Can I work on an e19 green card?

The code on your green card indicates to the Immigration Service how did you become a permanent resident; it has some utility in the agency's statistics and processing paperwork. But your rights and duties as a permanent resident are the same regardless of the category in which you are classified (with very few exceptions which are of no importance in this case). One of the rights you have as a permanent resident is the right to be work for any employer who would hire you; you do not need an extra document, like an employment authorization card, in order to accept employment or to obtain a Social Security number.