The three integers below can be expressed as the sum of two squares. The three integers are:
\begin{align}(1)&&11,572,060,353,961,555,386,606,814,001 \\(2)&&11,573,624,522,376,724,598,676,284,401 \\(3)&&11,575,215,560,569,326,509,742,400,801\end{align}
Can anyone tell me how to use SageMath (Computer Algebra System) to check whether any of these three integers is the sum of two fourth powers ?
If $n=(a_1^2+b_1^2)(a_2^2+b_2^2)(a_3^2+b_3^2)(a_4^2+b_4^2)(a_5^2+b_5^2)$, as the OP indicates is the case for one of the numbers, then it suffices to consider $16$ possibilities for $n=A^4+B^4$:
$$A^2+B^2i=(a_1+b_1i)(a_2\pm b_2i)(a_3\pm b_3i)(a_4\pm b_4i)(a_5\pm b_5i)$$
In other words, if you're given a factorization of the number of interest into a handful of primes congruent to $1$ mod $4$, then one approach to testing if the number is a sum of two fourth powers is to express each prime factor as a (unique) sum of two squares, and then test a relatively small number of combinations expressed as Gaussian integers.
Note, though, that finding the squares that sum to a large prime is a nontrivial step.