Q:

How important is the number of cores in RJ45 jacks with magnetics

While searching for RJ45 jacks with magnetics (MagJacks) in Digikey, we found that there is a selection criteria which says "Number of cores per jack".  Though we have been using MagJacks previously, this is the first time I noticed that criteria.  I compared the datasheets of few jacks with 1 core and 2 cores per jack, but couldn't find any notable difference.   

Is there any significance to this parameter? 
Which type of connector is better - the one with lesser number of cores or the one with more cores? 
For the same application, if I use a jack with more number of cores, what is the advantage which I can obtain? 

A:

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.