Properties

Label 5.5.14641.1-23.4-a3
Base field \(\Q(\zeta_{11})^+\)
Conductor norm \( 23 \)
CM no
Base change no
Q-curve no
Torsion order \( 10 \)
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

\({y}^2+\left(a^{4}+a^{3}-3a^{2}-3a\right){x}{y}+\left(a^{4}+a^{3}-4a^{2}-3a+2\right){y}={x}^{3}+\left(a^{4}+a^{3}-3a^{2}-3a+1\right){x}^{2}+\left(-7a^{4}+26a^{3}+4a^{2}-50a+12\right){x}-45a^{4}+84a^{3}+135a^{2}-274a+66\)
sage: E = EllipticCurve([K([0,-3,-3,1,1]),K([1,-3,-3,1,1]),K([2,-3,-4,1,1]),K([12,-50,4,26,-7]),K([66,-274,135,84,-45])])
 
gp: E = ellinit([Polrev([0,-3,-3,1,1]),Polrev([1,-3,-3,1,1]),Polrev([2,-3,-4,1,1]),Polrev([12,-50,4,26,-7]),Polrev([66,-274,135,84,-45])], K);
 
magma: E := EllipticCurve([K![0,-3,-3,1,1],K![1,-3,-3,1,1],K![2,-3,-4,1,1],K![12,-50,4,26,-7],K![66,-274,135,84,-45]]);
 

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: \((209a^4+282a^3-916a^2-698a+810)\) = \((-a^4+2a^2+1)^{10}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -41426511213649 \) = \(-23^{10}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{67578061913843073290}{41426511213649} a^{4} - \frac{20776798162111852984}{41426511213649} a^{3} + \frac{243672465429611249930}{41426511213649} a^{2} + \frac{116422572942295623941}{41426511213649} a - \frac{51184407321695715000}{41426511213649} \)
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: \(\Z/10\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-3 a^{4} + a^{3} + 22 a^{2} - 30 a + 7 : -101 a^{4} + 227 a^{3} + 108 a^{2} - 376 a + 94 : 1\right)$
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: \( 993.12606699865359490731750704130415623 \)
Tamagawa product: \( 10 \)
Torsion order: \(10\)
Leading coefficient: \( 0.820765345 \)
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\((j)_{-}\)
\((-a^4+2a^2+1)\) \(23\) \(10\) \(I_{10}\) Split multiplicative \(-1\) \(1\) \(10\) \(10\)

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.1.1

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.