Properties

Label 5.5.24217.1-29.1-a1
Base field 5.5.24217.1
Conductor norm \( 29 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 5.5.24217.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 3, -1, -5, 0, 1]))
 
gp: K = nfinit(Polrev([1, 3, -1, -5, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -1, -5, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(2a^{4}-9a^{2}+2\right){x}{y}+\left(2a^{4}-a^{3}-9a^{2}+3a+3\right){y}={x}^{3}+\left(-2a^{4}+9a^{2}+a-1\right){x}^{2}+\left(-6a^{4}-3a^{3}+23a^{2}+15a+1\right){x}-6a^{4}-7a^{3}+19a^{2}+20a+4\)
sage: E = EllipticCurve([K([2,0,-9,0,2]),K([-1,1,9,0,-2]),K([3,3,-9,-1,2]),K([1,15,23,-3,-6]),K([4,20,19,-7,-6])])
 
gp: E = ellinit([Polrev([2,0,-9,0,2]),Polrev([-1,1,9,0,-2]),Polrev([3,3,-9,-1,2]),Polrev([1,15,23,-3,-6]),Polrev([4,20,19,-7,-6])], K);
 
magma: E := EllipticCurve([K![2,0,-9,0,2],K![-1,1,9,0,-2],K![3,3,-9,-1,2],K![1,15,23,-3,-6],K![4,20,19,-7,-6]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a^4+a^3+10a^2-2a-4)\) = \((-2a^4+a^3+10a^2-2a-4)\)
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: \((-a^4+a^3+5a^2-2a-4)\) = \((-2a^4+a^3+10a^2-2a-4)\)
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: \( -\frac{3576764830}{29} a^{4} - \frac{6949922095}{29} a^{3} + 85021120 a^{2} + \frac{4914209131}{29} a + \frac{615642891}{29} \)
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: \(0\)
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: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 205.45966560079419072141895362146600669 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.32028110 \)
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\((j)_{-}\)
\((-2a^4+a^3+10a^2-2a-4)\) \(29\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 29.1-a consists of this curve only.

Base change

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

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