Considering that it was repaired by Apple, you can probably use My Support Profile to find its serial number. Log in with your Apple ID.
It is also stored in your last iTunes backup:
Open iTunes on the computer you used to sync the iPod and select Preferences…from the iTunes menu. (In Windows, choose Edit > Preferences.)
Select the Devices pane. Position the cursor over any of its backups. The iPod's serial number will appear in a popup window.
For a screenshot read: iOS: How to find the serial number, IMEI, MEID, CDN, and ICCID number
Scroll down to:
Additional Information >
Getting the information from your computer >
If you have access to your computer, but not your device