Properties

Label 2.2.56.1-242.1-a1
Base field \(\Q(\sqrt{14}) \)
Conductor norm \( 242 \)
CM no
Base change no
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{14}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 14 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, 0, 1]))
 
gp: K = nfinit(Polrev([-14, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-10a+29\right){x}-9a+29\)
sage: E = EllipticCurve([K([1,1]),K([1,-1]),K([1,1]),K([29,-10]),K([29,-9])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([1,-1]),Polrev([1,1]),Polrev([29,-10]),Polrev([29,-9])], K);
 
magma: E := EllipticCurve([K![1,1],K![1,-1],K![1,1],K![29,-10],K![29,-9]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-11a+44)\) = \((-a+4)\cdot(a+5)\cdot(a-5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 242 \) = \(2\cdot11\cdot11\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((484)\) = \((-a+4)^{4}\cdot(a+5)^{2}\cdot(a-5)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 234256 \) = \(2^{4}\cdot11^{2}\cdot11^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{704969}{484} \)
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 $\left(-\frac{329}{25} a + \frac{1207}{25} : -\frac{18938}{125} a + \frac{70854}{125} : 1\right)$
Height \(2.4183391352709074809368818561627269463\)
Torsion structure: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-a + 3 : 3 a - 12 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 2.4183391352709074809368818561627269463 \)
Period: \( 9.8313794013912849757676597475880577502 \)
Tamagawa product: \( 16 \)  =  \(2^{2}\cdot2\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 6.3542989382521078342040216286068828413 \)
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\((j)_{-}\)
\((-a+4)\) \(2\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((a+5)\) \(11\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((a-5)\) \(11\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 242.1-a consists of curves linked by isogenies of degrees dividing 4.

Base change

This elliptic curve is a \(\Q\)-curve.

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