Building on Knaccc's answer.
The first step to build an Elliptic curve library is to define a field of prime characteristic. In this case, the field is 2^255 - 19, I denote this with K. In most eprints it is denoted as F_p.
We then define a curve over this field of prime characteristics. The curve that we define in this case is a Montgomery ...
To this end, the field elements in the implementation of Monero seem
to be multiplied by 8 to ensure they lie in the smaller subgroup
The group elements are multiplied by 8 to ensure they lie in the correct subgroup. I'm phrasing it in this way to avoid ambiguity, so that "smaller subgroup" is not misinterpreted as one of the really small subgroups which ...