Gonality of the modular curve X1(N)X1(N)

In this paper we compute the gonality over QQ of the modular curve X1(N)X1(N) for all N⩽40N⩽40 and give upper bounds for each N⩽250N⩽250. This allows us to determine all N   for which X1(N)X1(N) has infinitely points of degree ⩽8. We conjecture that the modular units of Q(X1(N))Q(X1(N)) are freely generated by f2,…,f⌊N/2⌋+1f2,…,f⌊N/2⌋+1 where fkfk is obtained from the equation for X1(k)X1(k).