Properties

Label 5.5.24217.1-83.1-a1
Base field 5.5.24217.1
Conductor norm \( 83 \)
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

\({y}^2+\left(a^{4}-a^{3}-4a^{2}+4a+1\right){x}{y}+\left(3a^{4}-a^{3}-14a^{2}+3a+6\right){y}={x}^{3}+\left(-4a^{4}+a^{3}+19a^{2}-2a-6\right){x}^{2}+\left(-8a^{4}+2a^{3}+37a^{2}-4a-10\right){x}-6a^{4}+2a^{3}+27a^{2}-3a-9\)
sage: E = EllipticCurve([K([1,4,-4,-1,1]),K([-6,-2,19,1,-4]),K([6,3,-14,-1,3]),K([-10,-4,37,2,-8]),K([-9,-3,27,2,-6])])
 
gp: E = ellinit([Polrev([1,4,-4,-1,1]),Polrev([-6,-2,19,1,-4]),Polrev([6,3,-14,-1,3]),Polrev([-10,-4,37,2,-8]),Polrev([-9,-3,27,2,-6])], K);
 
magma: E := EllipticCurve([K![1,4,-4,-1,1],K![-6,-2,19,1,-4],K![6,3,-14,-1,3],K![-10,-4,37,2,-8],K![-9,-3,27,2,-6]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-3a^4+2a^3+14a^2-7a-7)\) = \((-3a^4+2a^3+14a^2-7a-7)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 83 \) = \(83\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((3a^4-2a^3-14a^2+7a+7)\) = \((-3a^4+2a^3+14a^2-7a-7)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 83 \) = \(83\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{42220647154}{83} a^{4} + \frac{18556175594}{83} a^{3} + \frac{202112507761}{83} a^{2} - \frac{45696285762}{83} a - \frac{103175812742}{83} \)
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: \( 271.34687258451646429352155153304014169 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.74367142 \)
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\((j)_{-}\)
\((-3a^4+2a^3+14a^2-7a-7)\) \(83\) \(1\) \(I_{1}\) 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 83.1-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.