RE: Shotgun Patterning
I had an 870, as does a close hunting buddy... patterned them together, (his was the 3 1/2" chamber, but was shooting 3" shells at the time), and we both found that the factory .665 choke worked well.
I tried a couple other chokes constrictions, a 650 showed promise with #6 lead, but I wasn't a fan of the lighter 6's.
12 pointer is right... I tried Remington Hevi-shot, (3" 1 5/8 load of #6) from the same gun and it was unreal. Add about 4x the pellet count at 40 yards, easily putting 30+ pellets in the turkey outline, with an average of 10 in the spine / brain areas.
The good thing is that Hevi-shot hits like the next larger lead counterpart, ie: Hevi #6 will hit as hard as Lead #5, with the added plus of getting more pellets when compared to the #5 counterpart.
Try a .665 choke with that Hevi-shot load out of your 870. The shells cost a bit more but I think you'll be very pleased. A tidbit more info... my gun preferred a non ported choke over the ported offering. Don't shy away due to the price, you'll only shoot a few shots a year to fill your tags and it's well worth the confidence when pulling the trigger.