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 $$0$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

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

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

gp: E = ellinit([1, a + 1, 0, 51*a + 69, -62*a + 339],K)

magma: E := ChangeRing(EllipticCurve([1, a + 1, 0, 51*a + 69, -62*a + 339]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 $$\mathfrak{N}$$ = $$(5 a - 2)$$ = $$\left(a\right) \cdot \left(a - 1\right)^{3}$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$54$$ = $$2 \cdot 3^{3}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(1613824 a + 4308992)$$ = $$\left(a\right)^{27} \cdot \left(a - 1\right)^{11}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$23776267862016$$ = $$2^{27} \cdot 3^{11}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$-\frac{241123607}{16384} a + \frac{59710933}{8192}$$ 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: $$0$$

sage: E.rank()

magma: Rank(E);

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$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$1$$ $$I_{27}$$ 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 $$p < 1000$$ 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.