Properties

Base field \(\Q(\zeta_{11})^+\)
Label 5.5.14641.1-121.1-b1
Conductor \((11,a^{4} + a^{3} - 3 a^{2} - 3 a - 2)\)
Conductor norm \( 121 \)
CM yes (\(-11\))
base-change yes: 121.b2
Q-curve yes
Torsion order \( 11 \)
Rank not available

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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

\( y^2 + \left(a^{3} + a^{2} - 2 a - 2\right) y = x^{3} + \left(a^{4} - a^{3} - 4 a^{2} + 3 a + 1\right) x^{2} + \left(9 a^{4} - 16 a^{3} - 23 a^{2} + 45 a - 10\right) x - 38 a^{4} + 67 a^{3} + 96 a^{2} - 188 a + 43 \)
magma: E := ChangeRing(EllipticCurve([0, a^4 - a^3 - 4*a^2 + 3*a + 1, a^3 + a^2 - 2*a - 2, 9*a^4 - 16*a^3 - 23*a^2 + 45*a - 10, -38*a^4 + 67*a^3 + 96*a^2 - 188*a + 43]),K);
sage: E = EllipticCurve(K, [0, a^4 - a^3 - 4*a^2 + 3*a + 1, a^3 + a^2 - 2*a - 2, 9*a^4 - 16*a^3 - 23*a^2 + 45*a - 10, -38*a^4 + 67*a^3 + 96*a^2 - 188*a + 43])
gp (2.8): E = ellinit([0, a^4 - a^3 - 4*a^2 + 3*a + 1, a^3 + a^2 - 2*a - 2, 9*a^4 - 16*a^3 - 23*a^2 + 45*a - 10, -38*a^4 + 67*a^3 + 96*a^2 - 188*a + 43],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()
\(\mathfrak{D}\) = \((11,11 a,11 a^{2} - 22,a^{3} + 6 a^{2} + a - 3,a^{4} - 2 a^{2} + 2 a + 7)\) = \( \left(a^{2} + a - 2\right)^{3} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 1331 \) = \( 11^{3} \)
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: \(\Z/11\Z\)
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]
Generator: $\left(a^{4} - 2 a^{3} - 2 a^{2} + 5 a - 1 : 3 a^{4} - 6 a^{3} - 8 a^{2} + 16 a - 2 : 1\right)$
magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp (2.8): elltors(E)[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\((j)_{-}\)
\( \left(a^{2} + a - 2\right) \) \(11\) \(2\) \(III\) Additive \(1\) \(2\) \(3\) \(0\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.

prime Image of Galois Representation
\(11\) 11B.1.1[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.b2, defined over \(\Q\), so it is also a \(\Q\)-curve.