Klein invariant modular function: Identities (formula 09.50.17.0003)

KleinInvariantJ

$Mathematica Notation$

Elliptic Functions

KleinInvariantJ[z]

Identities

Functional identities

http://functions.wolfram.com/09.50.17.0003.01

Input Form

64 KleinInvariantJ[z]^3 + 95232 KleinInvariantJ[2 z] KleinInvariantJ[z]^2 + 95232 KleinInvariantJ[2 z]^2 KleinInvariantJ[z] + 1510125 KleinInvariantJ[2 z] KleinInvariantJ[z] + 187500 KleinInvariantJ[z] + 64 KleinInvariantJ[2 z]^3 - 6000 KleinInvariantJ[z]^2 + 187500 KleinInvariantJ[2 z] - 6000 KleinInvariantJ[2 z]^2 - 110592 KleinInvariantJ[z]^2 KleinInvariantJ[2 z]^2 - 1953125 == 0

Standard Form

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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> <mn> 64 </mn> <mo> ⁢ </mo> <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", Rule[Editable, True]], ")"]], InterpretTemplate[Function[KleinInvariantJ[Slot[1]]]]] </annotation> </semantics> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 110592 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> J </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["J", "(", TagBox[RowBox[List["2", " ", "z"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[KleinInvariantJ[Slot[1]]]]] </annotation> </semantics> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <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", Rule[Editable, True]], ")"]], InterpretTemplate[Function[KleinInvariantJ[Slot[1]]]]] </annotation> </semantics> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 95232 </mn> <mo> ⁢ </mo> <semantics> <mrow> <mi> J </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["J", "(", TagBox[RowBox[List["2", " ", "z"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[KleinInvariantJ[Slot[1]]]]] </annotation> </semantics> <mo> ⁢ </mo> <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", Rule[Editable, True]], ")"]], InterpretTemplate[Function[KleinInvariantJ[Slot[1]]]]] </annotation> </semantics> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 6000 </mn> <mo> ⁢ </mo> <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", Rule[Editable, True]], ")"]], InterpretTemplate[Function[KleinInvariantJ[Slot[1]]]]] </annotation> </semantics> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 95232 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> J </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["J", "(", TagBox[RowBox[List["2", " ", "z"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[KleinInvariantJ[Slot[1]]]]] </annotation> </semantics> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mi> J </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["J", "(", TagBox["z", Rule[Editable, True]], ")"]], InterpretTemplate[Function[KleinInvariantJ[Slot[1]]]]] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mn> 1510125 </mn> <mo> ⁢ </mo> <semantics> <mrow> <mi> J </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["J", "(", TagBox[RowBox[List["2", " ", "z"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[KleinInvariantJ[Slot[1]]]]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <mrow> <mi> J </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["J", "(", TagBox["z", Rule[Editable, True]], ")"]], InterpretTemplate[Function[KleinInvariantJ[Slot[1]]]]] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mn> 187500 </mn> <mo> ⁢ </mo> <semantics> <mrow> <mi> J </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["J", "(", TagBox["z", Rule[Editable, True]], ")"]], InterpretTemplate[Function[KleinInvariantJ[Slot[1]]]]] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mn> 64 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> J </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["J", "(", TagBox[RowBox[List["2", " ", "z"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[KleinInvariantJ[Slot[1]]]]] </annotation> </semantics> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 6000 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> J </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["J", "(", TagBox[RowBox[List["2", " ", "z"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[KleinInvariantJ[Slot[1]]]]] </annotation> </semantics> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 187500 </mn> <mo> ⁢ </mo> <semantics> <mrow> <mi> J </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["J", "(", TagBox[RowBox[List["2", " ", "z"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[KleinInvariantJ[Slot[1]]]]] </annotation> </semantics> </mrow> <mo> - </mo> <mn> 1953125 </mn> </mrow> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <times /> <cn type='integer'> 64 </cn> <apply> <power /> <apply> <ci> KleinInvariantJ </ci> <ci> z </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 110592 </cn> <apply> <power /> <apply> <ci> KleinInvariantJ </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> KleinInvariantJ </ci> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 95232 </cn> <apply> <ci> KleinInvariantJ </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <ci> KleinInvariantJ </ci> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6000 </cn> <apply> <power /> <apply> <ci> KleinInvariantJ </ci> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 95232 </cn> <apply> <power /> <apply> <ci> KleinInvariantJ </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> KleinInvariantJ </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 1510125 </cn> <apply> <ci> KleinInvariantJ </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <apply> <ci> KleinInvariantJ </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 187500 </cn> <apply> <ci> KleinInvariantJ </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 64 </cn> <apply> <power /> <apply> <ci> KleinInvariantJ </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6000 </cn> <apply> <power /> <apply> <ci> KleinInvariantJ </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 187500 </cn> <apply> <ci> KleinInvariantJ </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1953125 </cn> </apply> <cn type='integer'> 0 </cn> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2001-10-29