12.57 isn't horrible for a battery just sitting. It's not fully charged (fully charged for a 12v flooded cell battery is about 12.7v, slightly higher for an AGM) but it's better than 75% charged, which should be plenty to crank your engine and certainly enough to operate the relay and starter solenoid.

I don't think your battery is your problem, but if you want to check, here's a good piece on testing Testing battery and charging system