Properties

Label 3.3.229.1-14.1-b1
Base field 3.3.229.1
Conductor norm \( 14 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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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

\({y}^2+\left(a^{2}-2\right){x}{y}+{y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(-501a^{2}-1104a-295\right){x}-14257a^{2}-30089a-6588\)
sage: E = EllipticCurve([K([-2,0,1]),K([-2,-1,1]),K([1,0,0]),K([-295,-1104,-501]),K([-6588,-30089,-14257])])
 
gp: E = ellinit([Polrev([-2,0,1]),Polrev([-2,-1,1]),Polrev([1,0,0]),Polrev([-295,-1104,-501]),Polrev([-6588,-30089,-14257])], K);
 
magma: E := EllipticCurve([K![-2,0,1],K![-2,-1,1],K![1,0,0],K![-295,-1104,-501],K![-6588,-30089,-14257]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a-3)\) = \((a+1)\cdot(-a^2+2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 14 \) = \(2\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-6636313a^2+4200712a+13397169)\) = \((a+1)^{3}\cdot(-a^2+2)^{24}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1532649851044531315208 \) = \(2^{3}\cdot7^{24}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{219768203370710958657491381}{1532649851044531315208} a^{2} - \frac{422256959586277880827223561}{1532649851044531315208} a - \frac{22234843138942680536190377}{1532649851044531315208} \)
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: $\left(7 a^{2} + \frac{75}{4} a + 9 : -\frac{23}{2} a^{2} - \frac{89}{4} a - \frac{7}{8} : 1\right)$
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: \( 1.5848783118414443266604255492729373906 \)
Tamagawa product: \( 24 \)  =  \(1\cdot( 2^{3} \cdot 3 )\)
Torsion order: \(2\)
Leading coefficient: \( 0.62839024075361314906521287939860880115 \)
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\((j)_{-}\)
\((a+1)\) \(2\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\((-a^2+2)\) \(7\) \(24\) \(I_{24}\) Split multiplicative \(-1\) \(1\) \(24\) \(24\)

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.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6, 8, 12 and 24.
Its isogeny class 14.1-b consists of curves linked by isogenies of degrees dividing 24.

Base change

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

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