I am trying to figure out how many stereoisomers this compound has. I know that there are two chiral carbon atoms, which allow $2\times2 = 4$ possible isomers. There is also a double bond which, as far as I know, doubles the number of stereoisomers, since there are now four possibilities for the cis-isomer, and four for the trans-isomer. So I thought the total number of stereoisomers was eight, but my teacher told me the answer was six. I have thought about it for a while, but I can't figure out why it should be six. What did I do wrong?
The answer lies in the double bond. The alcohol on the double bond undergoes keto-enol tautomerism. The greatly more stable aldehyde effectively makes the vinylic alcohol non-existent. This removes a factor for stereoisomers. six is an odd number of stereo isomers to have since 6 is factored to 2 and 3. This implies a stereoisomer with 3 forms. I think your teacher may be thinking of the trans, cis and aldehyde state of the alcohol to come up with six, but this would neglect a carbon stereo center. in this case 12 would be the correct answer. However you should ignore the tautomers and just assume aldehyde which means the answer should be 4.