# Properties

 Label 54983.a.54983.1 Conductor 54983 Discriminant -54983 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

 $y^2 + y = x^6 - x^5 - 3x^4 + x^3 + 3x^2 + x$ (homogenize, simplify) $y^2 + z^3y = x^6 - x^5z - 3x^4z^2 + x^3z^3 + 3x^2z^4 + xz^5$ (dehomogenize, simplify) $y^2 = 4x^6 - 4x^5 - 12x^4 + 4x^3 + 12x^2 + 4x + 1$ (minimize, homogenize)

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

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

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, 4, 12, 4, -12, -4, 4]))

## Invariants

 Conductor: $$N$$ = $$54983$$ = $$54983$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-54983$$ = $$- 54983$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$800$$ = $$2^{5} \cdot 5^{2}$$ $$I_4$$ = $$337216$$ = $$2^{6} \cdot 11 \cdot 479$$ $$I_6$$ = $$41495552$$ = $$2^{10} \cdot 7^{2} \cdot 827$$ $$I_{10}$$ = $$-225210368$$ = $$- 2^{12} \cdot 54983$$ $$J_2$$ = $$100$$ = $$2^{2} \cdot 5^{2}$$ $$J_4$$ = $$-3096$$ = $$- 2^{3} \cdot 3^{2} \cdot 43$$ $$J_6$$ = $$27848$$ = $$2^{3} \cdot 59^{2}$$ $$J_8$$ = $$-1700104$$ = $$- 2^{3} \cdot 7^{2} \cdot 4337$$ $$J_{10}$$ = $$-54983$$ = $$- 54983$$ $$g_1$$ = $$-10000000000/54983$$ $$g_2$$ = $$3096000000/54983$$ $$g_3$$ = $$-278480000/54983$$

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
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : -1 : 1)$$
$$(1 : 1 : 1)$$ $$(1 : -2 : 1)$$ $$(2 : 2 : 1)$$ $$(-1 : -2 : 3)$$ $$(2 : -3 : 1)$$ $$(-4 : -4 : 5)$$
$$(-1 : -25 : 3)$$ $$(-4 : -121 : 5)$$ $$(5 : 3721 : 24)$$ $$(5 : -17545 : 24)$$ $$(54 : 129735 : 11)$$ $$(54 : -131066 : 11)$$

magma: [C![-4,-121,5],C![-4,-4,5],C![-1,-25,3],C![-1,-2,3],C![-1,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,1,0],C![1,1,1],C![2,-3,1],C![2,2,1],C![5,-17545,24],C![5,3721,24],C![54,-131066,11],C![54,129735,11]];

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
$$(1 : -2 : 1) - (1 : -1 : 0)$$ $$z (x - z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - 3z^3$$ $$0.628570$$ $$\infty$$
$$(0 : -1 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - z^3$$ $$0.432242$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2 - xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.145844$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$3$$   (upper bound) Mordell-Weil rank: $$3$$ 2-Selmer rank: $$3$$ Regulator: $$0.032929$$ Real period: $$21.51843$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.708582$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$54983$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 176 T + 54983 T^{2} )$$

## 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 $$\Q$$.