Properties

Label 2.2.56.1-100.1-f1
Base field \(\Q(\sqrt{14}) \)
Conductor norm \( 100 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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Show commands: Magma / PariGP / SageMath

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

\({y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-1093a-4067\right){x}-57781a-216176\)
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([0,0]),K([-4067,-1093]),K([-216176,-57781])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([1,-1]),Polrev([0,0]),Polrev([-4067,-1093]),Polrev([-216176,-57781])], K);
 
magma: E := EllipticCurve([K![0,1],K![1,-1],K![0,0],K![-4067,-1093],K![-216176,-57781]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((10)\) = \((-a+4)^{2}\cdot(-a+3)\cdot(-a-3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 100 \) = \(2^{2}\cdot5\cdot5\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-62500)\) = \((-a+4)^{4}\cdot(-a+3)^{6}\cdot(-a-3)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 3906250000 \) = \(2^{4}\cdot5^{6}\cdot5^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{20720464}{15625} \)
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 $\left(\frac{3147}{7} a + \frac{11782}{7} : \frac{1236793}{49} a + \frac{661056}{7} : 1\right)$
Height \(6.6837746552226798720092008475408258980\)
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: $\left(15 a + \frac{107}{2} : -\frac{107}{4} a - 105 : 1\right)$
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: \( 6.6837746552226798720092008475408258980 \)
Period: \( 1.7726877650036809393831215444328774018 \)
Tamagawa product: \( 4 \)  =  \(1\cdot2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 3.1665768216075237616922964903708519553 \)
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\((j)_{-}\)
\((-a+4)\) \(2\) \(1\) \(IV\) Additive \(-1\) \(2\) \(4\) \(0\)
\((-a+3)\) \(5\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\((-a-3)\) \(5\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)

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 and 6.
Its isogeny class 100.1-f consists of curves linked by isogenies of degrees dividing 6.

Base change

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

Base field Curve
\(\Q\) 20.a2
\(\Q\) 15680.de2