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I got approved for a Citi Double Cash card a couple of years ago. When I applied I didn't have any credit cards but I did have student loans that I was (am) paying on. In fact the student loans were the only installment loans on my credit report at the time. My credit score was pretty similar to yours.
Did you try their preapproval page? I have a Cap1 quicksilver, Chase Sapphire Preferred, Amex PRG, Amex Blue Cash AU, and student loans. Experian score at 660. Tried their preapproval before but I'm sure it would be a denial as well.