Klein invariant modular function: Differentiation (formula 09.50.20.0008)

KleinInvariantJ

$Mathematica Notation$

Elliptic Functions

KleinInvariantJ[z]

Differentiation

Low-order differentiation

http://functions.wolfram.com/09.50.20.0008.01

Input Form

D[KleinInvariantJ[z], {z, 5}] == ((2048 I EllipticK[ModularLambda[z]]^6)/ (27 Pi^5 (-1 + ModularLambda[z])^2 ModularLambda[z]^2)) (-15 EllipticE[ModularLambda[z]]^4 (-2 + ModularLambda[z]) (1 + ModularLambda[z]) (-1 + 2 ModularLambda[z]) (1 - ModularLambda[z] + ModularLambda[z]^2)^2 + 60 EllipticE[ModularLambda[z]]^3 EllipticK[ModularLambda[z]] (1 - ModularLambda[z] + ModularLambda[z]^2) (6 - 18 ModularLambda[z] + 13 ModularLambda[z]^2 + 4 ModularLambda[z]^3 + 6 ModularLambda[z]^4 - 11 ModularLambda[z]^5 + 4 ModularLambda[z]^6) - 30 EllipticE[ModularLambda[z]]^2 EllipticK[ModularLambda[z]]^2 (-2 + ModularLambda[z]) (-23 + 92 ModularLambda[z] - 140 ModularLambda[z]^2 + 98 ModularLambda[z]^3 - 32 ModularLambda[z]^4 + 8 ModularLambda[z]^5 + 25 ModularLambda[z]^6 - 28 ModularLambda[z]^7 + 16 ModularLambda[z]^8) + 10 EllipticE[ModularLambda[z]] EllipticK[ModularLambda[z]]^3 (212 - 1060 ModularLambda[z] + 2175 ModularLambda[z]^2 - 2340 ModularLambda[z]^3 + 1380 ModularLambda[z]^4 - 402 ModularLambda[z]^5 + 36 ModularLambda[z]^6 - 12 ModularLambda[z]^7 + 123 ModularLambda[z]^8 - 112 ModularLambda[z]^9 + 32 ModularLambda[z]^10) - EllipticK[ModularLambda[z]]^4 (-2 + ModularLambda[z]) (-567 + 2835 ModularLambda[z] - 5774 ModularLambda[z]^2 + 6086 ModularLambda[z]^3 - 3440 ModularLambda[z]^4 + 926 ModularLambda[z]^5 - 131 ModularLambda[z]^6 + 149 ModularLambda[z]^7 - 20 ModularLambda[z]^8 - 64 ModularLambda[z]^9 + 64 ModularLambda[z]^10))

Standard Form

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MathML Form

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Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ) </mo> </mrow> <mn> 6 </mn> </msup> </mrow> <mrow> <mn> 27 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 5 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 15 </mn> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 2 </mn> </msup> <mo> - </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 60 </mn> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 2 </mn> </msup> <mo> - </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 11 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 13 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 18 </mn> <mo> ⁢ </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 30 </mn> <mo> ⁢ </mo> <msup> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 16 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 8 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 28 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 25 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 32 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 98 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 140 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 92 </mn> <mo> ⁢ </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> <mo> - </mo> <mn> 23 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 10 </mn> <mo> ⁢ </mo> <msup> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 32 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 10 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 112 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 9 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 123 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 8 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 36 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 402 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1380 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2340 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2175 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 1060 </mn> <mo> ⁢ </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> <mo> + </mo> <mn> 212 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 64 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 10 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 64 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 9 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 20 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 149 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 7 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 131 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 926 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3440 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 6086 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 5774 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2835 </mn> <mo> ⁢ </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> <mo> - </mo> <mn> 567 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 5 </cn> </degree> </bvar> <apply> <ci> KleinInvariantJ </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2048 </cn> <imaginaryi /> <apply> <power /> <apply> <ci> EllipticK </ci> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 27 </cn> <apply> <power /> <pi /> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -15 </cn> <apply> <plus /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> EllipticE </ci> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 60 </cn> <apply> <ci> EllipticK </ci> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 11 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 13 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 18 </cn> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <apply> <ci> EllipticE </ci> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 30 </cn> <apply> <power /> <apply> <ci> EllipticK </ci> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 28 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 25 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 98 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 140 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 92 </cn> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> -23 </cn> </apply> <apply> <power /> <apply> <ci> EllipticE </ci> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 10 </cn> <apply> <power /> <apply> <ci> EllipticK </ci> <apply> <ci> ModularLambda </ci> <ci> z 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</cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 402 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1380 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2340 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2175 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1060 </cn> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 212 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Rule Form

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Date Added to functions.wolfram.com (modification date)

2007-05-02