Properties

Label 4.4.11661.1-27.1-e1
Base field 4.4.11661.1
Conductor norm \( 27 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 4.4.11661.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, 5, -4, -2, 1]))
 
gp: K = nfinit(Polrev([3, 5, -4, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 5, -4, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-22a^{3}+46a^{2}+68a-105\right){x}-106a^{3}-64a^{2}+626a+732\)
sage: E = EllipticCurve([K([-2,-1,1,0]),K([0,-5,-1,1]),K([-3,0,1,0]),K([-105,68,46,-22]),K([732,626,-64,-106])])
 
gp: E = ellinit([Polrev([-2,-1,1,0]),Polrev([0,-5,-1,1]),Polrev([-3,0,1,0]),Polrev([-105,68,46,-22]),Polrev([732,626,-64,-106])], K);
 
magma: E := EllipticCurve([K![-2,-1,1,0],K![0,-5,-1,1],K![-3,0,1,0],K![-105,68,46,-22],K![732,626,-64,-106]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3-a^2-5a)\) = \((-a)\cdot(a+1)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 27 \) = \(3\cdot3^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((10a^3-49a^2+13a+69)\) = \((-a)^{5}\cdot(a+1)^{9}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 4782969 \) = \(3^{5}\cdot3^{9}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{43579931283769}{243} a^{3} + \frac{107777583890159}{243} a^{2} + \frac{123309264089110}{243} a - \frac{276240756518333}{243} \)
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 $\left(-7 a^{3} + 2 a^{2} + 35 a + 21 : -59 a^{3} + 30 a^{2} + 283 a + 126 : 1\right)$
Height \(0.021370245877703063977483871739899342419\)
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: \( 0.021370245877703063977483871739899342419 \)
Period: \( 408.46516301754559657854036357247184624 \)
Tamagawa product: \( 10 \)  =  \(5\cdot2\)
Torsion order: \(1\)
Leading coefficient: \( 3.23337916276245 \)
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\((j)_{-}\)
\((-a)\) \(3\) \(5\) \(I_{5}\) Split multiplicative \(-1\) \(1\) \(5\) \(5\)
\((a+1)\) \(3\) \(2\) \(III^{*}\) Additive \(1\) \(2\) \(9\) \(0\)

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 27.1-e 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.