I want to download my purchased software to a new computer that I own.

You don't have wrote which excat version of Acrobat Standard DC you have buyed. But...

... take a look here if this match with your version:

https://helpx.adobe.com/download-install/kb/acrobat-2020-downloads.html

https://helpx.adobe.com/download-install/kb/acrobat-2017-downloads.html

For serial number programs you may download ONLY if you bought direct from Adobe via download
-Log-in to your Adobe account and click this link https://account.adobe.com/products
--click Order history on the left and then the link to your order that will show on the right
--and finally click the link "Downloads and serial numbers" that will appear
---as always, write the serial number in a text file AND copy that text file and the
---download file(s) to an external drive (or two) to be sure you always have what you need
-if you did not buy direct from Adobe via download you must find your original install
-if you have an Education or volume license you MUST use your original media
--the link for an Education or volume license download is not available at all
-if you bought a disc and don't have a drive you will need to buy an external USB drive
-if you bought and lost a disc Adobe does not provide a replacement via download
-if you bought a download from a 3rd party vendor you will need to contact that vendor