# Properties

 Label 6081.b.164187.1 Conductor 6081 Discriminant 164187 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![2, 5, 0, -5, -2], R![1, 0, 1, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 5, 0, -5, -2]), R([1, 0, 1, 1]))

$y^2 + (x^3 + x^2 + 1)y = -2x^4 - 5x^3 + 5x + 2$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$6081$$ = $$3 \cdot 2027$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$164187$$ = $$3^{4} \cdot 2027$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$1608$$ = $$2^{3} \cdot 3 \cdot 67$$ $$I_4$$ = $$250788$$ = $$2^{2} \cdot 3 \cdot 20899$$ $$I_6$$ = $$92483880$$ = $$2^{3} \cdot 3 \cdot 5 \cdot 797 \cdot 967$$ $$I_{10}$$ = $$672509952$$ = $$2^{12} \cdot 3^{4} \cdot 2027$$ $$J_2$$ = $$201$$ = $$3 \cdot 67$$ $$J_4$$ = $$-929$$ = $$-1 \cdot 929$$ $$J_6$$ = $$4093$$ = $$4093$$ $$J_8$$ = $$-10087$$ = $$-1 \cdot 7 \cdot 11 \cdot 131$$ $$J_{10}$$ = $$164187$$ = $$3^{4} \cdot 2027$$ $$g_1$$ = $$4050375321/2027$$ $$g_2$$ = $$-279408827/6081$$ $$g_3$$ = $$18373477/18243$$
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![-13,112,6],C![-13,855,6],C![-4,5,1],C![-4,42,1],C![-2,0,1],C![-2,3,1],C![-1,-9,2],C![-1,-1,1],C![-1,0,1],C![-1,0,2],C![0,-2,1],C![0,1,1],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,0,1]];

Known rational points: (-13 : 112 : 6), (-13 : 855 : 6), (-4 : 5 : 1), (-4 : 42 : 1), (-2 : 0 : 1), (-2 : 3 : 1), (-1 : -9 : 2), (-1 : -1 : 1), (-1 : 0 : 1), (-1 : 0 : 2), (0 : -2 : 1), (0 : 1 : 1), (1 : -3 : 1), (1 : -1 : 0), (1 : 0 : 0), (1 : 0 : 1)

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: 0.00350425226552 Real period: 20.379078635919441225463040289 Tamagawa numbers: 4 (p = 3), 1 (p = 2027) 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 $$\Q$$.