# Properties

 Base field $$\Q(\zeta_{11})^+$$ Label 5.5.14641.1-43.1-b2 Conductor $$(43,-2 a^{4} + a^{3} + 6 a^{2} - 2 a - 1)$$ Conductor norm $$43$$ CM no base-change no Q-curve no Torsion order $$5$$ Rank not available

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Show commands for: Magma / SageMath / Pari/GP

## Base field $$\Q(\zeta_{11})^+$$

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 3, -4, -1, 1]);

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

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

## Weierstrass equation

$$y^2 + \left(a^{4} + a^{3} - 3 a^{2} - 3 a + 1\right) x y + \left(a^{4} + a^{3} - 4 a^{2} - 3 a + 2\right) y = x^{3} + \left(-a^{4} - a^{3} + 4 a^{2} + 2 a - 2\right) x^{2} + \left(-a^{4} - 4 a^{3} + a^{2} + 7 a + 1\right) x - 5 a^{4} - 4 a^{3} + 12 a^{2} + 5 a - 4$$
magma: E := ChangeRing(EllipticCurve([a^4 + a^3 - 3*a^2 - 3*a + 1, -a^4 - a^3 + 4*a^2 + 2*a - 2, a^4 + a^3 - 4*a^2 - 3*a + 2, -a^4 - 4*a^3 + a^2 + 7*a + 1, -5*a^4 - 4*a^3 + 12*a^2 + 5*a - 4]),K);

sage: E = EllipticCurve(K, [a^4 + a^3 - 3*a^2 - 3*a + 1, -a^4 - a^3 + 4*a^2 + 2*a - 2, a^4 + a^3 - 4*a^2 - 3*a + 2, -a^4 - 4*a^3 + a^2 + 7*a + 1, -5*a^4 - 4*a^3 + 12*a^2 + 5*a - 4])

gp: E = ellinit([a^4 + a^3 - 3*a^2 - 3*a + 1, -a^4 - a^3 + 4*a^2 + 2*a - 2, a^4 + a^3 - 4*a^2 - 3*a + 2, -a^4 - 4*a^3 + a^2 + 7*a + 1, -5*a^4 - 4*a^3 + 12*a^2 + 5*a - 4],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(43,-2 a^{4} + a^{3} + 6 a^{2} - 2 a - 1)$$ = $$\left(-a^{3} - a^{2} + 2 a + 3\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$43$$ = $$43$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(147008443,a + 27503793,a^{2} + 69606277,a^{3} - 3 a + 137461936,a^{4} - 4 a^{2} + 65140852)$$ = $$\left(-a^{3} - a^{2} + 2 a + 3\right)^{5}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$147008443$$ = $$43^{5}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$\frac{1668625015062}{147008443} a^{4} - \frac{5087619914002}{147008443} a^{3} - \frac{4613121597415}{147008443} a^{2} + \frac{13348817253799}{147008443} a - \frac{3093462765934}{147008443}$$ magma: jInvariant(E);  sage: E.j_invariant()  gp: E.j $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) magma: HasComplexMultiplication(E);  sage: E.has_cm(), E.cm_discriminant() $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

## Mordell-Weil group

Rank not available.

magma: Rank(E);

sage: E.rank()

Regulator: not available

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

sage: gens = E.gens(); gens

magma: Regulator(gens);

sage: E.regulator_of_points(gens)

## Torsion subgroup

Structure: $$\Z/5\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(2 a^{4} + 2 a^{3} - 5 a^{2} - 3 a + 2 : 3 a^{4} + 3 a^{3} - 5 a^{2} - 3 a : 1\right)$ magma: [piT(P) : P in Generators(T)];  sage: T.gens()  gp: T[3]

## Local data at primes of bad reduction

magma: LocalInformation(E);

sage: E.local_data()

prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$\left(-a^{3} - a^{2} + 2 a + 3\right)$$ $$43$$ $$5$$ $$I_{5}$$ Split multiplicative $$-1$$ $$1$$ $$5$$ $$5$$

## Galois Representations

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

prime Image of Galois Representation
$$5$$ 5B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 5.
Its isogeny class 43.1-b consists of curves linked by isogenies of degree 5.

## Base change

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