# Properties

 Label 18252.a.328536.1 Conductor 18252 Discriminant 328536 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![0, -3, 2, 2, -1], R![1, 1, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -3, 2, 2, -1]), R([1, 1, 0, 1]))

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

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$18252$$ = $$2^{2} \cdot 3^{3} \cdot 13^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$328536$$ = $$2^{3} \cdot 3^{5} \cdot 13^{2}$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$648$$ = $$2^{3} \cdot 3^{4}$$ $$I_4$$ = $$185796$$ = $$2^{2} \cdot 3^{2} \cdot 13 \cdot 397$$ $$I_6$$ = $$-2301624$$ = $$-1 \cdot 2^{3} \cdot 3^{2} \cdot 13 \cdot 2459$$ $$I_{10}$$ = $$1345683456$$ = $$2^{15} \cdot 3^{5} \cdot 13^{2}$$ $$J_2$$ = $$81$$ = $$3^{4}$$ $$J_4$$ = $$-1662$$ = $$-1 \cdot 2 \cdot 3 \cdot 277$$ $$J_6$$ = $$48772$$ = $$2^{2} \cdot 89 \cdot 137$$ $$J_8$$ = $$297072$$ = $$2^{4} \cdot 3^{2} \cdot 2063$$ $$J_{10}$$ = $$328536$$ = $$2^{3} \cdot 3^{5} \cdot 13^{2}$$ $$g_1$$ = $$14348907/1352$$ $$g_2$$ = $$-1817397/676$$ $$g_3$$ = $$329211/338$$
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![-2,3,1],C![-2,6,1],C![-1,-12,2],C![-1,-1,1],C![-1,2,1],C![-1,9,2],C![0,-1,1],C![0,0,1],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![9,-756,2],C![9,-17,2]];

Known rational points: (-2 : 3 : 1), (-2 : 6 : 1), (-1 : -12 : 2), (-1 : -1 : 1), (-1 : 2 : 1), (-1 : 9 : 2), (0 : -1 : 1), (0 : 0 : 1), (1 : -3 : 1), (1 : -1 : 0), (1 : 0 : 0), (1 : 0 : 1), (9 : -756 : 2), (9 : -17 : 2)

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

Number of rational Weierstrass points: $$0$$

## Invariants of the Jacobian:

Analytic rank*: $$2$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$2$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Regulator: 0.041338934918 Real period: 17.410898313879982931291375815 Tamagawa numbers: 3 (p = 2), 3 (p = 3), 1 (p = 13) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{3}\Z$$

### 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 $$\Q$$.