Properties

 Label 50000.a.200000.1 Conductor $50000$ Discriminant $200000$ Mordell-Weil group $$\Z \oplus \Z/{5}\Z$$ Sato-Tate group $F_{ac}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\C \times \C$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathsf{CM}$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

Learn more

Show commands: SageMath / Magma

Simplified equation

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

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

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

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

magma: X,pi:= SimplifiedModel(C);

Invariants

 Conductor: $$N$$ $$=$$ $$50000$$ $$=$$ $$2^{4} \cdot 5^{5}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$200000$$ $$=$$ $$2^{6} \cdot 5^{5}$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ $$=$$ $$0$$ $$=$$ $$0$$ $$I_4$$ $$=$$ $$0$$ $$=$$ $$0$$ $$I_6$$ $$=$$ $$0$$ $$=$$ $$0$$ $$I_{10}$$ $$=$$ $$8$$ $$=$$ $$2^{3}$$ $$J_2$$ $$=$$ $$0$$ $$=$$ $$0$$ $$J_4$$ $$=$$ $$0$$ $$=$$ $$0$$ $$J_6$$ $$=$$ $$0$$ $$=$$ $$0$$ $$J_8$$ $$=$$ $$0$$ $$=$$ $$0$$ $$J_{10}$$ $$=$$ $$200000$$ $$=$$ $$2^{6} \cdot 5^{5}$$ $$g_1$$ $$=$$ $$0$$ $$g_2$$ $$=$$ $$0$$ $$g_3$$ $$=$$ $$0$$

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_{10}$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

Rational points

All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : 1 : 1),\, (1 : -2 : 1)$$
All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : 1 : 1),\, (1 : -2 : 1)$$
All points: $$(1 : 0 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : -3 : 1),\, (1 : 3 : 1)$$

magma: [C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,0,0],C![1,1,1]]; // minimal model

magma: [C![0,-1,1],C![0,1,1],C![1,-3,1],C![1,0,0],C![1,3,1]]; // simplified model

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: $$\Z \oplus \Z/{5}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(1 : -2 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2z^3$$ $$0.821051$$ $$\infty$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$5$$
Generator $D_0$ Height Order
$$(1 : -2 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2z^3$$ $$0.821051$$ $$\infty$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$5$$
Generator $D_0$ Height Order
$$(1 : -3 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-3z^3$$ $$0.821051$$ $$\infty$$
$$(0 : 1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$z^3$$ $$0$$ $$5$$

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.821051$$ Real period: $$11.84775$$ Tamagawa product: $$5$$ Torsion order: $$5$$ Leading coefficient: $$1.945523$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$4$$ $$6$$ $$5$$ $$1$$
$$5$$ $$5$$ $$5$$ $$1$$ $$1$$

Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not the maximal image $\GSp(4,\F_\ell)$.

Prime $$\ell$$ mod-$$\ell$$ image Is torsion prime?
$$2$$ 2.36.1 no
$$3$$ 3.1296.1 no

Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $F_{ac}$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{U}(1)\times\mathrm{U}(1)$$

Decomposition of the Jacobian

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

Endomorphisms of the Jacobian

Not of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\zeta_{5})$$ with defining polynomial $$x^{4} - x^{3} + x^{2} - x + 1$$

Not of $$\GL_2$$-type over $$\overline{\Q}$$

Endomorphism ring over $$\overline{\Q}$$:

 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ the maximal order of $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\Q(\zeta_{5})$$ (CM) $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\C \times \C$$

Remainder of the endomorphism lattice by field

Over subfield $$F \simeq$$ $$\Q(\sqrt{5})$$ with generator $$a^{3} - a^{2}$$ with minimal polynomial $$x^{2} - x - 1$$:

 $$\End (J_{F})$$ $$\simeq$$ $$\Z [\frac{1 + \sqrt{5}}{2}]$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{5})$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$
Sato Tate group: F_{ab}
Of $$\GL_2$$-type, simple