Consider the cyclotomic field $L={{\mathbb{Q}}}(\zeta_8)={{\mathbb{Q}}}(\sqrt{2},i)$, where $\zeta_8$ is a primitive 8-th root of unity. Let $\Lambda={{\mathbb{Z}}}[\zeta_8]$ denote the ring of integers of $L$.

Question 1.Does there exist an abelian variety $A$ over ${{\mathbb{C}}}$ of dimensiontwowith ${\mathrm{End\,}} A\simeq \Lambda$ ?

I expect the answer "no". Note that I want the endomorphism ring to be exactly $\Lambda$, not just contain $\Lambda$. In particular, my abelian variety should be simple.

Question 2.Does there exist an abelian variety $A$ over ${{\mathbb{C}}}$ of dimensionfourwith ${\mathrm{End\,}} A\simeq \Lambda$ ?

I expect the answer "yes".

Question 3.Is it possible to find an abelian variety $A$ over ${{\mathbb{C}}}$ of dimension 2 or 4 with ${\mathrm{End\,}} A\simeq \Lambda$ and such that $A$ is definable (without endomorphisms) over a small number field, say over ${{\mathbb{Q}}}(i)$ or even over ${{\mathbb{Q}}}$?