Page 15 of

https://eprint.iacr.org/2014/653.pdf summarizes matrix size and CPU-time for a list of mersenne factorizations completed by the CADO group.

Sampling, 2^1123-1 was 124M matrix, and took 55 core-years.

2^1199-1 (the largest solved yet) was 270M matrix, and took 190 core-years.

Those are 76 bits apart; M1277 is 78 bits larger, so let's scale:

270/124 = 2.18, so M1277 might be 2.18 * 270M = ~520M matrix.

190/55 = 3.45, so M1277's matrix might take 3.45 * 190 = ~660 core-years.

I don't think any single machine is going to handle that, even if memory weren't a constraint. MPI splits the matrix in a way that also splits the memory requirement, so even if this matrix requires 200GB a cluster of 16 machines could solve it with normal amounts of memory.

A similar scaling of Greg's M1061 note of 3 CPU-centuries sieve time is sobering.

Edit to add: Table 2 page 11 of the above linked paper lists sieve data. For 2^1199-1: 37-bit large primes, mfbr = 109, mfba = 74, 13G raw relations. Not sure whether M1277 would best use 38 or 39 bit LP; let's say 39 and aim for 40G relations.