Properties

Base field \(\Q(\sqrt{-2}) \)
Label 2.0.8.1-54.4-a3
Conductor \((5 a - 2)\)
Conductor norm \( 54 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
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

\( y^2 + x y = x^{3} + \left(a + 1\right) x^{2} + \left(51 a + 69\right) x - 62 a + 339 \)
magma: E := ChangeRing(EllipticCurve([1, a + 1, 0, 51*a + 69, -62*a + 339]),K);
sage: E = EllipticCurve(K, [1, a + 1, 0, 51*a + 69, -62*a + 339])
gp (2.8): E = ellinit([1, a + 1, 0, 51*a + 69, -62*a + 339],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((5 a - 2)\) = \( \left(a\right) \cdot \left(a - 1\right)^{3} \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 54 \) = \( 2 \cdot 3^{3} \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((1613824 a + 4308992)\) = \( \left(a\right)^{27} \cdot \left(a - 1\right)^{11} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 23776267862016 \) = \( 2^{27} \cdot 3^{11} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( -\frac{241123607}{16384} a + \frac{59710933}{8192} \)
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: Trivial
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]

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\((j)_{-}\)
\( \left(a\right) \) \(2\) \(1\) \(I_{27}\) Non-split multiplicative \(1\) \(1\) \(27\) \(27\)
\( \left(a - 1\right) \) \(3\) \(1\) \(II^*\) Additive \(-1\) \(3\) \(11\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 9 and 27.
Its isogeny class 54.4-a consists of curves linked by isogenies of degrees dividing 27.

Base change

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