How to get "wc -l" to print just the number of lines without file name?

Try this way:

wc -l < file.txt

cat file.txt | wc -l

According to the man page (for the BSD version, I don't have a GNU version to check):

If no files are specified, the standard input is used and no file name is displayed. The prompt will accept input until receiving EOF, or [^D] in most environments.

To do this without the leading space, why not:

wc -l < file.txt | bc