Properties

Base field \(\Q(\sqrt{14}) \)
Label 2.2.56.1-5.2-b1
Conductor \((-a - 3)\)
Conductor norm \( 5 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
Rank not available

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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: x = polygen(QQ); K.<a> = NumberField(x^2 - 14)
 
gp: K = nfinit(a^2 - 14);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, 0, 1]);
 

Weierstrass equation

\( y^2 + x y = x^{3} - x^{2} + \left(8 a + 30\right) x - 27 a - 101 \)
sage: E = EllipticCurve(K, [1, -1, 0, 8*a + 30, -27*a - 101])
 
gp: E = ellinit([1, -1, 0, 8*a + 30, -27*a - 101],K)
 
magma: E := ChangeRing(EllipticCurve([1, -1, 0, 8*a + 30, -27*a - 101]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((-a - 3)\) = \( \left(-a - 3\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 5 \) = \( 5 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((387 a - 379)\) = \( \left(-a - 3\right)^{9} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 1953125 \) = \( 5^{9} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( -\frac{534684321}{1953125} a + \frac{3601913643}{1953125} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.

sage: E.rank()
 
magma: Rank(E);
 

Regulator: not available

sage: gens = E.gens(); gens
 
magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: E.regulator_of_points(gens)
 
magma: Regulator(gens);
 

Torsion subgroup

Structure: Trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

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\((j)_{-}\)
\( \left(-a - 3\right) \) \(5\) \(9\) \(I_{9}\) Split multiplicative \(-1\) \(1\) \(9\) \(9\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(3\) 3Nn

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 5.2-b consists of this curve only.

Base change

This curve is not the base-change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.