(Big deal. The singular ones are probably 0% of the doables.) EllipticK[( 6561 (Sqrt[2] + 5^(1/4))^8 (-1 + Sqrt[5])^4 (1 + Sqrt[5])^16)/( 1048576 (1 + Sqrt[2])^8 (Sqrt[2] + Sqrt[5])^8 (1 + Sqrt[ 10])^4)] == ( 4 (1 + Sqrt[2])^2 (1 + 5^(1/4))^4 (Sqrt[2] + Sqrt[5])^2 (1 + Sqrt[ 10]) Sqrt[2/\[Pi]] Gamma[1/4]^2)/( 9 (Sqrt[2] + 5^(1/4))^2 (-1 + Sqrt[5])^2 (1 + Sqrt[5])^6) EllipticK[ 1 - (6561 (Sqrt[2] + 5^(1/4))^8 (-1 + Sqrt[5])^4 (1 + Sqrt[5])^16)/( 1048576 (1 + Sqrt[2])^8 (Sqrt[2] + Sqrt[5])^8 (1 + Sqrt[ 10])^4)] == ( 8 (1 + Sqrt[2])^2 (1 + 5^(1/4))^4 (Sqrt[2] + Sqrt[5])^2 (1 + Sqrt[ 10]) Sqrt[2/\[Pi]] Gamma[1/4]^2)/( 45 (Sqrt[2] + 5^(1/4))^2 (-1 + Sqrt[5])^2 (1 + Sqrt[5])^6) EllipticK[( 6561 (Sqrt[2] + 5^(1/4))^8 (-1 + Sqrt[5])^4 (1 + Sqrt[5])^16)/( 1048576 (1 + Sqrt[2])^8 (Sqrt[2] + Sqrt[5])^8 (1 + Sqrt[10])^4)]/ EllipticK[ 1 - (6561 (Sqrt[2] + 5^(1/4))^8 (-1 + Sqrt[5])^4 (1 + Sqrt[5])^16)/( 1048576 (1 + Sqrt[2])^8 (Sqrt[2] + Sqrt[5])^8 (1 + Sqrt[ 10])^4)] == 5/2 The complementary parameter likewise factors into binomials: 1 - (6561 (Sqrt[2] + 5^(1/4))^8 (-1 + Sqrt[5])^4 (1 + Sqrt[5])^16)/( 1048576 (1 + Sqrt[2])^8 (Sqrt[2] + Sqrt[5])^8 (1 + Sqrt[10])^4) == ( 324 Sqrt[2] (Sqrt[2] + 5^(1/4))^4 (1 + Sqrt[5])^6)/((1 + Sqrt[ 2])^4 (1 + 5^(1/4))^12 (Sqrt[2] + Sqrt[5])^4 (1 + Sqrt[10])^2) I think Ramanujan actually understood why. —rwg