[thephantom1492]

If the battery got mesured at 12.71V, then it mesure full.

You should borrow a multimeter for this issue...

Also, how did you jump started? both wires on the battery or positive on battery and negative on engine?

I am trying to figure out why with a full battery the car don't start, but with a boost it does.

I might have went on the wrong track with my alternator idea...

See, if you have a battery that test good and have 12.71V then the alternator should work, and the car should have enought power to start. Meaning boosting shouln't do anything at all.

Now, why is boosting allowing to start and not the battery.

If you are jump starting with positive on battery and ground on the engine, it could point to a bad ground wire from battery to engine, or bad contact at the battery terminals. A multimeter would help ALOT to diagnose this... and confirm that you do have 12.71V at the battery.