I've been in the process of getting beta3 to build on Cygwin using 
--with-blas=atlas and Cygwin's GCC 6.3. Here is what I needed to do:

- Python3 was not building due to a 3.4.5-struct.patch in #22666. Erik sent 
me an revised patch for this.
- Pynac was not building, so I "reverted" 22764 by adding a symlink 
$SAGE_LOCAL/lib/python -> $SAGE_LOCAL/lib/python2.7. (This occurred also 
with Pynac 0.7.5.)
- Add gmp to the libraries in module_list.py for libs/ecl.pyx and 
rings/bernmm. This was likely caused by #22728. Fix at 
https://trac.sagemath.org/ticket/22878.

This also included the tickets #21399, #21459, #22810, #22628, along with 
singular (#22868), pynac (#22838), and NTL (#22869) upgrade tickets.

Best,
Travis

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