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First of all, it would have been wise to give your number in factorized form. The problem seems to be that Sage tries to be "too" clever by choosing a canonical polynomial for the extension GF(p^12):GF(p). You can generate a random irreducible polynomial and build the extension by hand
sage: p = 21888242871839275222246405745257275088696311157297823662689037894645226208583
sage: A = GF(p)
sage: B = A['x']
sage: poly = B.random_element(12, monic=True)
sage: while not poly.is_irreducible():
....: poly = B.random_element(12, monic=True)
sage: C = A.extension(poly, 'u')
sage: C
Finite Field in u of size 21888242871839275222246405745257275088696311157297823662689037894645226208583^12