# Properties

 Base field $$\Q(\zeta_{11})^+$$ Label 5.5.14641.1-121.1-a2 Conductor $$(11,a^{4} + a^{3} - 3 a^{2} - 3 a - 2)$$ Conductor norm $$121$$ CM no base-change yes: 121.c2 Q-curve yes Torsion order $$1$$ Rank not available

# Related objects

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

## Weierstrass equation

$$y^2 + \left(a^{2} - 1\right) x y + \left(a^{4} + a^{3} - 4 a^{2} - 2 a + 3\right) y = x^{3} + \left(-a^{3} + a^{2} + 3 a - 3\right) x^{2} + \left(2 a^{4} - 5 a^{3} - 6 a^{2} + 13 a - 1\right) x + 16 a^{4} - 31 a^{3} - 40 a^{2} + 86 a - 22$$
magma: E := ChangeRing(EllipticCurve([a^2 - 1, -a^3 + a^2 + 3*a - 3, a^4 + a^3 - 4*a^2 - 2*a + 3, 2*a^4 - 5*a^3 - 6*a^2 + 13*a - 1, 16*a^4 - 31*a^3 - 40*a^2 + 86*a - 22]),K);

sage: E = EllipticCurve(K, [a^2 - 1, -a^3 + a^2 + 3*a - 3, a^4 + a^3 - 4*a^2 - 2*a + 3, 2*a^4 - 5*a^3 - 6*a^2 + 13*a - 1, 16*a^4 - 31*a^3 - 40*a^2 + 86*a - 22])

gp (2.8): E = ellinit([a^2 - 1, -a^3 + a^2 + 3*a - 3, a^4 + a^3 - 4*a^2 - 2*a + 3, 2*a^4 - 5*a^3 - 6*a^2 + 13*a - 1, 16*a^4 - 31*a^3 - 40*a^2 + 86*a - 22],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(11,a^{4} + a^{3} - 3 a^{2} - 3 a - 2)$$ = $$\left(a^{2} + a - 2\right)^{2}$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$121$$ = $$11^{2}$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(121,121 a,121 a^{2} - 242,11 a^{3} + 66 a^{2} + 11 a - 33,11 a^{4} - 22 a^{2} + 22 a + 77)$$ = $$\left(a^{2} + a - 2\right)^{8}$$ magma: Discriminant(E);  sage: E.discriminant()  gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$214358881$$ = $$11^{8}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp (2.8): norm(E.disc) $$j$$ = $$-121$$ magma: jInvariant(E);  sage: E.j_invariant()  gp (2.8): 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()

magma: Generators(E); // includes torsion

sage: E.gens()

Regulator: not available

magma: Regulator(Generators(E));

sage: E.regulator_of_points(E.gens())

## Torsion subgroup

Structure: Trivial magma: TorsionSubgroup(E);  sage: E.torsion_subgroup().gens()  gp (2.8): elltors(E)[2] magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp (2.8): elltors(E)[1]

## 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$$(j)_{-}$$
$$\left(a^{2} + a - 2\right)$$ $$11$$ $$1$$ $$IV^*$$ Additive $$-1$$ $$2$$ $$8$$ $$0$$

## Galois Representations

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

prime Image of Galois Representation
$$11$$ 11B.1.10[5]

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 11.
Its isogeny class 121.1-a consists of curves linked by isogenies of degree 11.

## Base change

This curve is the base-change of elliptic curves 121.c2, defined over $$\Q$$, so it is also a $$\Q$$-curve.