I'm not sure if JC23 is the same thing as a JA number. If it is, it might help you identify who made your machine. The plastic outside is a bit unusual. Most machines from that period were metal. Can you post a picture? Many times we may not know a particular brand but will recognize the machine itself. SewUSA carries a lot of manuals, they may have one that works even if it doesn't say "Bradford" on it.
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http://www.sewusa.com/Sewing_Machine...on_Manuals.htm
Rodney