Properties

 Label 832.a.832.1 Conductor 832 Discriminant -832 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

 $y^2 + (x^3 + x)y = x^5 - x^3 + x^2 + 2x - 1$ (homogenize, simplify) $y^2 + (x^3 + xz^2)y = x^5z - x^3z^3 + x^2z^4 + 2xz^5 - z^6$ (dehomogenize, simplify) $y^2 = x^6 + 4x^5 + 2x^4 - 4x^3 + 5x^2 + 8x - 4$ (minimize, homogenize)

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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 2, 1, -1, 0, 1], R![0, 1, 0, 1]);

sage: X = HyperellipticCurve(R([-4, 8, 5, -4, 2, 4, 1]))

magma: X,pi:= SimplifiedModel(C);

Invariants

 Conductor: $$N$$ $$=$$ $$832$$ $$=$$ $$2^{6} \cdot 13$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-832$$ $$=$$ $$- 2^{6} \cdot 13$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ $$=$$ $$2176$$ $$=$$ $$2^{7} \cdot 17$$ $$I_4$$ $$=$$ $$-8384$$ $$=$$ $$- 2^{6} \cdot 131$$ $$I_6$$ $$=$$ $$-6349824$$ $$=$$ $$- 2^{10} \cdot 3^{2} \cdot 13 \cdot 53$$ $$I_{10}$$ $$=$$ $$-3407872$$ $$=$$ $$- 2^{18} \cdot 13$$ $$J_2$$ $$=$$ $$272$$ $$=$$ $$2^{4} \cdot 17$$ $$J_4$$ $$=$$ $$3170$$ $$=$$ $$2 \cdot 5 \cdot 317$$ $$J_6$$ $$=$$ $$51008$$ $$=$$ $$2^{6} \cdot 797$$ $$J_8$$ $$=$$ $$956319$$ $$=$$ $$3 \cdot 7 \cdot 13 \cdot 31 \cdot 113$$ $$J_{10}$$ $$=$$ $$-832$$ $$=$$ $$- 2^{6} \cdot 13$$ $$g_1$$ $$=$$ $$-23262937088/13$$ $$g_2$$ $$=$$ $$-996749440/13$$ $$g_3$$ $$=$$ $$-58965248/13$$

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: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (-2 : 5 : 1)$$

magma: [C![-2,5,1],C![1,-1,0],C![1,0,0]];

Number of rational Weierstrass points: $$1$$

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
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 + xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 + z^3$$ $$0$$ $$8$$

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$21.14821$$ Tamagawa product: $$1$$ Torsion order: $$8$$ Leading coefficient: $$0.330440$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$6$$ $$6$$ $$1$$ $$1 + T + 2 T^{2}$$
$$13$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 2 T + 13 T^{2} )$$

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 $$\Q$$.