Indeed we get
, so this function is doubly periodic (...really? Read on), and thus, what's lacking to consider it an elliptic function? Well, the periods must be a basis for the real
dimensional linear space
, and in this case we get
, which screws the whole thing up.
The complete, long and deep explanation may be well beyond what I'd be willing to explain by this means: We need a free abelian group in
which is also a maximal order there and etc.
Making it short, though, we can simply say: that the above ratio is real and not complex non-real means the function
has actually just one single period (!) which, automatically, disqualifies it from being an elliptic function.