Los Angeles to Rome FCO is on sale with all three alliances, including Alitalia nonstop flights for travel August to March 2018.
Alitalia nonstop flights for $525 look like the best deal. Other low fares I found through Momondo.com show ticket prices as low as $495 (tip: watch out for lowest fare tickets with airport changes or long layovers).
Low fare dates for tickets around $570 round trip are searchable on Google Flights, but many itineraries do not offer online booking links. I search Google Flights, then plug low fare dates into Momondo for online travel agency searches.
Momondo search for Alitalia flights
Los Angeles LAX – Rome FCO – Los Angeles LAX nonstop
Momondo.com search United Airlines flights
Los Angeles LAX – Chicago ORD – Rome FCO – Chicago ORD – Los Angeles LAX
Momondo.com search British Airways/American Airlines flights
Los Angeles LAX – London LHR – Rome FCO – Charlotte CLT – Los Angeles LAX