Properties

 Label 100069.a.100069.1 Conductor 100069 Discriminant 100069 Mordell-Weil group $$\Z \times \Z$$ 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: Magma / SageMath

Simplified equation

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

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

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

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([-4, 4, 0, 8, 4, 4, 1]))

Invariants

 Conductor: $$N$$ = $$100069$$ = $$100069$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$100069$$ = $$100069$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ = $$1984$$ = $$2^{6} \cdot 31$$ $$I_4$$ = $$86848$$ = $$2^{6} \cdot 23 \cdot 59$$ $$I_6$$ = $$60326400$$ = $$2^{9} \cdot 3 \cdot 5^{2} \cdot 1571$$ $$I_{10}$$ = $$409882624$$ = $$2^{12} \cdot 100069$$ $$J_2$$ = $$248$$ = $$2^{3} \cdot 31$$ $$J_4$$ = $$1658$$ = $$2 \cdot 829$$ $$J_6$$ = $$-7104$$ = $$- 2^{6} \cdot 3 \cdot 37$$ $$J_8$$ = $$-1127689$$ = $$- 563 \cdot 2003$$ $$J_{10}$$ = $$100069$$ = $$100069$$ $$g_1$$ = $$938120019968/100069$$ $$g_2$$ = $$25289460736/100069$$ $$g_3$$ = $$-436924416/100069$$

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

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

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

Known points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (2 : 5 : 1),\, (2 : -13 : 1)$$

magma: [C![1,-1,0],C![1,0,0],C![2,-13,1],C![2,5,1]];

Number of rational Weierstrass points: $$0$$

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: $$\Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.467543$$ $$\infty$$
$$D_0 - 2 \cdot(1 : -1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^2z$$ $$0.492648$$ $$\infty$$

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.210375$$ Real period: $$7.311107$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$1.538079$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$100069$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 6 T + 100069 T^{2} )$$

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 $$\Q$$.