Here's the public SageMathCloud worksheet: https://cloud.sagemath.com/projects/49a2531d-9d02-42c9-9db6-...
I wish they would have supplied those matrices they printed out in machine-readable format so we could easily put those in Sage too.
I wrote the det code in Sage that you're calling there (though in Sage-6.4 it'll likely be replaced by a call to FLINT). It computes det(A) in a very interesting way, which is asymptotically massively faster than Mathematica. To compute det(A), choose a random vector v and solve Ax = v using a p-adic lifting algorithm (the one in https://cs.uwaterloo.ca/~astorjoh/iml.html). One can prove --using Cramer's rule--that the lcm of the denominators of the entries of x will then be a divisor d of det(A), and with high probability one expects that det(A)/d is a tiny integer. One can then provably (using the Hadamard bound) find det(A) by working modulo a few additional primes and using the Chinese Remainder theorem.