# Properties

 Label 1497.a.1497.1 Conductor 1497 Discriminant 1497 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

## Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 1, -2, 1, -1], R![1, 1, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 1, -2, 1, -1]), R([1, 1, 0, 1]))

$y^2 + (x^3 + x + 1)y = -x^4 + x^3 - 2x^2 + x - 1$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$1497$$ = $$3 \cdot 499$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$1497$$ = $$3 \cdot 499$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$712$$ = $$2^{3} \cdot 89$$ $$I_4$$ = $$8260$$ = $$2^{2} \cdot 5 \cdot 7 \cdot 59$$ $$I_6$$ = $$2199752$$ = $$2^{3} \cdot 131 \cdot 2099$$ $$I_{10}$$ = $$6131712$$ = $$2^{12} \cdot 3 \cdot 499$$ $$J_2$$ = $$89$$ = $$89$$ $$J_4$$ = $$244$$ = $$2^{2} \cdot 61$$ $$J_6$$ = $$-60$$ = $$-1 \cdot 2^{2} \cdot 3 \cdot 5$$ $$J_8$$ = $$-16219$$ = $$-1 \cdot 7^{2} \cdot 331$$ $$J_{10}$$ = $$1497$$ = $$3 \cdot 499$$ $$g_1$$ = $$5584059449/1497$$ $$g_2$$ = $$172012436/1497$$ $$g_3$$ = $$-158420/499$$
Alternative geometric invariants: G2

## Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$ $$C_2$$ (GAP id : [2,1]) magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $$C_2$$ (GAP id : [2,1])

## Rational points

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);

This curve is locally solvable everywhere.

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

All rational points: (1 : -2 : 1), (1 : -1 : 0), (1 : -1 : 1), (1 : 0 : 0)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

Number of rational Weierstrass points: $$0$$

## Invariants of the Jacobian:

Analytic rank: $$1$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$1$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Regulator: 0.0217721370501 Real period: 12.141841992069544964622865474 Tamagawa numbers: 1 (p = 3), 1 (p = 499) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\mathrm{trivial}$$

### Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{USp}(4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{USp}(4)$$

### Decomposition

Simple over $$\overline{\Q}$$

### Endomorphisms

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 $$\Q$$.