Hardy's Taxicab

Another Cambridge mathematician, Littlewood, said that every positive integer was one of Ramanujan's personal friends.

Hardy wrote:

I remember once going to see [Ramanujan] when he was lying ill at Putney. I had ridden in taxi cab number **** and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen.

"No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

Vleeptron PizzaQ Honor Code -- no Googling, etc., just pure, solitary brainwork. (Oh, okay, you can program your computer to find the answer -- but remember that Ramanujan got the answer just lying in a hospital bed.)

What was the number of Hardy's taxicab?