Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











KleinInvariantJ






Mathematica Notation

Traditional Notation









Elliptic Functions > KleinInvariantJ[z] > Series representations > Generalized power series > Expansions at generic point z==z0 > For the function itself





http://functions.wolfram.com/09.50.06.0003.01









  


  










Input Form





KleinInvariantJ[z] \[Proportional] KleinInvariantJ[Subscript[z, 0]] + ((8 I EllipticK[w]^2 (-1 + w) w (1 - ModularLambda[Subscript[z, 0]] + ModularLambda[Subscript[z, 0]]^2)^2 (2 - 3 ModularLambda[Subscript[z, 0]] - 3 ModularLambda[Subscript[z, 0]]^2 + 2 ModularLambda[Subscript[z, 0]]^ 3))/(27 ((-EllipticE[1 - w]) EllipticK[w] + EllipticK[1 - w] (-EllipticE[w] + EllipticK[w])) (-1 + ModularLambda[Subscript[z, 0]])^3 ModularLambda[Subscript[z, 0]]^ 3)) (z - Subscript[z, 0]) - 8 EllipticK[w]^3 (1 - w) w (1 - ModularLambda[Subscript[z, 0]] + ModularLambda[Subscript[z, 0]]^2) ((EllipticE[w] ModularLambda[Subscript[z, 0]] (2 - 7 ModularLambda[Subscript[z, 0]] + 7 ModularLambda[Subscript[z, 0]]^2 - 7 ModularLambda[Subscript[z, 0]]^4 + 7 ModularLambda[Subscript[z, 0]]^5 - 2 ModularLambda[Subscript[z, 0]]^6) + EllipticK[w] w (2 w (3 - 10 ModularLambda[Subscript[z, 0]] + 9 ModularLambda[Subscript[z, 0]]^2 + ModularLambda[Subscript[z, 0]]^3 + 2 ModularLambda[Subscript[z, 0]]^4 - 3 ModularLambda[Subscript[z, 0]]^5 + ModularLambda[Subscript[z, 0]]^6) - 6 + 18 ModularLambda[Subscript[z, 0]] - 11 ModularLambda[Subscript[z, 0]]^2 - 9 ModularLambda[Subscript[z, 0]]^3 - 4 ModularLambda[Subscript[z, 0]]^4 + 13 ModularLambda[Subscript[z, 0]]^5 - 9 ModularLambda[Subscript[z, 0]]^6 + 2 ModularLambda[Subscript[z, 0]]^7))/ (27 (EllipticK[1 - w] (EllipticE[w] - EllipticK[w]) + EllipticE[1 - w] EllipticK[w])^2 (ModularLambda[Subscript[z, 0]] - 1)^4 ModularLambda[Subscript[z, 0]]^4)) (z - Subscript[z, 0])^2 + \[Ellipsis] /; (z -> Subscript[z, 0]) && w = InverseEllipticNomeQ[E^(I Pi Subscript[z, 0])]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["KleinInvariantJ", "[", "z", "]"]], "\[Proportional]", RowBox[List[RowBox[List["KleinInvariantJ", "[", SubscriptBox["z", "0"], "]"]], "+", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["8", " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["EllipticK", "[", "w", "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "w"]], ")"]], " ", "w", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "+", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "2"]]], ")"]], "2"], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["3", " ", RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]]]], "-", RowBox[List["3", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "2"]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "3"]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["27", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["EllipticE", "[", RowBox[List["1", "-", "w"]], "]"]]]], " ", RowBox[List["EllipticK", "[", "w", "]"]]]], "+", RowBox[List[RowBox[List["EllipticK", "[", RowBox[List["1", "-", "w"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["EllipticE", "[", "w", "]"]]]], "+", RowBox[List["EllipticK", "[", "w", "]"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]]]], ")"]], "3"], " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "3"]]], ")"]]]], RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], "-", RowBox[List["8", " ", SuperscriptBox[RowBox[List["EllipticK", "[", "w", "]"]], "3"], " ", RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]], " ", "w", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "+", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "2"]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["EllipticE", "[", "w", "]"]], " ", RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["7", " ", RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]]]], "+", RowBox[List["7", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "2"]]], "-", RowBox[List["7", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "4"]]], "+", RowBox[List["7", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "5"]]], "-", RowBox[List["2", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "6"]]]]], ")"]]]], "+", RowBox[List[RowBox[List["EllipticK", "[", "w", "]"]], " ", "w", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "w", RowBox[List["(", RowBox[List["3", "-", RowBox[List["10", " ", RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]]]], "+", RowBox[List["9", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "2"]]], "+", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "3"], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "4"]]], "-", RowBox[List["3", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "5"]]], "+", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "6"]]], ")"]]]], "-", "6", "+", RowBox[List["18", " ", RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]]]], "-", RowBox[List["11", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "2"]]], "-", RowBox[List["9", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "3"]]], "-", RowBox[List["4", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "4"]]], "+", RowBox[List["13", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "5"]]], "-", RowBox[List["9", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "6"]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "7"]]]]], ")"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List["27", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["EllipticK", "[", RowBox[List["1", "-", "w"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", "w", "]"]], "-", RowBox[List["EllipticK", "[", "w", "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["EllipticE", "[", RowBox[List["1", "-", "w"]], "]"]], " ", RowBox[List["EllipticK", "[", "w", "]"]]]]]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "-", "1"]], ")"]], "4"], " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "4"]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "2"]]], "+", "\[Ellipsis]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", SubscriptBox["z", "0"]]], ")"]], "\[And]", "w"]]]], "=", RowBox[List["InverseEllipticNomeQ", "[", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", SubscriptBox["z", "0"]]]], "]"]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <semantics> <mrow> <mi> J </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;J&quot;, &quot;(&quot;, TagBox[&quot;z&quot;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[KleinInvariantJ[Slot[1]]]]] </annotation> </semantics> <mo> &#8733; </mo> <mrow> <semantics> <mrow> <mi> J </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;J&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[KleinInvariantJ[Slot[1]]]]] </annotation> </semantics> <mo> + </mo> <mrow> <mfrac> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> w </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> w </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 2 </mn> </msup> <mo> - </mo> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 27 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 3 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> w </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 2 </mn> </msup> <mo> - </mo> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mn> 27 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> w </mi> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 7 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 13 </mn> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 11 </mn> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 18 </mn> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> w </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 6 </mn> </msup> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 3 </mn> </msup> <mo> + </mo> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 10 </mn> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mtext> </mtext> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mi> w </mi> </mrow> </mrow> <mo> = </mo> <mrow> <msup> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> </msup> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Set </ci> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> KleinInvariantJ </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <ci> KleinInvariantJ </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 8 </cn> <imaginaryi /> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> w </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> w </ci> <cn type='integer'> -1 </cn> </apply> <ci> w </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 27 </cn> <apply> <plus /> <apply> <times /> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> EllipticK </ci> <ci> w </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> EllipticE </ci> <ci> w </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> EllipticE </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <apply> <ci> EllipticK </ci> <ci> w </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <ci> w </ci> <apply> <plus /> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 27 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> EllipticE </ci> <ci> w </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> EllipticK </ci> <ci> w </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> EllipticE </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <apply> <ci> EllipticK </ci> <ci> w </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> EllipticE </ci> <ci> w </ci> </apply> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> w </ci> <apply> <ci> EllipticK </ci> <ci> w </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 13 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 11 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 18 </cn> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> w </ci> <apply> <plus /> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 6 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 10 </cn> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -6 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> w </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <ci> w </ci> </apply> </apply> <apply> <ci> InverseEllipticNomeQ </ci> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["KleinInvariantJ", "[", "z", "]"]], "\[Proportional]", RowBox[List[RowBox[List["KleinInvariantJ", "[", SubscriptBox["z", "0"], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["8", " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["EllipticK", "[", "w", "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "w"]], ")"]], " ", "w", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "+", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "2"]]], ")"]], "2"], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["3", " ", RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]]]], "-", RowBox[List["3", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "2"]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "3"]]]]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], RowBox[List["27", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["EllipticE", "[", RowBox[List["1", "-", "w"]], "]"]]]], " ", RowBox[List["EllipticK", "[", "w", "]"]]]], "+", RowBox[List[RowBox[List["EllipticK", "[", RowBox[List["1", "-", "w"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["EllipticE", "[", "w", "]"]]]], "+", RowBox[List["EllipticK", "[", "w", "]"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]]]], ")"]], "3"], " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "3"]]]], "-", FractionBox[RowBox[List["8", " ", SuperscriptBox[RowBox[List["EllipticK", "[", "w", "]"]], "3"], " ", RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]], " ", "w", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "+", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["EllipticE", "[", "w", "]"]], " ", RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["7", " ", RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]]]], "+", RowBox[List["7", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "2"]]], "-", RowBox[List["7", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "4"]]], "+", RowBox[List["7", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "5"]]], "-", RowBox[List["2", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "6"]]]]], ")"]]]], "+", RowBox[List[RowBox[List["EllipticK", "[", "w", "]"]], " ", "w", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "w", " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["10", " ", RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]]]], "+", RowBox[List["9", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "2"]]], "+", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "3"], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "4"]]], "-", RowBox[List["3", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "5"]]], "+", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "6"]]], ")"]]]], "-", "6", "+", RowBox[List["18", " ", RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]]]], "-", RowBox[List["11", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "2"]]], "-", RowBox[List["9", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "3"]]], "-", RowBox[List["4", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "4"]]], "+", RowBox[List["13", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "5"]]], "-", RowBox[List["9", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "6"]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "7"]]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "2"]]], RowBox[List["27", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["EllipticK", "[", RowBox[List["1", "-", "w"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", "w", "]"]], "-", RowBox[List["EllipticK", "[", "w", "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["EllipticE", "[", RowBox[List["1", "-", "w"]], "]"]], " ", RowBox[List["EllipticK", "[", "w", "]"]]]]]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "-", "1"]], ")"]], "4"], " ", SuperscriptBox[RowBox[List["ModularLambda", "[", SubscriptBox["z", "0"], "]"]], "4"]]]], "+", "\[Ellipsis]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", SubscriptBox["z", "0"]]], ")"]], "&&", "w"]]]], "=", RowBox[List["InverseEllipticNomeQ", "[", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", SubscriptBox["z", "0"]]]], "]"]]]]]]










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





2007-05-02





© 1998- Wolfram Research, Inc.