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Properties

Label 5.5.65657.1-29.1-a1
Base field 5.5.65657.1
Conductor norm \( 29 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 5.5.65657.1

Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 2 x^{2} + 5 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 5, 2, -5, -1, 1]))
 
gp: K = nfinit(Polrev([1, 5, 2, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 2, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(-a^{4}+2a^{3}+5a^{2}-6a-5\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(3a^{4}-4a^{3}-13a^{2}+10a+8\right){x}^{2}+\left(4a^{4}-4a^{3}-21a^{2}+10a+22\right){x}+5a^{4}-6a^{3}-24a^{2}+16a+21\)
sage: E = EllipticCurve([K([-5,-6,5,2,-1]),K([8,10,-13,-4,3]),K([-1,-1,1,0,0]),K([22,10,-21,-4,4]),K([21,16,-24,-6,5])])
 
gp: E = ellinit([Polrev([-5,-6,5,2,-1]),Polrev([8,10,-13,-4,3]),Polrev([-1,-1,1,0,0]),Polrev([22,10,-21,-4,4]),Polrev([21,16,-24,-6,5])], K);
 
magma: E := EllipticCurve([K![-5,-6,5,2,-1],K![8,10,-13,-4,3],K![-1,-1,1,0,0],K![22,10,-21,-4,4],K![21,16,-24,-6,5]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a^4+3a^3+8a^2-7a-4)\) = \((-2a^4+3a^3+8a^2-7a-4)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 29 \) = \(29\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-2a^4+3a^3+8a^2-7a-4)\) = \((-2a^4+3a^3+8a^2-7a-4)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -29 \) = \(-29\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{12183}{29} a^{4} - \frac{19048}{29} a^{3} - \frac{55150}{29} a^{2} + \frac{66157}{29} a + \frac{70545}{29} \)
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: \(1\)
Generator $\left(-a^{4} + a^{3} + 5 a^{2} - 3 a - 4 : -a^{4} + a^{3} + 5 a^{2} - 2 a - 5 : 1\right)$
Height \(0.012463742166385614723572298802834068269\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.012463742166385614723572298802834068269 \)
Period: \( 6352.3095014081826769397085423735992377 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.54493280 \)
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\((j)_{-}\)
\((-2a^4+3a^3+8a^2-7a-4)\) \(29\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 29.1-a consists of curves linked by isogenies of degree 3.

Base change

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

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