First of all, F1 students will not file a Form 1040. They will file a Form 1040NR.
1040 is for residents and citizens. You can only get the education credit if you file 1040.
1040NR is for non residents.
F1 students are Non Residents.

Non Residents do not receive the credit for education.

You should contact a tax professional or an immigration lawyer for your case.  Internal Revenue Service and Homeland Security do exchange information.

I can give you the contact info for a reputed CPA firm in Maryland who gives 30 minutes free consultation.

Let me know if you prefer.

Thanks