Battery Light On After Dead Battery
2012 Toyota Corolla LE 1.8L Automatic with almost 75k miles. The car wouldn't start last week. I removed the battery from the car and charged it over night (NOCO Genius G26000). Installed the battery and car started right up, but battery light on (light was NOT on before dead battery). The next day when I tried to start it, it barely started. I hopped onto the highway for 40 minutes and then ran some errands that day with no problem, but two days later when trying to start the car, just clicking noise. Went to Autozone and bought a Duralast Platinum AGM battery and installed it. The car starts up fine now, but battery light still on. I noticed that it seems that the blower for the heat/AC is weak. Only the fastest speed (4) makes a noise (air blowing). 1 and 2 are silent and 3 barely makes any noise. The wiper blades also seem to move slower than normal. When I rotate the steering wheel to back out of a parking spot or making a u-turn, there is some resistance turning the wheel.
Is the alternator the culprit? I guess not driving the car much for the past four months has killed the battery and the alternator. If it is the alternator, should I get a new one or a remanufactured one? Was looking at RockAuto, they have only remanufactured Denso. Local parts stores only have after-market.
BTW, the old battery was a Diehard Gold (E5 - May 2015).