x^3z^3 + x^2z^4 + xz^5$ (dehomogenize, simplify) $y^2 = x^6 - 2x^4 - 10x^3 + 5x^2 + 6x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 1, -3, -1], R![1, 1, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 1, -3, -1]), R([1, 1, 0, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, 6, 5, -10, -2, 0, 1]))

## Invariants

 Conductor: $$N$$ = $$61553$$ = $$61553$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-61553$$ = $$- 61553$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$520$$ = $$2^{3} \cdot 5 \cdot 13$$ $$I_4$$ = $$80356$$ = $$2^{2} \cdot 20089$$ $$I_6$$ = $$2059752$$ = $$2^{3} \cdot 3 \cdot 19 \cdot 4517$$ $$I_{10}$$ = $$-252121088$$ = $$- 2^{12} \cdot 61553$$ $$J_2$$ = $$65$$ = $$5 \cdot 13$$ $$J_4$$ = $$-661$$ = $$- 661$$ $$J_6$$ = $$12173$$ = $$7 \cdot 37 \cdot 47$$ $$J_8$$ = $$88581$$ = $$3 \cdot 29527$$ $$J_{10}$$ = $$-61553$$ = $$- 61553$$ $$g_1$$ = $$-1160290625/61553$$ $$g_2$$ = $$181527125/61553$$ $$g_3$$ = $$-51430925/61553$$

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

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

## 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

Known points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : -1 : 1)$$ $$(1 : -1 : 1)$$
$$(-2 : -1 : 1)$$ $$(-1 : 1 : 2)$$ $$(-1 : 2 : 1)$$ $$(1 : -2 : 1)$$ $$(-1 : -4 : 2)$$ $$(3 : -6 : 1)$$
$$(-2 : 10 : 1)$$ $$(-2 : 18 : 3)$$ $$(-2 : -19 : 3)$$ $$(3 : -25 : 1)$$ $$(24 : 266217 : 119)$$ $$(24 : -2305064 : 119)$$

magma: [C![-2,-19,3],C![-2,-1,1],C![-2,10,1],C![-2,18,3],C![-1,-4,2],C![-1,-1,1],C![-1,1,2],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![3,-25,1],C![3,-6,1],C![24,-2305064,119],C![24,266217,119]];

Number of rational Weierstrass points: $$0$$

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: $$\Z \times \Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : -1 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2$$ $$0.597353$$ $$\infty$$
$$(0 : 0 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.362615$$ $$\infty$$
$$(0 : -1 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.148079$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$3$$   (upper bound) Mordell-Weil rank: $$3$$ 2-Selmer rank: $$3$$ Regulator: $$0.029513$$ Real period: $$20.66583$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.609926$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$61553$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 246 T + 61553 T^{2} )$$

## 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 $$\Q$$.