Properties

 Label 691.a.691.1 Conductor 691 Discriminant -691 Mordell-Weil group $$\Z/{8}\Z$$ 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: SageMath / Magma

Simplified equation

 $y^2 + (x + 1)y = x^5 - x^3 - x^2$ (homogenize, simplify) $y^2 + (xz^2 + z^3)y = x^5z - x^3z^3 - x^2z^4$ (dehomogenize, simplify) $y^2 = 4x^5 - 4x^3 - 3x^2 + 2x + 1$ (minimize, homogenize)

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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -1, -1, 0, 1], R![1, 1]);

sage: X = HyperellipticCurve(R([1, 2, -3, -4, 0, 4]))

magma: X,pi:= SimplifiedModel(C);

Invariants

 Conductor: $$N$$ $$=$$ $$691$$ $$=$$ $$691$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-691$$ $$=$$ $$-691$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ $$=$$ $$416$$ $$=$$ $$2^{5} \cdot 13$$ $$I_4$$ $$=$$ $$-13184$$ $$=$$ $$- 2^{7} \cdot 103$$ $$I_6$$ $$=$$ $$-1301312$$ $$=$$ $$- 2^{6} \cdot 20333$$ $$I_{10}$$ $$=$$ $$-2830336$$ $$=$$ $$- 2^{12} \cdot 691$$ $$J_2$$ $$=$$ $$52$$ $$=$$ $$2^{2} \cdot 13$$ $$J_4$$ $$=$$ $$250$$ $$=$$ $$2 \cdot 5^{3}$$ $$J_6$$ $$=$$ $$601$$ $$=$$ $$601$$ $$J_8$$ $$=$$ $$-7812$$ $$=$$ $$- 2^{2} \cdot 3^{2} \cdot 7 \cdot 31$$ $$J_{10}$$ $$=$$ $$-691$$ $$=$$ $$-691$$ $$g_1$$ $$=$$ $$-380204032/691$$ $$g_2$$ $$=$$ $$-35152000/691$$ $$g_3$$ $$=$$ $$-1625104/691$$

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

Rational points

All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -1 : 1)$$

magma: [C![0,-1,1],C![0,0,1],C![1,-1,1],C![1,0,0]];

Number of rational Weierstrass points: $$2$$

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

This curve is locally solvable everywhere.

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);

Mordell-Weil group of the Jacobian

Group structure: $$\Z/{8}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : -1 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)$$ $$(-x + z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$8$$

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$18.81256$$ Tamagawa product: $$1$$ Torsion order: $$8$$ Leading coefficient: $$0.293946$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$691$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 28 T + 691 T^{2} )$$

Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{USp}(4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{USp}(4)$$

Decomposition of the Jacobian

Simple over $$\overline{\Q}$$

Endomorphisms of the Jacobian

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 $$\Q$$.