Properties

Base field \(\Q(\sqrt{89}) \)
Label 2.2.89.1-81.1-a1
Conductor \( \left(9\right) \)
Conductor norm \( 81 \)
CM yes (\(-267\))
base-change no
Q-curve yes
Torsion order \( 3 \)
Rank not available

Related objects

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Show commands for: Magma / SageMath / Pari/GP

Base field \(\Q(\sqrt{89}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 22 \); class number \(1\).

magma: K<a> := NumberField(x^2 - x - 22);
sage: K.<a> = NumberField(x^2 - x - 22)
gp (2.8): K = nfinit(a^2 - a - 22);

Weierstrass equation

\( y^2 + y = x^{3} + \left(-1590 a - 8580\right) x + 92750 a + 359875 \)
magma: E := ChangeRing(EllipticCurve([0, 0, 1, -1590*a - 8580, 92750*a + 359875]),K);
sage: E = EllipticCurve(K, [0, 0, 1, -1590*a - 8580, 92750*a + 359875])
gp (2.8): E = ellinit([0, 0, 1, -1590*a - 8580, 92750*a + 359875],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \( \left(9\right) \) = \( \left(3\right)^{2} \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 81 \) = \( 9^{2} \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \( \left(27\right) \) = \( \left(3\right)^{3} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 729 \) = \( 9^{3} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( -2086403563729465344000 a - 8798344145175011328000 \)
magma: jInvariant(E);
sage: E.j_invariant()
gp (2.8): E.j
\( \text{End} (E) \) = \(\Z[(1+\sqrt{-267})/2]\)   ( Complex Multiplication )
magma: HasComplexMultiplication(E);
sage: E.has_cm(), E.cm_discriminant()
\( \text{ST} (E) \) = $N(\mathrm{U}(1))$

Mordell-Weil rank and generators

Rank not available.
magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: not available

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

Torsion subgroup

Structure: \(\Z/3\Z\)
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: $\left(75 : -53 a + 276 : 1\right)$
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 ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(3\right) \) 9 \(2\) \( III \) Additive 2 3 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
\(89\) 89B.34.19[2]

For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -267 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -267 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 89 and 267.
Its isogeny class 81.1-a consists of 4 curves linked by isogenies of degrees dividing 267.

Base change

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