Properties

 Label 1397.a.1397.1 Conductor 1397 Discriminant 1397 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, 0, 0, -1, 0, 1], R![1]);

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

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

Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$1397$$ = $$11 \cdot 127$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$1397$$ = $$11 \cdot 127$$

G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$96$$ = $$2^{5} \cdot 3$$ $$I_4$$ = $$0$$ = $$0$$ $$I_6$$ = $$-576000$$ = $$-1 \cdot 2^{9} \cdot 3^{2} \cdot 5^{3}$$ $$I_{10}$$ = $$5722112$$ = $$2^{12} \cdot 11 \cdot 127$$ $$J_2$$ = $$12$$ = $$2^{2} \cdot 3$$ $$J_4$$ = $$6$$ = $$2 \cdot 3$$ $$J_6$$ = $$1004$$ = $$2^{2} \cdot 251$$ $$J_8$$ = $$3003$$ = $$3 \cdot 7 \cdot 11 \cdot 13$$ $$J_{10}$$ = $$1397$$ = $$11 \cdot 127$$ $$g_1$$ = $$248832/1397$$ $$g_2$$ = $$10368/1397$$ $$g_3$$ = $$144576/1397$$
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,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-1,1],C![1,0,0],C![1,0,1]];

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

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

Number of rational Weierstrass points: $$1$$

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: 0.00880400573902 Real period: 24.985911685696936774276819761 Tamagawa numbers: 1 (p = 11), 1 (p = 127) 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 $$\Q$$.