Pay off car loan immediately or let it establish on my credit report?

So, I just bought a car. I put a decent chunk down and financed the rest. I couldn’t make myself pay cash, it was just too much to part with all at once, as it would have depleted my emergency fund. Interest rate isn’t terrible with today’s rates (under 6%). Then right after taking out the loan, before the first payment is even due, I had two unexpected windfalls, which just happen to be enough to completely cover the car loan.

I have no other high interest rate debt to pay off. There is no prepayment penalty, I have an emergency fund and my retirement accounts are funded for the year. So I’m looking to throw the windfalls at paying for the car loan in full.

Is it better credit-wise to pay off the car loan immediately upon being notified which bank has the loan? Or to let it get established on your credit report for a few months before paying it in full? I have an excellent credit score (over 800), but not a whole lot on my credit report, as I only have a mortgage, a few cards that I pay off every month, and haven’t had a car loan since 2008, when we paid off my husband’s current car (my former car was really old, and has been paid off even longer).

I was thinking about paying everything except $5000 off immediately, and then paying $1000/month until the remainder is paid off, to cut down on the interest, but give it a few months to show payment history. But I’m open to changing that if there is a better option credit wise.