Consider also the size and quality of your PSU. If it's a decent 500W model, it will run a HD5770. If it is a chokemax (has"max in its name but isn't Enermax, or has a little voltage switch on it), it is not to be trusted for more than 75% of its label, so you might be limited to a HD5670.
Yeah, that's a problem. On that PSU the best card I would be sure about recommending is an HD5570 but perhaps an HD5670 would be ok. For a card better than that you will need to upgrade the power supply. A 400w PSU of a quality brand should be good for the HD5770.