You found this out, but ideals matter because each individual ideal must occur an even number of times during the NFS algebraic square root, not just the prime that the ideal lies over. If the norm of a given relation contains a factor p, only one of the ideals over p in the algebraic factor base gets its count incremented. For an algebraic polynomial of degree d there are as many as d different entries in the algebraic factor base for each prime p, and you must sieve them individually.
