In this article we study the endomorphism algebras of abelian varieties A defined over a given number field K with large cyclic 2-torsion fields. A key step in doing so is to provide criteria for all the endomorphisms of A to be defined over K(A[2]), the field extension generated by its 2-torsion. When \(K= \mathbb {Q}\) and \(\textrm{Gal}(\mathbb {Q}(A[2])/\mathbb {Q})\) is cyclic of prime order \(p = 2 \dim (A) +1\) , we prove that there are only finitely many possibilities for the geometric endomorphism algebra \(\textrm{End}(A) \otimes \mathbb {Q}\) . In fact, when \(\dim (A) \not \in \{3,5,9,21,33,81\}\) , we show \(\textrm{End}(A) \otimes \mathbb {Q}\) is a proper subfield of the p-th cyclotomic field. In particular, when \(g=2\) , \(\textrm{End}(A) \otimes \mathbb {Q}\) is isomorphic to either \(\mathbb {Q}\) or \(\mathbb {Q}(\sqrt{5})\) .
