Most of the 13 footers will be over 1500lbs loaded so finding a 4 cylinder that will pull this is difficult. The toyota/nissan 4 cylinder engines in their small trucks will do it as will the Subaru Forester/Outback/Legacy's. The trucks don't require trailer brakes
but the soob's do. We pull our trailer fine with our 2.4cyl Toyota.
The small 4cyl SUV's (Elements/Rav4's/CRV's) are not rated to pull enough for the 13' footers though some members have done so. There are many threads on this forum about TV's but realistically you should never exceed what the car manual says for liability reasons.
Saying this, some of the new 6 cylinder SUV's like the RAV4's get excellent gas mileage so don't restrict yourself to just 4 cylinders. Sometimes a big engine in small car will get better mileage because it doesn't have to work hard.