Properties

Label 5.5.14641.1-23.4-a1
Base field \(\Q(\zeta_{11})^+\)
Conductor norm \( 23 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
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

\({y}^2+\left(a^{4}+a^{3}-3a^{2}-2a\right){x}{y}+\left(a^{4}+a^{3}-4a^{2}-2a+3\right){y}={x}^{3}+\left(-a^{4}-a^{3}+5a^{2}+3a-5\right){x}^{2}+\left(-2681a^{4}-1331a^{3}+7125a^{2}+587a-4049\right){x}-146736a^{4}-108973a^{3}+354020a^{2}+126719a-132130\)
sage: E = EllipticCurve([K([0,-2,-3,1,1]),K([-5,3,5,-1,-1]),K([3,-2,-4,1,1]),K([-4049,587,7125,-1331,-2681]),K([-132130,126719,354020,-108973,-146736])])
 
gp: E = ellinit([Polrev([0,-2,-3,1,1]),Polrev([-5,3,5,-1,-1]),Polrev([3,-2,-4,1,1]),Polrev([-4049,587,7125,-1331,-2681]),Polrev([-132130,126719,354020,-108973,-146736])], K);
 
magma: E := EllipticCurve([K![0,-2,-3,1,1],K![-5,3,5,-1,-1],K![3,-2,-4,1,1],K![-4049,587,7125,-1331,-2681],K![-132130,126719,354020,-108973,-146736]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^4+2a^2+1)\) = \((-a^4+2a^2+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 23 \) = \(23\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((2a^4-9a^2+2a+4)\) = \((-a^4+2a^2+1)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -529 \) = \(-23^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{1665599353208772437646840723861648544}{529} a^{4} - \frac{515871876593405186030634717346598316}{529} a^{3} + \frac{5986748941379712064585808940738486553}{529} a^{2} + \frac{2844175551769835948036386085428144797}{529} a - \frac{1271720281023855543211042033983127776}{529} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-\frac{27}{2} a^{4} + \frac{9}{2} a^{3} + \frac{79}{2} a^{2} - \frac{115}{4} a - \frac{109}{4} : \frac{239}{8} a^{4} + 29 a^{3} - \frac{281}{4} a^{2} - \frac{449}{8} a + \frac{57}{8} : 1\right)$
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: \( 0.31780034143956915037034160225321733000 \)
Tamagawa product: \( 2 \)
Torsion order: \(2\)
Leading coefficient: \( 0.820765345 \)
Analytic order of Ш: \( 625 \) (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\((j)_{-}\)
\((-a^4+2a^2+1)\) \(23\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(2\) 2B
\(5\) 5B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 5 and 10.
Its isogeny class 23.4-a consists of curves linked by isogenies of degrees dividing 10.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.