Properties

Base field \(\Q(\sqrt{14}) \)
Label 2.2.56.1-10.1-c2
Conductor \((a + 2)\)
Conductor norm \( 10 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
Rank not available

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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

\( y^2 + \left(a + 1\right) x y = x^{3} + \left(1285 a - 4941\right) x - 48650 a + 182431 \)
sage: E = EllipticCurve(K, [a + 1, 0, 0, 1285*a - 4941, -48650*a + 182431])
 
gp: E = ellinit([a + 1, 0, 0, 1285*a - 4941, -48650*a + 182431],K)
 
magma: E := ChangeRing(EllipticCurve([a + 1, 0, 0, 1285*a - 4941, -48650*a + 182431]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((a + 2)\) = \( \left(-a + 4\right) \cdot \left(-a + 3\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 10 \) = \( 2 \cdot 5 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((-2 a + 6)\) = \( \left(-a + 4\right)^{2} \cdot \left(-a + 3\right) \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 20 \) = \( 2^{2} \cdot 5 \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( \frac{7114676554418062503}{10} a + \frac{26620682081199569989}{10} \)
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\((j)_{-}\)
\( \left(-a + 4\right) \) \(2\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\( \left(-a + 3\right) \) \(5\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

Galois Representations

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

prime Image of Galois Representation
\(5\) 5B.4.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 10.1-c consists of curves linked by isogenies of degree 5.

Base change

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