lines, and each line pencil has exactly elements. In particular, the number v is congruent to either 1 or 3 modulo 6. In fact, the following classical theorem holds.
Theorem. There exists a Steiner triple system on v points if and only if .
The number of isomorphism types of the small Steiner triple systems is well known.
It's commonly assumed that the number of isomorphism types of Steiner triple systems with v=19 points is about 109. But unfortunately, this is out of the range of todays computers.