Properties

Base field \(\Q(\sqrt{-2}) \)
Label 2.0.8.1-1458.4-e2
Conductor \((27 a)\)
Conductor norm \( 1458 \)
CM no
base-change yes: 1728.z2,54.b2
Q-curve yes
Torsion order \( 9 \)
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

\( y^2 + x y + y = x^{3} - x^{2} - 14 x + 29 \)
magma: E := ChangeRing(EllipticCurve([1, -1, 1, -14, 29]),K);
 
sage: E = EllipticCurve(K, [1, -1, 1, -14, 29])
 
gp (2.8): E = ellinit([1, -1, 1, -14, 29],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((27 a)\) = \( \left(a\right) \cdot \left(-a - 1\right)^{3} \cdot \left(a - 1\right)^{3} \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 1458 \) = \( 2 \cdot 3^{6} \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((124416)\) = \( \left(a\right)^{18} \cdot \left(-a - 1\right)^{5} \cdot \left(a - 1\right)^{5} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 15479341056 \) = \( 2^{18} \cdot 3^{10} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( -\frac{1167051}{512} \)
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: \(\Z/9\Z\)
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: $\left(-3 : -5 : 1\right)$
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\((j)_{-}\)
\( \left(a\right) \) \(2\) \(18\) \(I_{18}\) Split multiplicative \(-1\) \(1\) \(18\) \(18\)
\( \left(-a - 1\right) \) \(3\) \(3\) \(IV\) Additive \(1\) \(3\) \(5\) \(0\)
\( \left(a - 1\right) \) \(3\) \(3\) \(IV\) Additive \(1\) \(3\) \(5\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3 and 9.
Its isogeny class 1458.4-e consists of curves linked by isogenies of degrees dividing 9.

Base change

This curve is the base-change of elliptic curves 1728.z2, 54.b2, defined over \(\Q\), so it is also a \(\Q\)-curve.