A triangular number is a positive integer that can be expressed in the form $t_n=1+2+3+\cdots+n$, for some positive integer $n$. The three smallest triangular numbers that are also perfect squares are $t_1=1=1^2, t_8=36=6^2,$ and $t_{49}=1225=35^2$. What is the sum of the digits of the fourth smallest triangular number that is also a perfect square? $\textbf{(A)} ~6 \qquad$ $\textbf{(B)} ~9 \qquad$ $\textbf{(C)} ~12 \qquad$ $\textbf{(D)} ~18 \qquad$ $\textbf{(E)} ~27$