The 2 short segments (TLV-FCO and FCO-TLV) are in Business class. It's a A320. J configuration is 2-2 vs 3-3 in Y.
The 2 long segments (FCO-JFK and JFK-FCO) are in Premium Economy. They are mostly flown by their A330. Configuration is 2-3-2 compared to economy which is 2-4-2. They also fly a 772 to JFK sometimes in which the seat configuration is 2-4-2 vs. 3-4-3 in economy.
2 free checked bags allowed in these cabins.
Bookable on Expedia, Orbitz and Priceline etc. (Don't change any settings when searching. Just search economy.)
Sample dates 7/10 - 7/17 or 8/9 - 8-24 MOST DATES ARE AVAILABLE
Available to BOS and MIA as well from $1151
(note that some of the search results may have 1 or both of the short segments in Y instead of J. So double check before booking)