Specifically, EllipticE[m] == EllipticK[m] + (Pi^2 q Inactive[D][Log[EllipticTheta[4, 0, q]], q])/EllipticK[m] or EllipticE[m] == EllipticK[m] - (I Pi (Derivative[1][DedekindEta][-((I Log[q])/(2 Pi))]/ DedekindEta[-((I Log[q])/(2 Pi))] - Derivative[1][DedekindEta][-((I Log[q^2])/(2 Pi))]/ DedekindEta[-((I Log[q^2])/(2 Pi))]))/EllipticK[m] E.g., associated with the 5th singular value, Out[332]= EllipticE[1/2 - Sqrt[-2 + Sqrt[5]]] == EllipticK[1/2 - Sqrt[-2 + Sqrt[5]]] + ( 160 Sqrt[5] \[Pi]^2 - 5^(1/4) (-4 + Sqrt[10 (1 + Sqrt[5])]) Gamma[1/20]^2 Gamma[9/ 20]^2)/(3200 \[Pi] EllipticK[1/2 - Sqrt[-2 + Sqrt[5]]]) The harmless little quartic (actually octic) surd required enhanced interrogation. In[337]:= Log[EllipticNomeQ[1/2 - Sqrt[-2 + Sqrt[5]]]]/\[Pi] // FunctionExpand Out[337]= -(EllipticK[1/2 + Sqrt[-2 + Sqrt[5]]]/ EllipticK[1/2 - Sqrt[-2 + Sqrt[5]]]) In[339]:= N[%^2] Out[339]= 5. Express K as Gammas via etas: In[344]:= EllipticK[(16*DedekindEta[\[Tau]]^8*DedekindEta[4*\[Tau]]^16)/ DedekindEta[2*\[Tau]]^24] == (Pi*DedekindEta[2*\[Tau]]^10)/ (2*DedekindEta[\[Tau]]^4* DedekindEta[4*\[Tau]]^4) /. \[Tau] -> Sqrt[-5]/2 Out[344]= EllipticK[( 16 DedekindEta[(I Sqrt[5])/2]^8 DedekindEta[2 I Sqrt[5]]^16)/ DedekindEta[I Sqrt[5]]^24] == (\[Pi] DedekindEta[I Sqrt[5]]^10)/( 2 DedekindEta[(I Sqrt[5])/2]^4 DedekindEta[2 I Sqrt[5]]^4) Are these the droids we're looking for? In[345]:= RootApproximant[N[%[[1, 1]], 69]] Out[345]= Root[1 - 72 #1 + 88 #1^2 - 32 #1^3 + 16 #1^4 &, 1] In[347]:= ToRadicals[%%] // FullSimplify Out[347]= 1/2 - Sqrt[-2 + Sqrt[5]] Yup. Replace the etas from the table. In[350]:= %344 /. etavals Out[350]= EllipticK[((1 + Sqrt[5])^9 (5 - Sqrt[5] - 2 Sqrt[-110 + 50 Sqrt[5]])^2 (1 + 2/GoldenRatio^(3/2)))/( 20480 GoldenRatio)] == ( 2 2^(3/4) Sqrt[ GoldenRatio/((5 - Sqrt[5] - 2 Sqrt[-110 + 50 Sqrt[5]]) (1 + 2/ GoldenRatio^(3/2)) \[Pi])] Gamma[1/20] Gamma[9/20])/( 5^(3/8) (1 + Sqrt[5])^(15/4)) In[351]:= FullSimplify[Rule @@ %] Out[351]= EllipticK[1/2 - Sqrt[-2 + Sqrt[5]]] -> (Gamma[1/20] Gamma[9/20])/( 4 (5 (85 + 38 Sqrt[5]))^(1/4) Sqrt[(25 - 11 Sqrt[5]) \[Pi]]) Dispel yucky surds. In[387]:= Denominator[%351[[2]]]/Sqrt[\[Pi]] Out[387]= 4 Sqrt[25 - 11 Sqrt[5]] (5 (85 + 38 Sqrt[5]))^(1/4) In[388]:= Log[Prepend[{2, 5, GoldenRatio}, %]] Out[388]= {Log[4 Sqrt[25 - 11 Sqrt[5]] (5 (85 + 38 Sqrt[5]))^(1/4)], Log[2], Log[5], Log[GoldenRatio]} In[389]:= %.FindIntegerNullVector[%] Out[389]= -20 Log[2] - 5 Log[5] + 8 Log[4 Sqrt[25 - 11 Sqrt[5]] (5 (85 + 38 Sqrt[5]))^(1/4)] + 2 Log[GoldenRatio] In[401]:= FullSimplify /@ (%351[[1]] -> %351[[2]]*E^(%389/8) /. GoldenRatio -> \[Phi]) Out[401]= EllipticK[1/2 - Sqrt[-2 + Sqrt[5]]] -> ( 5^(3/8) \[Phi]^(1/4) Gamma[9/20] Gamma[21/20])/Sqrt[2 \[Pi]] This must be my yucky day! Back to E: In[402]:= %332 /. % /. \[Phi] -> GoldenRatio Out[402]= EllipticE[1/2 - Sqrt[-2 + Sqrt[5]]] == ( 160 Sqrt[5] \[Pi]^2 - 5^(1/4) (-4 + Sqrt[10 (1 + Sqrt[5])]) Gamma[1/20]^2 Gamma[9/ 20]^2)/(1600 5^(3/8) GoldenRatio^(1/4) Sqrt[2 \[Pi]] Gamma[9/20] Gamma[21/20]) + ( 5^(3/8) GoldenRatio^(1/4) Gamma[9/20] Gamma[21/20])/Sqrt[2 \[Pi]] In[403]:= FullSimplify /@ % Out[403]= EllipticE[1/2 - Sqrt[-2 + Sqrt[5]]] == ( 5^(1/4) (2 Sqrt[10] + 5 Sqrt[1 + Sqrt[5]]) Gamma[1/20] Gamma[9/ 20]^2 + (20 Sqrt[2] \[Pi]^2)/Gamma[21/20])/( 40 2^(3/4) 5^(7/8) (1 + Sqrt[5])^(1/4) Sqrt[\[Pi]] Gamma[9/20]) which is the best I could do. --rwg