possible orders of elliptic functions

The order of a non-trivial elliptic function cannot be zero. This is asimple consequence of Liouville’s theorem. Were the order of an ellipticfunction zero, then the function would have no poles. By definition,an elliptic function has no essential singularities and is doublyperiodic. Hence, if the degree were zero, the function would becontinuous everywhere and hence, being doubly periodic, would bebounded (since continuous functions on a compact domain (like theclosure of the fundamental parallelogram) are bounded). By Liouville’stheorem, this would imply that the function is constant.

The order of an elliptic function cannot be 1. This follows from thefact that the residues at the poles of an elliptic function within afundamental parallelogram must sum to zero — if the function were ofdegree 1, it would have exactly one first-order pole in thefundamental parallelgram but any first-order pole must have a non-zeroresidue.

Any number greater than one is possible as the order of an ellipticfunction. As an example of an elliptic function of order two, we maytake the Weierstass $\mathrm{\wp }$-function, which has a single pole of order 2in the fundamental domain. The $n$-th derivative of this functionwill have a single pole of order $n+2$ in the fundamental domain,hence be of order $n+2$, so is an example showing that, for everyinteger greater than 2, there exists an elliptic function having thatinteger as its order.