Properties

Base field \(\Q(\sqrt{-2}) \)
Label 2.0.8.1-108.2-a1
Conductor \((-6 a - 6)\)
Conductor norm \( 108 \)
CM no
base-change no
Q-curve yes
Torsion order \( 4 \)
Rank \( 0 \)

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

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^2 + 2)
 
gp (2.8): K = nfinit(a^2 + 2);
 

Weierstrass equation

\( y^2 + a x y = x^{3} + \left(-a + 1\right) x^{2} + \left(-59 a + 4\right) x + 122 a - 261 \)
magma: E := ChangeRing(EllipticCurve([a, -a + 1, 0, -59*a + 4, 122*a - 261]),K);
 
sage: E = EllipticCurve(K, [a, -a + 1, 0, -59*a + 4, 122*a - 261])
 
gp (2.8): E = ellinit([a, -a + 1, 0, -59*a + 4, 122*a - 261],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((-6 a - 6)\) = \( \left(a\right)^{2} \cdot \left(-a - 1\right)^{2} \cdot \left(a - 1\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 108 \) = \( 2^{2} \cdot 3^{3} \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((14256 a - 1296)\) = \( \left(a\right)^{8} \cdot \left(-a - 1\right)^{9} \cdot \left(a - 1\right)^{4} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 408146688 \) = \( 2^{8} \cdot 3^{13} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( -\frac{18202756}{81} a - \frac{253086988}{81} \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp (2.8): E.j
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
magma: HasComplexMultiplication(E);
 
sage: E.has_cm(), E.cm_discriminant()
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \( 0 \)
magma: Rank(E);
 
sage: E.rank()
 
magma: Generators(E); // includes torsion
 
sage: E.gens()
 

Regulator: 1

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

Torsion subgroup

Structure: \(\Z/4\Z\)
magma: TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp (2.8): elltors(E)[2]
 
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp (2.8): elltors(E)[1]
 
Generator: $\left(3 a + 5 : -4 a - 3 : 1\right)$
magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp (2.8): elltors(E)[3]
 

Local data at primes of bad reduction

magma: LocalInformation(E);
 
sage: E.local_data()
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(a\right) \) \(2\) \(1\) \(IV^*\) Additive \(-1\) \(2\) \(8\) \(0\)
\( \left(-a - 1\right) \) \(3\) \(4\) \(I_{3}^*\) Additive \(-1\) \(2\) \(9\) \(3\)
\( \left(a - 1\right) \) \(3\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) 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, 4, 6 and 12.
Its isogeny class 108.2-a consists of curves linked by isogenies of degrees dividing 12.

Base change

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