# Properties

 Label 39497.a.39497.1 Conductor 39497 Discriminant 39497 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, 1, -1, 1, -1, 1], R![1]);

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

$y^2 + y = x^6 - x^5 + x^4 - x^3 + x^2 - x$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$39497$$ = $$127 \cdot 311$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$39497$$ = $$127 \cdot 311$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-480$$ = $$-1 \cdot 2^{5} \cdot 3 \cdot 5$$ $$I_4$$ = $$18240$$ = $$2^{6} \cdot 3 \cdot 5 \cdot 19$$ $$I_6$$ = $$-1683456$$ = $$-1 \cdot 2^{12} \cdot 3 \cdot 137$$ $$I_{10}$$ = $$161779712$$ = $$2^{12} \cdot 127 \cdot 311$$ $$J_2$$ = $$-60$$ = $$-1 \cdot 2^{2} \cdot 3 \cdot 5$$ $$J_4$$ = $$-40$$ = $$-1 \cdot 2^{3} \cdot 5$$ $$J_6$$ = $$-744$$ = $$-1 \cdot 2^{3} \cdot 3 \cdot 31$$ $$J_8$$ = $$10760$$ = $$2^{3} \cdot 5 \cdot 269$$ $$J_{10}$$ = $$39497$$ = $$127 \cdot 311$$ $$g_1$$ = $$-777600000/39497$$ $$g_2$$ = $$8640000/39497$$ $$g_3$$ = $$-2678400/39497$$
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,-182,5],C![-2,57,5],C![-1,-3,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-14,3],C![1,-13,3],C![1,-1,0],C![1,-1,1],C![1,0,1],C![1,1,0],C![2,-7,1],C![2,6,1]];

Known rational points: (-2 : -182 : 5), (-2 : 57 : 5), (-1 : -3 : 1), (-1 : 2 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -14 : 3), (1 : -13 : 3), (1 : -1 : 0), (1 : -1 : 1), (1 : 0 : 1), (1 : 1 : 0), (2 : -7 : 1), (2 : 6 : 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: 0.0394019851843 Real period: 14.578300309156177036554972858 Tamagawa numbers: 1 (p = 127), 1 (p = 311) 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 $$\Q$$.