Name: Alma Phoenix & Cottonwood, AZ USDA zone 9b, Sunset 13 & ??
Mary, I have no doubt yours is a chiltepin, too. In all my reading and viewing photos of these peppers for a few years (since I planted the first one), I've seen both round and pointy ones. No need to second guess.
Alma
In matters of style, swim with the current; in matters of principle, stand like a rock. – Thomas Jefferson
