1 week is the statutory minimum notice for employees of between 1 month and 2 years continuous service. Contractually the notice period may be greater even if the term is "missing" from your copy of the contract.
The general (although by no means universal) rule is that if you ask your employer if you can leave before your notice is up, you should not expect to get paid for the part of your notice you don't serve. On the other hand, if your employer asks you to leave early, they would have to pay you for the full notice period.