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Properties

Base field \(\Q(\zeta_{11})^+\)
Label 5.5.14641.1-121.1-a1
Conductor \((11,a^{4} + a^{3} - 3 a^{2} - 3 a - 2)\)
Conductor norm \( 121 \)
CM no
base-change yes: 121.c1
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

\( y^2 + \left(a^{4} - 3 a^{2} + 1\right) x y + \left(a^{2} + a - 1\right) y = x^{3} + \left(a^{4} - a^{3} - 3 a^{2} + 2 a\right) x^{2} + \left(-15 a^{4} + 38 a^{3} - 10 a^{2} - 19 a\right) x + 94 a^{4} - 262 a^{3} + 77 a^{2} + 174 a - 46 \)
magma: E := ChangeRing(EllipticCurve([a^4 - 3*a^2 + 1, a^4 - a^3 - 3*a^2 + 2*a, a^2 + a - 1, -15*a^4 + 38*a^3 - 10*a^2 - 19*a, 94*a^4 - 262*a^3 + 77*a^2 + 174*a - 46]),K);
 
sage: E = EllipticCurve(K, [a^4 - 3*a^2 + 1, a^4 - a^3 - 3*a^2 + 2*a, a^2 + a - 1, -15*a^4 + 38*a^3 - 10*a^2 - 19*a, 94*a^4 - 262*a^3 + 77*a^2 + 174*a - 46])
 
gp (2.8): E = ellinit([a^4 - 3*a^2 + 1, a^4 - a^3 - 3*a^2 + 2*a, a^2 + a - 1, -15*a^4 + 38*a^3 - 10*a^2 - 19*a, 94*a^4 - 262*a^3 + 77*a^2 + 174*a - 46],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()
 
\(\mathfrak{D}\) = \((11,11 a,11 a^{2} - 22,11 a^{3} - 33 a,a^{4} + 8 a^{3} + 2 a^{2} - 23 a - 6)\) = \( \left(a^{2} + a - 2\right)^{4} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 14641 \) = \( 11^{4} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( -24729001 \)
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: \(\Z/11\Z\)
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: $\left(a^{2} - 3 a + 2 : -a^{3} + 3 a - 1 : 1\right)$
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\((j)_{-}\)
\( \left(a^{2} + a - 2\right) \) \(11\) \(1\) \(IV\) Additive \(-1\) \(2\) \(4\) \(0\)

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]

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.c1, defined over \(\Q\), so it is also a \(\Q\)-curve.