If the game was purchased recently, try reloading your games library via the main menu (2nd entry: Reload my games) or click the refresh icon in the top left of your games library. Origin requires a restart or refresh of your games library before if will show product codes on the game details page after a purchase.
Hope that works for you. If not, please contact a Game Advisor at https://help.ea.com/contact-us and they will be able to provide you with your code.
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