# Properties

 Label 50625.a.253125.1 Conductor $50625$ Discriminant $253125$ Mordell-Weil group $$\Z$$ 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

# Related objects

Show commands: SageMath / Magma

## Simplified equation

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

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

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

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

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$50625$$ $$=$$ $$3^{4} \cdot 5^{4}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$253125$$ $$=$$ $$3^{4} \cdot 5^{5}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$0$$ $$=$$ $$0$$ $$I_4$$ $$=$$ $$0$$ $$=$$ $$0$$ $$I_6$$ $$=$$ $$0$$ $$=$$ $$0$$ $$I_{10}$$ $$=$$ $$324$$ $$=$$ $$2^{2} \cdot 3^{4}$$ $$J_2$$ $$=$$ $$0$$ $$=$$ $$0$$ $$J_4$$ $$=$$ $$0$$ $$=$$ $$0$$ $$J_6$$ $$=$$ $$0$$ $$=$$ $$0$$ $$J_8$$ $$=$$ $$0$$ $$=$$ $$0$$ $$J_{10}$$ $$=$$ $$253125$$ $$=$$ $$3^{4} \cdot 5^{5}$$ $$g_1$$ $$=$$ $$0$$ $$g_2$$ $$=$$ $$0$$ $$g_3$$ $$=$$ $$0$$

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

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

magma: [C![1,-1,1],C![1,0,0],C![1,1,1]]; // 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: $$\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2$$ $$0.189560$$ $$\infty$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2$$ $$0.189560$$ $$\infty$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$2xz^2 + z^3$$ $$0.189560$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.189560$$ Real period: $$5.175122$$ Tamagawa product: $$2$$ Torsion order: $$1$$ Leading coefficient: $$1.961993$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$3$$ $$4$$ $$4$$ $$1$$ $$1$$
$$5$$ $$4$$ $$5$$ $$2$$ $$1$$

## 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
$$3$$ 3.6480.5 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