Conventional corn here so we get ours killed before planting corn into it.  If I was rup corn I would be letting it stand I think and no tilling into it.  That is what we do with beans.  When you kill it the plants aren't taking in any water and the field has a lot of cover so that can happen.  Only our 3rd year with covers and its a learning process for sure.  Here after it is dead we go in with a turbo till or field cultivator to try to get the ground to dry.  Its been wet here and cold... personally only 20 acres in on Monday and it was the worst conditions we've ever planted in.  This was not a cover field either.  Nothing will be easy this spring it seems but if it was my ground once it was dry enough to work I would be doing some sort of tillage to dry and warm that soil.  My favorite thing to do with covers is plant beans into 18 inch cereal rye and then come back later and kill it with rup.  Cleanest bean field you'll ever have all season long and a fantastic seed bed.  These are my experiences here in SEMN.