Klein invariant modular function: Representations through equivalent functions (formula 09.50.27.0005)

KleinInvariantJ

$Mathematica Notation$

Elliptic Functions

KleinInvariantJ[z]

Representations through equivalent functions

With related functions

Involving other related functions

http://functions.wolfram.com/09.50.27.0005.01

Input Form

KleinInvariantJ[z] == (DedekindEta[z]^24 + 256 DedekindEta[2 z]^24)^3/ (1728 DedekindEta[z]^48 DedekindEta[2 z]^24)

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["KleinInvariantJ", "[", "z", "]"]], "\[Equal]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["DedekindEta", "[", "z", "]"]], "24"], "+", RowBox[List["256", " ", SuperscriptBox[RowBox[List["DedekindEta", "[", RowBox[List["2", " ", "z"]], "]"]], "24"]]]]], ")"]], "3"], RowBox[List["1728", " ", SuperscriptBox[RowBox[List["DedekindEta", "[", "z", "]"]], "48"], " ", SuperscriptBox[RowBox[List["DedekindEta", "[", RowBox[List["2", " ", "z"]], "]"]], "24"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <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> <mo> ⩵ </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <semantics> <mrow> <mi> η </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Eta]", "(", TagBox["z", Rule[Editable, True]], ")"]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> <mn> 24 </mn> </msup> <mo> + </mo> <mrow> <mn> 256 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> η </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Eta]", "(", TagBox[RowBox[List["2", " ", "z"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> <mn> 24 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mrow> <mn> 1728 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> η </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Eta]", "(", TagBox["z", Rule[Editable, True]], ")"]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> <mn> 48 </mn> </msup> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> η </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Eta]", "(", TagBox[RowBox[List["2", " ", "z"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> <mn> 24 </mn> </msup> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> KleinInvariantJ </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <ci> DedekindEta </ci> <ci> z </ci> </apply> <cn type='integer'> 24 </cn> </apply> <apply> <times /> <cn type='integer'> 256 </cn> <apply> <power /> <apply> <ci> DedekindEta </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 24 </cn> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1728 </cn> <apply> <power /> <apply> <ci> DedekindEta </ci> <ci> z </ci> </apply> <cn type='integer'> 48 </cn> </apply> <apply> <power /> <apply> <ci> DedekindEta </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 24 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KleinInvariantJ", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["DedekindEta", "[", "z", "]"]], "24"], "+", RowBox[List["256", " ", SuperscriptBox[RowBox[List["DedekindEta", "[", RowBox[List["2", " ", "z"]], "]"]], "24"]]]]], ")"]], "3"], RowBox[List["1728", " ", SuperscriptBox[RowBox[List["DedekindEta", "[", "z", "]"]], "48"], " ", SuperscriptBox[RowBox[List["DedekindEta", "[", RowBox[List["2", " ", "z"]], "]"]], "24"]]]]]]]]

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

2001-10-29