uh well there's this thing called complex multiplication which you might first see for elliptic curves (but exists in general for abelian varieties)
The idea is for some elliptic curves End(E) = integers - where all your endomorphisms are given by $[n]: P\mapsto nP$
but for some elliptic curves you have extra endomorphisms. for example for $y^2=x^3-x $ over $\mathbb{C}$, you have the automorphism $(x,y) \mapsto (-x,iy)$ these curves are said to have complex multiplication
