Properties

Label 3.3.229.1-8.1-b6
Base field 3.3.229.1
Conductor norm \( 8 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Base field 3.3.229.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -4, 0, 1]))
 
gp: K = nfinit(Polrev([-1, -4, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -4, 0, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-131a^{2}-464a-415\right){x}-2913a^{2}-7754a-4350\)
sage: E = EllipticCurve([K([1,0,0]),K([4,0,-1]),K([-3,1,1]),K([-415,-464,-131]),K([-4350,-7754,-2913])])
 
gp: E = ellinit([Polrev([1,0,0]),Polrev([4,0,-1]),Polrev([-3,1,1]),Polrev([-415,-464,-131]),Polrev([-4350,-7754,-2913])], K);
 
magma: E := EllipticCurve([K![1,0,0],K![4,0,-1],K![-3,1,1],K![-415,-464,-131],K![-4350,-7754,-2913]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2)\) = \((a+1)\cdot(a^2-a-3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 8 \) = \(2\cdot4\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-4a^2+12a+12)\) = \((a+1)^{2}\cdot(a^2-a-3)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -4096 \) = \(-2^{2}\cdot4^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{304458198456664063}{2} a^{2} + \frac{1237813476398146963}{32} a + \frac{19170794207295639649}{32} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(\frac{3}{4} a^{2} - \frac{29}{4} a - \frac{27}{2} : -\frac{7}{8} a^{2} + \frac{25}{8} a + \frac{33}{4} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 1.2341965007087875840566161024071329934 \)
Tamagawa product: \( 2 \)  =  \(2\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 1.0194750053561877888775265541138080627 \)
Analytic order of Ш: \( 25 \) (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\((j)_{-}\)
\((a+1)\) \(2\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((a^2-a-3)\) \(4\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B
\(3\) 3B
\(5\) 5B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 5, 6, 10, 15 and 30.
Its isogeny class 8.1-b consists of curves linked by isogenies of degrees dividing 30.

Base change

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

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