Properties

Label 5.5.65657.1-27.1-a1
Base field 5.5.65657.1
Conductor norm \( 27 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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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

\({y}^2+\left(a^{3}-4a\right){x}{y}+\left(-2a^{4}+3a^{3}+9a^{2}-8a-7\right){y}={x}^{3}+\left(2a^{4}-3a^{3}-9a^{2}+10a+6\right){x}^{2}+\left(2a^{4}-a^{3}-15a^{2}+10a+16\right){x}+a^{3}-4a^{2}+4a+3\)
sage: E = EllipticCurve([K([0,-4,0,1,0]),K([6,10,-9,-3,2]),K([-7,-8,9,3,-2]),K([16,10,-15,-1,2]),K([3,4,-4,1,0])])
 
gp: E = ellinit([Polrev([0,-4,0,1,0]),Polrev([6,10,-9,-3,2]),Polrev([-7,-8,9,3,-2]),Polrev([16,10,-15,-1,2]),Polrev([3,4,-4,1,0])], K);
 
magma: E := EllipticCurve([K![0,-4,0,1,0],K![6,10,-9,-3,2],K![-7,-8,9,3,-2],K![16,10,-15,-1,2],K![3,4,-4,1,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2a^4-3a^3-9a^2+9a+7)\) = \((-a^4+a^3+4a^2-2a-2)^{3}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 27 \) = \(3^{3}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-4a^4+9a^3+21a^2-27a-20)\) = \((-a^4+a^3+4a^2-2a-2)^{11}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 177147 \) = \(3^{11}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -169969542 a^{4} + 461869972 a^{3} + 56647563 a^{2} - 437225155 a - 98969253 \)
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: \( 503.66026422848228643103090862786614596 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.96560918 \)
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\((j)_{-}\)
\((-a^4+a^3+4a^2-2a-2)\) \(3\) \(1\) \(II^{*}\) Additive \(-1\) \(3\) \(11\) \(0\)

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 27.1-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.