Properties

 Label 961.a.961.3 Conductor 961 Discriminant 961 Mordell-Weil group $$\Z/{5}\Z$$ Sato-Tate group $G_{3,3}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{RM}$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type yes

Related objects

Show commands for: Magma / SageMath

Simplified equation

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

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

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, -2, 1, 6, 6, 4, 1]))

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

Invariants

 $$N$$ = $$961$$ = $$31^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); $$\Delta$$ = $$961$$ = $$31^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$520$$ = $$2^{3} \cdot 5 \cdot 13$$ $$I_4$$ = $$6724$$ = $$2^{2} \cdot 41^{2}$$ $$I_6$$ = $$1481672$$ = $$2^{3} \cdot 89 \cdot 2081$$ $$I_{10}$$ = $$3936256$$ = $$2^{12} \cdot 31^{2}$$ $$J_2$$ = $$65$$ = $$5 \cdot 13$$ $$J_4$$ = $$106$$ = $$2 \cdot 53$$ $$J_6$$ = $$-672$$ = $$- 2^{5} \cdot 3 \cdot 7$$ $$J_8$$ = $$-13729$$ = $$- 13729$$ $$J_{10}$$ = $$961$$ = $$31^{2}$$ $$g_1$$ = $$1160290625/961$$ $$g_2$$ = $$29110250/961$$ $$g_3$$ = $$-2839200/961$$

Automorphism group

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

Rational points

magma: [C![1,-1,0],C![1,0,0]];

Points: $$(1 : 0 : 0),\, (1 : -1 : 0)$$

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

Number of rational Weierstrass points: $$0$$

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.

Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));

Group structure: $$\Z/{5}\Z$$

Generator Height Order
$$x^2 + xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$5$$

BSD invariants

 Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$11.23219$$ Tamagawa product: $$1$$ Torsion order: $$5$$ Leading coefficient: $$0.449287$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$31$$ $$2$$ $$2$$ $$1$$ $$( 1 - T )^{2}$$

Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $G_{3,3}$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$

Decomposition of the Jacobian

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

Endomorphisms of the Jacobian

Of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z [\frac{1 + \sqrt{5}}{2}]$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{5})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.