What is the serial number tied to?
It's also possible that is a classic jazz neck put on a P bass. A new waterslide decal to change it to a "P Bass " on the headstock. It's not uncommon for necks to get switched out, some people prefer a thinner neck on their P Basses. There's something strange about the font and color of the "Fender" on the headstock too, The classic 70s series is a black "Fender" outlined in gold. This is all with no outline - this is the color of the Fender Squier Series or Traditional series. These basses were known as the Black Label Fenders. I know because I had a Fender Jazz bass Squier series.