# Properties

 Label 62563.a.62563.1 Conductor 62563 Discriminant 62563 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

 $y^2 + (x^3 + x + 1)y = -3x^4 + x^3 + 3x^2 - x$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = -3x^4z^2 + x^3z^3 + 3x^2z^4 - xz^5$ (dehomogenize, simplify) $y^2 = x^6 - 10x^4 + 6x^3 + 13x^2 - 2x + 1$ (minimize, homogenize)

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

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

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, -2, 13, 6, -10, 0, 1]))

## Invariants

 Conductor: $$N$$ = $$62563$$ = $$62563$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$62563$$ = $$62563$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$2056$$ = $$2^{3} \cdot 257$$ $$I_4$$ = $$234340$$ = $$2^{2} \cdot 5 \cdot 11717$$ $$I_6$$ = $$110702184$$ = $$2^{3} \cdot 3 \cdot 1709 \cdot 2699$$ $$I_{10}$$ = $$256258048$$ = $$2^{12} \cdot 62563$$ $$J_2$$ = $$257$$ = $$257$$ $$J_4$$ = $$311$$ = $$311$$ $$J_6$$ = $$21365$$ = $$5 \cdot 4273$$ $$J_8$$ = $$1348521$$ = $$3 \cdot 107 \cdot 4201$$ $$J_{10}$$ = $$62563$$ = $$62563$$ $$g_1$$ = $$1121154893057/62563$$ $$g_2$$ = $$5279098423/62563$$ $$g_3$$ = $$1411136885/62563$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$
$$(-1 : 1 : 1)$$ $$(1 : 0 : 3)$$ $$(1 : -3 : 1)$$ $$(2 : -5 : 1)$$ $$(2 : -6 : 1)$$ $$(3 : -12 : 2)$$
$$(-4 : 15 : 1)$$ $$(3 : -35 : 2)$$ $$(1 : -37 : 3)$$ $$(-4 : 52 : 1)$$ $$(-7 : 4368 : 20)$$ $$(-7 : -9225 : 20)$$

magma: [C![-7,-9225,20],C![-7,4368,20],C![-4,15,1],C![-4,52,1],C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-37,3],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![1,0,3],C![2,-6,1],C![2,-5,1],C![3,-35,2],C![3,-12,2]];

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
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0.392712$$ $$\infty$$
$$D_0 - 2 \cdot(1 : -1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.387128$$ $$\infty$$
$$(0 : -1 : 1) + (2 : -5 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$(-x + 2z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2 - z^3$$ $$0.282553$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$3$$   (upper bound) Mordell-Weil rank: $$3$$ 2-Selmer rank: $$3$$ Regulator: $$0.029057$$ Real period: $$21.40327$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.621921$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$62563$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 120 T + 62563 T^{2} )$$

## 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 $$\Q$$.