# Properties

 Label 65167.b.65167.1 Conductor 65167 Discriminant 65167 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, 1, -2, -5, -1, 1], R![1, 1, 0, 1]);

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

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

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$65167$$ = $$65167$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$65167$$ = $$65167$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$2440$$ = $$2^{3} \cdot 5 \cdot 61$$ $$I_4$$ = $$136036$$ = $$2^{2} \cdot 71 \cdot 479$$ $$I_6$$ = $$103702632$$ = $$2^{3} \cdot 3 \cdot 11 \cdot 73 \cdot 5381$$ $$I_{10}$$ = $$266924032$$ = $$2^{12} \cdot 65167$$ $$J_2$$ = $$305$$ = $$5 \cdot 61$$ $$J_4$$ = $$2459$$ = $$2459$$ $$J_6$$ = $$5693$$ = $$5693$$ $$J_8$$ = $$-1077579$$ = $$-1 \cdot 3^{2} \cdot 17 \cdot 7043$$ $$J_{10}$$ = $$65167$$ = $$65167$$ $$g_1$$ = $$2639363440625/65167$$ $$g_2$$ = $$69768284875/65167$$ $$g_3$$ = $$529591325/65167$$
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![-5,30,3],C![-5,113,3],C![-3,14,1],C![-3,15,1],C![-2,3,1],C![-2,6,1],C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-7,2],C![1,-6,2],C![1,-1,0],C![1,0,0],C![2,-6,1],C![2,-5,1]];

Known rational points: (-5 : 30 : 3), (-5 : 113 : 3), (-3 : 14 : 1), (-3 : 15 : 1), (-2 : 3 : 1), (-2 : 6 : 1), (-1 : 0 : 1), (-1 : 1 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -7 : 2), (1 : -6 : 2), (1 : -1 : 0), (1 : 0 : 0), (2 : -6 : 1), (2 : -5 : 1)

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

Number of rational Weierstrass points: $$0$$

## Invariants of the Jacobian:

Analytic rank*: $$3$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$3$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Regulator: 0.0315730779935 Real period: 19.721382025078483163202399814 Tamagawa numbers: 1 (p = 65167) 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 $$\Q$$.