What, exactly, is an "elliptic integral singular value"? Weisstein (http://mathworld.wolfram.com/EllipticIntegralSingularValue.html) when converted from modulus- to parameterspeak, says it's a value m=ModularLambda[√r] such that EllipticK[1-m]/EllipticK[m] ==√r, where r seems to be restricted to a positive rational. E.g., "the seventh singular value": ModularLambda[I Sqrt[7]] == 1/16 (8 - 3 Sqrt[7]) EllipticK[1/16 (8 - 3 Sqrt[7])] == Gamma[1/7] Gamma[2/7] Gamma[4/7]/(4 7^(1/4) π) EllipticK[1/2 + 3 Sqrt[7]/16]/ EllipticK[1/16 (8 - 3 Sqrt[7])] == Sqrt[7] Why rational r? Why √ ? Why positive? E.g., ModularLambda[2 I E^(-I π/6)] == -7 + 4 Sqrt[3] EllipticK[-7 + 4 Sqrt[3]] == 3^(1/4) Sqrt[2 + Sqrt[3]] Gamma[1/3]^3/(8 2^(1/3) π) EllipticK[8 - 4 Sqrt[3]]/EllipticK[-7 + 4 Sqrt[3]] == 2 E^(-I π/6) But things can get weird: ModularLambda[4 (-1)^(1/3)] == (-2 + Sqrt[2] - Sqrt[3] + Sqrt[6])^4 This is the same m as ModularLambda[2 I Sqrt[3]] == (-2 + Sqrt[2] - Sqrt[3] + Sqrt[6])^4 i.e., the "twelfth singular value" instead of period ratio 4 E^(-I π/6). EllipticK[(-2 + Sqrt[2] - Sqrt[3] + Sqrt[6])^4] == 3^(1/4) (1 + 2 Sqrt[2] - Sqrt[3]) (10 - 6 Sqrt[2] - 5 Sqrt[3] + 4 Sqrt[6]) Gamma[1/3]^3/ (16 2^(5/6) π) (Not quite the same as yesterday's.) EllipticK[1 - (-2 + Sqrt[2] - Sqrt[3] + Sqrt[6])^4]/ EllipticK[(-2 + Sqrt[2] - Sqrt[3] + Sqrt[6])^4] == 2 Sqrt[3] Even weirder: ModularLambda[I 3 E^(-I π/6)] == 1/2 (1 - Sqrt[1 - 4 (16001 - 6350 2^(1/3) (1 - I Sqrt[3]) - 5040 2^(2/3) (1 + I Sqrt[3]))]) EllipticK[1/2 (1 - Sqrt[ 1 - 4 (16001 - 6350 2^(1/3) (1 - I Sqrt[3]) - 5040 2^(2/3) (1 + I Sqrt[3]))])] == ((-1)^(1/12) 3^(1/4) Sqrt[5/9 + (2 2^(1/3))/9 - (2 I 2^(1/3))/(3 Sqrt[3])] Gamma[1/3]^3)/(4 2^(1/3) π) EllipticK[ 1/2 + 1/2 Sqrt[1 - 4 (16001 - 6350 2^(1/3) (1 - I Sqrt[3]) - 5040 2^(2/3) (1 + I Sqrt[3]))]]/ EllipticK[ 1/2 - 1/2 Sqrt[1 - 4 (16001 - 6350 2^(1/3) (1 - I Sqrt[3]) - 5040 2^(2/3) (1 + I Sqrt[3]))]] == Sqrt[7] E^(I ArcCot[3 Sqrt[3]]) instead of 3 E^(-I π/6) ! And sadly, ModularLambda[I GoldenRatio] seems transcendental. --rwg Elsewhere I said *rwg>I'm having no luck finding a table of special values of EllipticE. This may be due to a simpler formula than above to produce them from EllipticK. It looks like they're something K + something/K.* Indeed, see (40) and (41) of http://mathworld.wolfram.com/EllipticIntegralSingularValue.html . This requires the elliptic α function. But closed forms require finding 𝜗' for particular q, which is equivalent to the problem I was attacking with "logderiveta".