# Properties

 Base field 5.5.65657.1 Label 5.5.65657.1-5.1-b2 Conductor $$(5,a^{2} - a - 2)$$ Conductor norm $$5$$ CM no base-change no Q-curve no Torsion order $$1$$ Rank $$0$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field 5.5.65657.1

Generator $$a$$, with minimal polynomial $$x^{5} - x^{4} - 5 x^{3} + 2 x^{2} + 5 x + 1$$; class number $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^5 - x^4 - 5*x^3 + 2*x^2 + 5*x + 1)

gp: K = nfinit(a^5 - a^4 - 5*a^3 + 2*a^2 + 5*a + 1);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 2, -5, -1, 1]);

## Weierstrass equation

$$y^2+\left(-a^{4}+2a^{3}+4a^{2}-5a-2\right)xy+\left(a^{4}-a^{3}-4a^{2}+2a+3\right)y=x^{3}+\left(-2a^{4}+3a^{3}+9a^{2}-10a-8\right)x^{2}+\left(86a^{4}+73a^{3}-426a^{2}-515a-104\right)x+1231a^{4}+500a^{3}-5653a^{2}-5184a-905$$
sage: E = EllipticCurve(K, [-a^4 + 2*a^3 + 4*a^2 - 5*a - 2, -2*a^4 + 3*a^3 + 9*a^2 - 10*a - 8, a^4 - a^3 - 4*a^2 + 2*a + 3, 86*a^4 + 73*a^3 - 426*a^2 - 515*a - 104, 1231*a^4 + 500*a^3 - 5653*a^2 - 5184*a - 905])

gp: E = ellinit([-a^4 + 2*a^3 + 4*a^2 - 5*a - 2, -2*a^4 + 3*a^3 + 9*a^2 - 10*a - 8, a^4 - a^3 - 4*a^2 + 2*a + 3, 86*a^4 + 73*a^3 - 426*a^2 - 515*a - 104, 1231*a^4 + 500*a^3 - 5653*a^2 - 5184*a - 905],K)

magma: E := ChangeRing(EllipticCurve([-a^4 + 2*a^3 + 4*a^2 - 5*a - 2, -2*a^4 + 3*a^3 + 9*a^2 - 10*a - 8, a^4 - a^3 - 4*a^2 + 2*a + 3, 86*a^4 + 73*a^3 - 426*a^2 - 515*a - 104, 1231*a^4 + 500*a^3 - 5653*a^2 - 5184*a - 905]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(5,a^{2} - a - 2)$$ = $$\left(-a^{2} + a + 2\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$5$$ = $$5$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(5,a + 3,a^{4} - a^{3} - 4 a^{2} + 2 a + 4,-a^{4} + a^{3} + 5 a^{2} - 3 a - 1,-a^{4} + 2 a^{3} + 4 a^{2} - 6 a + 1)$$ = $$\left(-a^{2} + a + 2\right)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$5$$ = $$5$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$-\frac{660212871515640604}{5} a^{4} + \frac{1794053534107468711}{5} a^{3} + \frac{220022592976562427}{5} a^{2} - \frac{1698345910262749894}{5} a - \frac{384439131601574358}{5}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

## Mordell-Weil group

Rank: $$0$$

sage: E.rank()

magma: Rank(E);

sage: gens = E.gens(); gens

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;

sage: E.regulator_of_points(gens)

magma: Regulator(gens);

## Torsion subgroup

Structure: Trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T);

## Local data at primes of bad reduction

sage: E.local_data()

magma: LocalInformation(E);

prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$\left(-a^{2} + a + 2\right)$$ $$5$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ except those listed.

prime Image of Galois Representation
$$3$$ 3B
$$5$$ 5B.1.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3, 5 and 15.
Its isogeny class 5.1-b consists of curves linked by isogenies of degrees dividing 15.

## Base change

This curve is not the base-change of an elliptic curve defined over $$\Q$$. It is not a $$\Q$$-curve.