WARNING: This page is about a different car, the 2023 Audi S3, 2023 Audi RS 3, 2023 Audi A3 Quattro, and 2023 Audi A3. However, it is still accessible from the selected car via links, so may be relevant.
The engine number ("engine code" and "serial number") is located at the back of the joint between the cylinder block and the oil pan upper section -arrow-.