IIRC the first 4 screw S&W revolvers were the Combat Magnum (pre 19) that began shipping in early 1956 in the K-260,000 range,

your serial is very near the end of the no dash run
For reference have these two in my collection that flank your serial
Model 19 Serial K 422034 Shipped May 1961
Model 15-1 Serial K 436822 Shipped June 1961

Enjoy