Properties

 Label 6511.a.6511.1 Conductor 6511 Discriminant 6511 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, 0, 2, 1], R![1, 1, 0, 1]);

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

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

Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$6511$$ = $$17 \cdot 383$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$6511$$ = $$17 \cdot 383$$

G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$584$$ = $$2^{3} \cdot 73$$ $$I_4$$ = $$48676$$ = $$2^{2} \cdot 43 \cdot 283$$ $$I_6$$ = $$5691560$$ = $$2^{3} \cdot 5 \cdot 7 \cdot 20327$$ $$I_{10}$$ = $$26669056$$ = $$2^{12} \cdot 17 \cdot 383$$ $$J_2$$ = $$73$$ = $$73$$ $$J_4$$ = $$-285$$ = $$-1 \cdot 3 \cdot 5 \cdot 19$$ $$J_6$$ = $$1301$$ = $$1301$$ $$J_8$$ = $$3437$$ = $$7 \cdot 491$$ $$J_{10}$$ = $$6511$$ = $$17 \cdot 383$$ $$g_1$$ = $$2073071593/6511$$ $$g_2$$ = $$-110869845/6511$$ $$g_3$$ = $$6933029/6511$$
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![-3,1,2],C![-3,30,2],C![-2,-6,3],C![-2,5,3],C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,0,1]];

Known rational points: (-3 : 1 : 2), (-3 : 30 : 2), (-2 : -6 : 3), (-2 : 5 : 3), (-1 : 0 : 1), (-1 : 1 : 1), (0 : -1 : 1), (0 : 0 : 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: 0.0104912241237 Real period: 27.158650474279064281716897663 Tamagawa numbers: 1 (p = 17), 1 (p = 383) 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 $$\Q$$.