Installed the same in my garage. I installed a 240v line with 6 guage wire on a 50amp breaker for the NEMA 14-50 outlet. Good news is that LADWP will rebate up to $500 for a charger so this will end up costing me about $80 for the outlet install and about $100 for the charger above the $500 total $180 for a 40 amp level 2 charger. But man does that elec meter wheel spin when its plugged in. I calculated it will cost about $2.50 per charge at my rate. Cheaper than a gallon of gas.
LADWP also has a Time of Use rate plan that you can sign up for that lowers your electric rate between 8:00 p.m. 10:00 a.m., Monday through Friday, all day Saturday and Sunday, regular rates other times. With one big exception it also raises the rate substantially above standard rate during peak period which is Sep-June 1:00 p.m. 5:00 p.m., Monday through Friday. If you are home during those hours you would want to try and chill down the house more than you normally would prior to 1:00pm and then try and minimize AC use until 5:00pm.
Note that if you stay with the standard rate plan most of your EV charging will likely be in the Tier 3 rate which is anything above 1050 kWh, so with standard (non-Time of Use) rate you will be paying a bit more for your EV charging than for most of your other electric use.
LADWP also has an additional 0.025 discount for EV charging, although it requires having a second meter installed. I have no idea how much they charge for installing the second meter I am guessing it is not free, whereas I think installing the first TOU meter is probably free because that just replaces your original meter, although where I live (Georgia) it turns out I already had a TOU meter and a guy just had to come out and reprogram it, using his laptop while sitting in his truck!
Note also that any of these discount rates only reduce the rate portion of your bill and sales tax, other taxes and tariffs are based on kWh hours and the rate discount has no effect on them. So while you will usually save money it may not be as much as you estimate using a back of the envelope calculation based only on the percentage reduction of electric rate.