# Properties

 Label 62233.a.62233.1 Conductor 62233 Discriminant -62233 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

# Learn more about

Show commands for: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + x + 1)y = -3x^4 + x^3 + 5x^2 - 5x$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = -3x^4z^2 + x^3z^3 + 5x^2z^4 - 5xz^5$ (dehomogenize, simplify) $y^2 = x^6 - 10x^4 + 6x^3 + 21x^2 - 18x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -5, 5, 1, -3]), R([1, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -5, 5, 1, -3], R![1, 1, 0, 1]);

sage: X = HyperellipticCurve(R([1, -18, 21, 6, -10, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$62233$$ $$=$$ $$62233$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-62233$$ $$=$$ $$-62233$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$3336$$ $$=$$ $$2^{3} \cdot 3 \cdot 139$$ $$I_4$$ $$=$$ $$257892$$ $$=$$ $$2^{2} \cdot 3 \cdot 21491$$ $$I_6$$ $$=$$ $$244275816$$ $$=$$ $$2^{3} \cdot 3 \cdot 29 \cdot 350971$$ $$I_{10}$$ $$=$$ $$-254906368$$ $$=$$ $$- 2^{12} \cdot 62233$$ $$J_2$$ $$=$$ $$417$$ $$=$$ $$3 \cdot 139$$ $$J_4$$ $$=$$ $$4559$$ $$=$$ $$47 \cdot 97$$ $$J_6$$ $$=$$ $$54933$$ $$=$$ $$3 \cdot 18311$$ $$J_8$$ $$=$$ $$530645$$ $$=$$ $$5 \cdot 106129$$ $$J_{10}$$ $$=$$ $$-62233$$ $$=$$ $$-62233$$ $$g_1$$ $$=$$ $$-12608989261857/62233$$ $$g_2$$ $$=$$ $$-330580899567/62233$$ $$g_3$$ $$=$$ $$-9552244437/62233$$

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

Known points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : -1 : 1)$$ $$(-1 : -2 : 1)$$
$$(1 : -2 : 1)$$ $$(-1 : 3 : 1)$$ $$(2 : -5 : 1)$$ $$(2 : -6 : 1)$$ $$(-3 : 14 : 1)$$ $$(-3 : 15 : 1)$$
$$(-4 : -27 : 3)$$ $$(5 : -53 : 2)$$ $$(-4 : 100 : 3)$$ $$(5 : -100 : 2)$$ $$(11 : -891 : 6)$$ $$(11 : -1052 : 6)$$
$$(73 : -251641 : 52)$$ $$(73 : -475376 : 52)$$

magma: [C![-4,-27,3],C![-4,100,3],C![-3,14,1],C![-3,15,1],C![-1,-2,1],C![-1,3,1],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![2,-6,1],C![2,-5,1],C![5,-100,2],C![5,-53,2],C![11,-1052,6],C![11,-891,6],C![73,-475376,52],C![73,-251641,52]];

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
$$(0 : 0 : 1) + (2 : -6 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$(-x + 2z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-3xz^2$$ $$0.562559$$ $$\infty$$
$$(1 : -2 : 1) - (1 : -1 : 0)$$ $$z (x - z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2z^3$$ $$0.285327$$ $$\infty$$
$$D_0 - 2 \cdot(1 : -1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.212996$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$3$$   (upper bound) Mordell-Weil rank: $$3$$ 2-Selmer rank: $$3$$ Regulator: $$0.026759$$ Real period: $$21.72187$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.581268$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$62233$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 146 T + 62233 T^{2} )$$

## 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 $$\Q$$.