The Honda Civic LX has a MSRP of $18,490 - so the price you have is pretty decent. The Corolla LE is not considered the base model, or did you mean to type Corolla L? It is the Corolla LE, then that price is pretty good - LE trim levels start at $18,565 MSRP. The Corolla L MSRP runs $16,950. Equipment level wise - the Honda Lx and Corolla LE are pretty much tied. Note that the Corolla comes with 2 years / 25K miles of free maintenance standard, not sure what you meant by one free oil change - you get those already for that period of time.
Keep in mind, with an economy car like these, there isn't much margin to be made - so discounts are few and far between. The prices you have might be different, depending on your market - some would consider those prices high, some would see them as a good deal - all depends on where you live at. Where I live at (DC Metro area) - the Corolla LE is at a great price, but the Honda Civic LX I could get a for atleast a grand cheaper.
Reliability between the two is also a wash - these two brands have historically outlasted most of their competitors, especially at this price point. Sure there will also be some people with horror stories on these cars, but the majority have been trouble-free. Overall fuel economy, cabin size, ride, safety equipment, etc. - pretty close between these
two.
I've owned both brands over the years - can't say one is definitely superior to the other. Best bet is to see if you can get an extended test drive on both, see which one fits your body and needs the best. Also couldn't hurt to look at other competitors - these two are considered the "benchmark" for their class - but offerings from their competitors tend are pretty intense. Some, you can get more car for the money - some offer features that Honda and Toyota does not have at this price, including:
Ford Focus SE
Dodge Dart SXT
Hyundai Elantra SE
Nissan Sentra SV
Mazda 3i
Subaru Impreza 2.0i
If it was me - I would lean more toward the Honda Civic LX compared the Corolla LE - as the biggest issue would be the choice of transaxle. Civic has a conventional transaxle where as the Toyota has CVT. CVT have shown pretty decent reliability with commuter cars, having been used for decades in larger transportation trucks and busses - but the Toyota CVT just never sat well with me. If you compared the Civic LX to the Corolla L - that's a different story. Both will have 5-speed manual transaxles standard, Toyota will lose some overall content, but the core components (powertrain, airbags, ABS/ECT/EBD, LED lamps, brakes, etc.) are identical, and would be almost $1500 cheaper as well.
Another thing to factor in would be insurance costs. In my case and where I live at, the Honda's premium would run about $600-$700 more per year - that's roughly two car payments. Costs of maintenance would also be slightly higher (you don't get two years / 25K miles free like the Corolla) and depreciation for Hondas tend to be higher in my area. But like I mentioned, depends on where you live - if I moved back to the mid-eastern US - the Honda would actually be cheaper to insure and depreciate less.
