# 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

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] $\left(75 : -53 a + 276 : 1\right)$ 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 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.