Elliptic curve $E$ over a field $K$ is said to have complex multiplication if $E$ has endomorphism except for multiplication by $n$($n∈\Bbb{Z}$) maps.
We usually consider $K$ as $ \Bbb{C}$.In this case, elliptic curve which has completely multiplication is very rare.
My question is, what happen in the case $K＝$(p-adic field) ? Here, p-adic field means finite extension of $ \Bbb{Q}_p$. For example, let $p$ be an odd prime, $E:y^2＝x^3-x$, ($x,y$)→($-x, \sqrt{-1}y$)(notice $\sqrt{-1}∈\Bbb{Q}_p$), so it has complex multiplication. But are elliptic curves over such fields are rare ? Although definition of 'rare' is ambiguous, if you know some thing to remark, I appreciated if you could hear some remarks.