Klein invariant modular function: Differentiation (formula 09.50.20.0006)

KleinInvariantJ

$Mathematica Notation$

Elliptic Functions

KleinInvariantJ[z]

Differentiation

Low-order differentiation

http://functions.wolfram.com/09.50.20.0006.01

Input Form

D[KleinInvariantJ[z], {z, 3}] == ((128 I EllipticK[ModularLambda[z]]^4)/ (27 Pi^3 (-1 + ModularLambda[z])^2 ModularLambda[z]^2)) (3 EllipticE[ModularLambda[z]]^2 (-2 + ModularLambda[z]) (1 + ModularLambda[z]) (-1 + 2 ModularLambda[z]) (1 - ModularLambda[z] + ModularLambda[z]^2)^2 - 6 EllipticK[ModularLambda[z]] EllipticE[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) + 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))

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "3"]], "}"]]], RowBox[List["KleinInvariantJ", "[", "z", "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["128", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["ModularLambda", "[", "z", "]"]], "]"]], "4"]]], RowBox[List["27", " ", SuperscriptBox["\[Pi]", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["ModularLambda", "[", "z", "]"]]]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["ModularLambda", "[", "z", "]"]], "2"]]]], RowBox[List["(", RowBox[List[RowBox[List["3", SuperscriptBox[RowBox[List["EllipticE", "[", RowBox[List["ModularLambda", "[", "z", "]"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["ModularLambda", "[", "z", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["ModularLambda", "[", "z", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["ModularLambda", "[", "z", "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["ModularLambda", "[", "z", "]"]], "+", SuperscriptBox[RowBox[List["ModularLambda", "[", "z", "]"]], "2"]]], ")"]], "2"]]], "-", RowBox[List["6", " ", RowBox[List["EllipticK", "[", RowBox[List["ModularLambda", "[", "z", "]"]], "]"]], RowBox[List["EllipticE", "[", RowBox[List["ModularLambda", "[", "z", "]"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["ModularLambda", "[", "z", "]"]], "+", SuperscriptBox[RowBox[List["ModularLambda", "[", "z", "]"]], "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["6", "-", RowBox[List["18", " ", RowBox[List["ModularLambda", "[", "z", "]"]]]], "+", RowBox[List["13", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", "z", "]"]], "2"]]], "+", RowBox[List["4", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", "z", "]"]], "3"]]], "+", RowBox[List["6", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", "z", "]"]], "4"]]], "-", RowBox[List["11", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", "z", "]"]], "5"]]], "+", RowBox[List["4", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", "z", "]"]], "6"]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["ModularLambda", "[", "z", "]"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["ModularLambda", "[", "z", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "23"]], "+", RowBox[List["92", " ", RowBox[List["ModularLambda", "[", "z", "]"]]]], "-", RowBox[List["140", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", "z", "]"]], "2"]]], "+", RowBox[List["98", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", "z", "]"]], "3"]]], "-", RowBox[List["32", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", "z", "]"]], "4"]]], "+", RowBox[List["8", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", "z", "]"]], "5"]]], "+", RowBox[List["25", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", "z", "]"]], "6"]]], "-", RowBox[List["28", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", "z", "]"]], "7"]]], "+", RowBox[List["16", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", "z", "]"]], "8"]]]]], ")"]]]]]], ")"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mn> 3 </mn> </msup> <semantics> <mrow> <mi> J </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["J", "(", TagBox["z", Identity, Rule[Editable, True]], ")"]], InterpretTemplate[Function[KleinInvariantJ[Slot[1]]]]] </annotation> </semantics> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> </mfrac> <mo>  </mo> <mrow> <mfrac> <mrow> <mn> 128 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mrow> <mn> 27 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 3 </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> </mrow> <mo> - </mo> <mn> 23 </mn> </mrow> <mo> ) </mo> </mrow> <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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 6 </mn> <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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> </mrow> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </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]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 3 </cn> </degree> </bvar> <apply> <ci> KleinInvariantJ </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 128 </cn> <imaginaryi /> <apply> <power /> <apply> <ci> EllipticK </ci> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 27 </cn> <apply> <power /> <pi /> <cn type='integer'> 3 </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 /> <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> EllipticK </ci> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <ci> EllipticE </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> <ci> EllipticK </ci> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> EllipticE </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> <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> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02