nt.number theory - elliptic curve with j-invariant T

The idea for (2) is the following: the modular curve $Y(ell^n)$ classifying elliptic curves
over ${mathbb C}$ together with an isomorphism $({mathbb Z}/ell^n)^2 cong E[ell^n]$
identifying the standard symplectic pairing on the left (i.e. $langle (a_1,a_2),(b_1 ,b_2)rangle
= e^{2pi i (a_1b_2-a_2b_1)/ell^n}$) with the Weil pairing on the right,
is irreducible. (It is isomorphic to $mathcal H/Gamma(ell^n)$, where
$mathcal H$ is the complex upper half-plane and $Gamma(ell^n)$ is the congruence
subgroup of full level $ell^n$.)

(3) follows from (2) and the irreducibility of cyclotomic polynomials over ${mathbb Q}$.