You should be able to easily fly at 12500' in any old bog standard C172 - at least we do it regularly in the USA. Even nipped to 13,500 for brief periods. All perfectly legal of course even without 02.
In Europe I understand that oxygen or European physiology is is different though, so you may need to bring supplemental oxegen with you above 10,000'......or else you suddenly keel over and die.