We write E_k for \Eta(x^k). One more expression for \Eta(i*x):
# # [ 16 ] # | E | # 1 | 4 |1/4 4 2 2 # \Eta(x) = -- | --- - 4 x U | \where U = E E E # E [ U ] 8 4 2 # 4 #
Expressions for \Eta(W*x) where W is the third root of unity are
# # [ R 3 ]1/3 # \Eta(x) = | -- - 3 x E | \where # | E 9 | # [ 9 ] # # [ 3 12 12 ]1/3 # R = | 27 x E + E | # [ 9 3 ] #
and
# # [ 1 / 4 3 3 4 \ ]1/3 # \Eta(x) = | -- | E + 9 x E E - 3 x E | | # | E \ 3 3 27 9 / | # [ 9 ] #
These follow from relations given in Somos' eta07.gp
The following clean relations for fifth and seventh roots do not seem to be solvable (for E1):
E1^5*E25 +5*x*E1^4*E25^2 +15*x^2*E1^3*E25^3 +25*x^3*E1^2*E25^4 \ +25*x^4*E1*E25^5 -1*E5^6
It's clearly *possible*: P(1,5*a,5*b) mod P(1,5,25) will be at worst quartic in eta(q). The only trick is to find a remainder smaller than the entire math-fun archive. --rwg
E1^7*E49 +147*x^8*E1^3*E49^5 +21*x^4*E1^5*E49^3 +343*x^10*E1^2*E49^6 \ +343*x^12*E1*E49^7 +35*x^3*E1^2*E7^4*E49^2 +49*x^5*E1*E7^4*E49^3 \ +49*x^6*E1^4*E49^4 +7*x*E1^3*E7^4*E49 +7*x^2*E1^6*E49^2 -1*E7^8
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