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Properties

Label 5.5.14641.1-121.1-a1
Base field \(\Q(\zeta_{11})^+\)
Conductor norm \( 121 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 11 \)
Rank \( 0 \)

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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 \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 3, 3, -4, -1, 1]))
 
gp: K = nfinit(Polrev([-1, 3, 3, -4, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 3, -4, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{4}-3a^{2}+1\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(a^{4}-a^{3}-3a^{2}+2a\right){x}^{2}+\left(-15a^{4}+38a^{3}-10a^{2}-19a\right){x}+94a^{4}-262a^{3}+77a^{2}+174a-46\)
sage: E = EllipticCurve([K([1,0,-3,0,1]),K([0,2,-3,-1,1]),K([-1,1,1,0,0]),K([0,-19,-10,38,-15]),K([-46,174,77,-262,94])])
 
gp: E = ellinit([Polrev([1,0,-3,0,1]),Polrev([0,2,-3,-1,1]),Polrev([-1,1,1,0,0]),Polrev([0,-19,-10,38,-15]),Polrev([-46,174,77,-262,94])], K);
 
magma: E := EllipticCurve([K![1,0,-3,0,1],K![0,2,-3,-1,1],K![-1,1,1,0,0],K![0,-19,-10,38,-15],K![-46,174,77,-262,94]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^4-a^3-2a^2+3a+1)\) = \((a^2+a-2)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 121 \) = \(11^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((2a^4-6a^3-7a^2+20a-1)\) = \((a^2+a-2)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -14641 \) = \(-11^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -24729001 \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(0\)
Torsion structure: \(\Z/11\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(a^{2} - 3 a + 2 : -a^{3} + 3 a - 1 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 17090.609925524151197841516899107713710 \)
Tamagawa product: \( 1 \)
Torsion order: \(11\)
Leading coefficient: \( 1.16731165 \)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\((a^2+a-2)\) \(11\) \(1\) \(IV\) Additive \(-1\) \(2\) \(4\) \(0\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) 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 elliptic curve is a \(\Q\)-curve. It is the base change of the following elliptic curve:

Base field Curve
\(\Q\) 121.c1