Properties

Label 2.2.56.1-56.1-d5
Base field \(\Q(\sqrt{14}) \)
Conductor norm \( 56 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 4 \)
Rank \( 2 \)

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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

\({y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-75{x}-361\)
sage: E = EllipticCurve([K([0,1]),K([1,0]),K([0,1]),K([-75,0]),K([-361,0])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([1,0]),Polrev([0,1]),Polrev([-75,0]),Polrev([-361,0])], K);
 
magma: E := EllipticCurve([K![0,1],K![1,0],K![0,1],K![-75,0],K![-361,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2a)\) = \((-a+4)^{3}\cdot(-2a+7)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 56 \) = \(2^{3}\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((224)\) = \((-a+4)^{10}\cdot(-2a+7)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 50176 \) = \(2^{10}\cdot7^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1443468546}{7} \)
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: \(2\)
Generators $\left(-\frac{6}{4225} a - \frac{27423}{4225} : \frac{752147}{274625} a + \frac{1826}{274625} : 1\right)$ $\left(\frac{6}{4225} a - \frac{27423}{4225} : \frac{755723}{274625} a - \frac{3634}{274625} : 1\right)$
Heights \(2.9610675796985463489395591937715670450\) \(2.9610675796985463489395591937715670450\)
Torsion structure: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-\frac{13}{2} : \frac{11}{4} a : 1\right)$ $\left(-2 a + 1 : -a + 14 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(2\)
Regulator: \( 8.0908383878021160530200337268514296456 \)
Period: \( 2.6362935287269056242494966545455530267 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 2.8503177441011686065450864444986300260 \)
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\((j)_{-}\)
\((-a+4)\) \(2\) \(2\) \(III^{*}\) Additive \(-1\) \(3\) \(10\) \(0\)
\((-2a+7)\) \(7\) \(2\) \(I_{2}\) 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
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 56.1-d consists of curves linked by isogenies of degrees dividing 8.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 392.d1
\(\Q\) 448.e1