Properties

Base field \(\Q(\sqrt{-2}) \)
Label 2.0.8.1-51.4-a1
Conductor \((a - 7)\)
Conductor norm \( 51 \)
CM no
base-change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{-2}) \)

Generator \(a\), with minimal polynomial \( x^{2} + 2 \); class number \(1\).

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

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((a - 7)\) = \( \left(a - 1\right) \cdot \left(2 a + 3\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 51 \) = \( 3 \cdot 17 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((-14 a + 47)\) = \( \left(a - 1\right)^{2} \cdot \left(2 a + 3\right)^{2} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 2601 \) = \( 3^{2} \cdot 17^{2} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( \frac{1437952}{2601} a - \frac{95168}{2601} \)
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: \( 1 \)

sage: E.rank()
 
magma: Rank(E);
 

Generator: $\left(-a - 1 : 2 : 1\right)$

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

Height: 0.03202960608770522

sage: [P.height() for P in gens]
 
magma: [Height(P):P in gens];
 

Regulator: 0.0320296060877

sage: E.regulator_of_points(gens)
 
magma: Regulator(gens);
 

Torsion subgroup

Structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Generator: $\left(\frac{1}{2} : -\frac{1}{4} a - \frac{1}{2} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(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\((j)_{-}\)
\( \left(a - 1\right) \) \(3\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\( \left(2 a + 3\right) \) \(17\) \(2\) \(I_{2}\) 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.
Its isogeny class 51.4-a consists of curves linked by isogenies of degree 2.

Base change

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