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The solution is to explicitly declare T as a finite field with 241^32 elements:

sage: K.<x> = GF(241)[]
sage: T.<a> = GF(241^32, modulus=x^32+44)
sage: J.<s> = T[]
sage: G = s^32 + 7
sage: G.roots()
[(221*a^13, 1),
 (190*a^13, 1),
 (51*a^29, 1),
 (20*a^29, 1)]

I would be great if Sage were able to detect that your construction is equivalent to mine, but it is unfortunately not the case (yet)!