Properties

 Base field $$\Q(\zeta_{11})^+$$ Label 5.5.14641.1-121.1-b2 Conductor $$(11,a^{4} + a^{3} - 3 a^{2} - 3 a - 2)$$ Conductor norm $$121$$ CM yes ($$-11$$) base-change yes: 121.b1 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

$$y^2 + \left(a^{4} - 4 a^{2} + 3\right) y = x^{3} + \left(-a^{4} + 4 a^{2} + a - 2\right) x^{2} + \left(-4 a^{4} + 9 a^{3} - a^{2} - 3 a\right) x - 13 a^{4} + 34 a^{3} - 6 a^{2} - 25 a + 5$$
magma: E := ChangeRing(EllipticCurve([0, -a^4 + 4*a^2 + a - 2, a^4 - 4*a^2 + 3, -4*a^4 + 9*a^3 - a^2 - 3*a, -13*a^4 + 34*a^3 - 6*a^2 - 25*a + 5]),K);
sage: E = EllipticCurve(K, [0, -a^4 + 4*a^2 + a - 2, a^4 - 4*a^2 + 3, -4*a^4 + 9*a^3 - a^2 - 3*a, -13*a^4 + 34*a^3 - 6*a^2 - 25*a + 5])
gp (2.8): E = ellinit([0, -a^4 + 4*a^2 + a - 2, a^4 - 4*a^2 + 3, -4*a^4 + 9*a^3 - a^2 - 3*a, -13*a^4 + 34*a^3 - 6*a^2 - 25*a + 5],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() $$\mathfrak{D}$$ = $$(121,121 a,121 a^{2} - 242,121 a^{3} - 363 a,11 a^{4} + 88 a^{3} + 22 a^{2} - 253 a - 66)$$ = $$\left(a^{2} + a - 2\right)^{9}$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$2357947691$$ = $$11^{9}$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$-32768$$ magma: jInvariant(E); sage: E.j_invariant() gp (2.8): E.j $$\text{End} (E)$$ = $$\Z[(1+\sqrt{-11})/2]$$ ( Complex Multiplication ) magma: HasComplexMultiplication(E); sage: E.has_cm(), E.cm_discriminant() $$\text{ST} (E)$$ = $N(\mathrm{U}(1))$

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$$(j)_{-}$$
$$\left(a^{2} + a - 2\right)$$ $$11$$ $$2$$ $$III^*$$ Additive $$1$$ $$2$$ $$9$$ $$0$$

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]

For all other primes $$p$$, the image is the normalizer of a split Cartan subgroup if $$\left(\frac{ -11 }{p}\right)=+1$$ or the normalizer of a nonsplit Cartan subgroup if $$\left(\frac{ -11 }{p}\right)=-1$$.

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 11.
Its isogeny class 121.1-b consists of curves linked by isogenies of degree11.

Base change

This curve is the base-change of elliptic curves 121.b1, defined over $$\Q$$, so it is also a $$\Q$$-curve.