Hi @Gibbs,
Welcome to the Dyson Community!
A serial number is unique to each individual Dyson machine, so replacement batteries won’t include this - only the one originally supplied with the vacuum. The machine’s original serial number could also be located behind the clear bin, inside the handle or behind the filter - depending on the model. We would only be able to guarantee a replacement battery is genuine when purchased directly from Dyson, or from one of our authorised retailers.
Can you let me know which country you’re based in? I should be able to help guide you to the right replacement part on the Dyson website.