Very cool, thank you! I looked into UCF a bit and on this page:
http://doc.sagemath.org/html/en/reference/number_fields/sage/rings/number_field/number_field.html

It says "Doing arithmetic in towers of relative fields that depends on 
canonical coercions is currently VERY SLOW. It is much better to explicitly 
coerce all elements into a common field, then do arithmetic with them there 
(which is quite fast).", and on another page, it says "arithmetical 
operations are quite expensive, so the use of internally represented 
cyclotomics is not recommended for doing arithmetic over number fields, 
such as calculations with matrices of cyclotomics." 

What exactly does that mean? To me, it reads like it's saying that I 
shouldn't be using UCF with algebraic operations, because that is slow 
somehow. So, would it be better to use something else? Or am I reading that 
incorrectly? Sorry for the confusion


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