Here is a quick search using snappy:

In[29]: C = OrientableCuspedCensus(num_cusps = 1)

In[30]: square = [M.name() for M in C if abs(M.cusp_info()[0].modulus - 1j) < 0.00002]; square

Out[30]: ['m130', 'm135', 'm139', 'v1859', 'v3318', 't07829', 't12033', 't12035', 't12036', 't12038', 't12040', 't12041', 't12043', 't12045', 't12050', 'o9_17193', 'o9_19556', 'o9_21441', 'o9_22828', 'o9_31519', 'o9_31521', 'o9_35959', 'o9_41335', 'o9_42724']

A couple of the t's are double covers of the m's. I'll guess that it is not too hard to construct manifolds with square cusp. Finally, I don't see why there should be any particular pattern to these manifolds (beyond some covering relations, sometimes).

Here is a quick search using snappy:

In[29]: C = OrientableCuspedCensus(num_cusps = 1)

In[30]: square = [M.name() for M in C if abs(M.cusp_info()[0].modulus - 1j) < 0.00002]; square

Out[30]: ['m130', 'm135', 'm139', 'v1859', 'v3318', 't07829', 't12033', 't12035', 't12036', 't12038', 't12040', 't12041', 't12043', 't12045', 't12050', 'o9_17193', 'o9_19556', 'o9_21441', 'o9_22828', 'o9_31519', 'o9_31521', 'o9_35959', 'o9_41335', 'o9_42724']

A couple of the t's are double covers of the m's. I'll guess that it is not too hard to construct manifolds with square cusp. Finally, I don't see why there should be any particular pattern to these manifolds (beyond some covering relations, sometimes).

EDIT:

I vaguely recall that Craig Hodgson (Melbourne) showed me at some point a list of hyperbolic manifolds with particularly nice cusp shapes, including square and hexagonal. I believe that his list was made by trawling the snappy census, but perhaps he has further thoughts about these manifolds.

Here is a quick search using snappy:

In[29]: C = OrientableCuspedCensus(num_cusps = 1)

In[30]: square = [M.name() for M in C if abs(M.cusp_info()[0].modulus - 1j) < 0.00002]; square

Out[30]: ['m130', 'm135', 'm139', 'v1859', 'v3318', 't07829', 't12033', 't12035', 't12036', 't12038', 't12040', 't12041', 't12043', 't12045', 't12050', 'o9_17193', 'o9_19556', 'o9_21441', 'o9_22828', 'o9_31519', 'o9_31521', 'o9_35959', 'o9_41335', 'o9_42724']

A couple of the t's are double covers of the m's. I'll guess that it is not to hard to construct manifolds with square cusp. Finally, I don't see why there should be any particular pattern to these manifolds (beyond some covering relations, sometimes).

Here is a quick search using snappy:

In[29]: C = OrientableCuspedCensus(num_cusps = 1)

In[30]: square = [M.name() for M in C if abs(M.cusp_info()[0].modulus - 1j) < 0.00002]; square

Out[30]: ['m130', 'm135', 'm139', 'v1859', 'v3318', 't07829', 't12033', 't12035', 't12036', 't12038', 't12040', 't12041', 't12043', 't12045', 't12050', 'o9_17193', 'o9_19556', 'o9_21441', 'o9_22828', 'o9_31519', 'o9_31521', 'o9_35959', 'o9_41335', 'o9_42724']

A couple of the t's are double covers of the m's. I'll guess that it is not too hard to construct manifolds with square cusp. Finally, I don't see why there should be any particular pattern to these manifolds (beyond some covering relations, sometimes).