  This is the second in a series of three papers on systematic analysis of rank
1 Coulomb branch geometries of four dimensional $\mathcal{N}$=2 SCFTs. In the
first paper we developed a strategy for classifying physical rank-1 CB
geometries of $\mathcal{N}$=2 SCFTs. Here we show how to carry out this
strategy computationally to construct the Seiberg-Witten curves and one-forms
for all the rank-1 SCFTs. Explicit expressions are given for all cases, with
the exception of the $N_f$=4 SU(2) gauge theory and the En SCFTs which were
previously constructed. Our classification includes all known rank-1 theories
plus a new one with an abelian flavor group, plus nine additional theories
whose existence is more speculative. Four of those, reported in our first
paper, depend on the assumption of new frozen rank-1 SCFTs. Here we also also
show that the assumption of the existence of certain rank-0 $\mathcal{N}$=2
SCFTs leads to five additional consistent rank-1 CB geometries.
