How important is the number of cores in RJ45 jacks with magnetics
It's important if you are doing gigabit ethernet, and for cost/benefit reasons.

Basically, traditional 10/100 ethernet networks only use two of the 4 CAT-5/6 pairs. As such, extremely budget minded magjacks intended for only 10/100 use can simply omit the other two cores (and their associated costs).

The odd numbers seem to relate to additional cores used for filtering purposes. a jack with 1 core basically just has a common-mode inductor, and does not actually provide a ethernet "magjack" in the sense it's normally understood.

A 3-core jack has two-pairs of isolation transformers, and either one common-mode choke for all four pairs, or a choke on one of the two pairs (the TX pair, from the ones I looked at).

It seems similar for the larger numbers - 4 cores would either be all four pairs, or two pairs with individual common-mode inductors (note that digikey has both "4" and "4 (two ports only)" categories for the 4-core option), 5 is 4 pairs with one shared CM choke (or a more exotic topology), etc...



The huge number of cores parts look to be exotic stuff. the 12-pair is a gigabit magjack with per-pair common-mode and DC-tap-off chokes, and the 48 pair is actually a MRJ21 connector, which is an exotic connector that actually carries four separate gigabit ethernet connections (e.g. 16 pairs) over a single connection, and is used for extremely high-density network switches.