Properties

 Label 7403.a.7403.1 Conductor 7403 Discriminant -7403 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, -2, 1, 1], R![1]);

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

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

Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$7403$$ = $$11 \cdot 673$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-7403$$ = $$-1 \cdot 11 \cdot 673$$

G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$1280$$ = $$2^{8} \cdot 5$$ $$I_4$$ = $$35584$$ = $$2^{8} \cdot 139$$ $$I_6$$ = $$14821376$$ = $$2^{11} \cdot 7237$$ $$I_{10}$$ = $$-30322688$$ = $$-1 \cdot 2^{12} \cdot 11 \cdot 673$$ $$J_2$$ = $$160$$ = $$2^{5} \cdot 5$$ $$J_4$$ = $$696$$ = $$2^{3} \cdot 3 \cdot 29$$ $$J_6$$ = $$224$$ = $$2^{5} \cdot 7$$ $$J_8$$ = $$-112144$$ = $$-1 \cdot 2^{4} \cdot 43 \cdot 163$$ $$J_{10}$$ = $$-7403$$ = $$-1 \cdot 11 \cdot 673$$ $$g_1$$ = $$-104857600000/7403$$ $$g_2$$ = $$-2850816000/7403$$ $$g_3$$ = $$-5734400/7403$$
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,-399,9],C![-2,-330,9],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],C![2,-6,1],C![2,5,1]];

Known rational points: (-2 : -399 : 9), (-2 : -330 : 9), (-1 : -1 : 1), (-1 : 0 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -1 : 1), (1 : 0 : 0), (1 : 0 : 1), (2 : -6 : 1), (2 : 5 : 1)

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

Number of rational Weierstrass points: $$1$$

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: 0.0153381034338 Real period: 23.405457129865924662429560935 Tamagawa numbers: 1 (p = 11), 1 (p = 673) 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 $$\Q$$.