Properties

Label 2.2.56.1-56.1-b2
Base field \(\Q(\sqrt{14}) \)
Conductor \((2a)\)
Conductor norm \( 56 \)
CM no
Base change yes: 56.a4,3136.p4
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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Show commands for: Magma / Pari/GP / 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(Pol(Vecrev([-14, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, 0, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-30a+112\right){x}-944a+3532\)
sage: E = EllipticCurve([K([0,1]),K([1,0]),K([0,1]),K([112,-30]),K([3532,-944])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([1,0])),Pol(Vecrev([0,1])),Pol(Vecrev([112,-30])),Pol(Vecrev([3532,-944]))], K);
 
magma: E := EllipticCurve([K![0,1],K![1,0],K![0,1],K![112,-30],K![3532,-944]]);
 

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: \((-28)\) = \((-a+4)^{4}\cdot(-2a+7)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 784 \) = \(2^{4}\cdot7^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{432}{7} \)
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: \(1\)
Generator $\left(-\frac{226}{25} a + \frac{808}{25} : \frac{7534}{125} a - \frac{28072}{125} : 1\right)$
Height \(1.93305723333979\)
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: $\left(-2 a + 6 : 12 a - 44 : 1\right)$
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: \( 1.93305723333979 \)
Period: \( 24.4747121231892 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 3.16110044380836 \)
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 \(-1\) \(3\) \(4\) \(0\)
\((-2a+7)\) \(7\) \(2\) \(I_{2}\) 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, 4 and 8.
Its isogeny class 56.1-b consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base change of elliptic curves 56.a4, 3136.p4, defined over \(\Q\), so it is also a \(\Q\)-curve.