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Let us consider the "simpler" situation for the degree of the cyclotomic polynomial:
p = ZZ( (2^128-3)/76439 )
F = GF(p)
S.<W> = PolynomialRing( F )
K.<w> = GF( p^2 )
R.<z> = PolynomialRing( K, sparse=True )
# N = 9817501343317677957515279627338949795623675802414662483433914174328
N = 2^3 * 17 * (1000.next_prime())
f = cyclotomic_polynomial( N, 'z' )
len(f.monomials())
This code constructs $f$ having
4985
monomials. The value of $N$ is
sage: N
137224
and it is "small", when compared to the "bigger" number
sage: 9817501343317677957515279627338949795623675802414662483433914174328.factor()
2^3 * 17 * 72187509877335867334671173730433454379585851488343106495837604223
For this smaller $N$, the polynomial $f$ was computed and stored. It is of course no place to store the corresponding data for the "bigger" number instead. And the code for the computation of the cyclotomic polynomial was designed for such practical situations.
