# Properties

 Label 1499.a.1499.1 Conductor 1499 Discriminant 1499 Mordell-Weil group $$\Z$$ 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

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

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

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

magma: X,pi:= SimplifiedModel(C);

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

## Invariants

 Conductor: $$N$$ = $$1499$$ = $$1499$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$1499$$ = $$1499$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$424$$ = $$2^{3} \cdot 53$$ $$I_4$$ = $$5668$$ = $$2^{2} \cdot 13 \cdot 109$$ $$I_6$$ = $$401960$$ = $$2^{3} \cdot 5 \cdot 13 \cdot 773$$ $$I_{10}$$ = $$6139904$$ = $$2^{12} \cdot 1499$$ $$J_2$$ = $$53$$ = $$53$$ $$J_4$$ = $$58$$ = $$2 \cdot 29$$ $$J_6$$ = $$516$$ = $$2^{2} \cdot 3 \cdot 43$$ $$J_8$$ = $$5996$$ = $$2^{2} \cdot 1499$$ $$J_{10}$$ = $$1499$$ = $$1499$$ $$g_1$$ = $$418195493/1499$$ $$g_2$$ = $$8634866/1499$$ $$g_3$$ = $$1449444/1499$$

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

All points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -1 : 1),\, (1 : -3 : 2),\, (1 : -6 : 2)$$

magma: [C![0,-1,1],C![0,0,1],C![1,-6,2],C![1,-3,2],C![1,-1,0],C![1,-1,1],C![1,0,0]];

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: $$\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : 0 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$(-x + z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0.009040$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.009040$$ Real period: $$25.97726$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.234846$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$1499$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 36 T + 1499 T^{2} )$$

## 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 $$\Q$$.