Weierstrass invariants: Representations through equivalent functions (formula 09.19.27.0013)

WeierstrassInvariants

$Mathematica Notation$

Elliptic Functions

WeierstrassInvariants[{w₁,w₃}]

Representations through equivalent functions

With related functions

Involving elliptic integrals and modular functions

http://functions.wolfram.com/09.19.27.0013.01

Input Form

Subscript[g, 2]^3 (2 m^3 - 3 m^2 - 3 m + 2)^2 - 108 Subscript[g, 3]^2 (m^2 - m + 1)^3 == 0 /; m == InverseEllipticNomeQ[E^((I Pi Subscript[\[Omega], 3])/ Subscript[\[Omega], 1])] && {Subscript[\[Omega], 1], Subscript[\[Omega], 3]} == WeierstrassHalfPeriods[{Subscript[g, 2], Subscript[g, 3]}]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["g", "2", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["m", "3"]]], "-", RowBox[List["3", " ", SuperscriptBox["m", "2"]]], "-", RowBox[List["3", " ", "m"]], "+", "2"]], ")"]], "2"]]], "-", RowBox[List["108", " ", SubsuperscriptBox["g", "3", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["m", "2"], "-", "m", "+", "1"]], ")"]], "3"]]]]], "\[Equal]", "0"]], "/;", RowBox[List[RowBox[List["m", "\[Equal]", RowBox[List["InverseEllipticNomeQ", "[", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", SubscriptBox["\[Omega]", "3"]]], SubscriptBox["\[Omega]", "1"]]], "]"]]]], "\[And]", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "\[Equal]", RowBox[List["WeierstrassHalfPeriods", "[", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]], "]"]]]]]]]]]]

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> <mrow> <msubsup> <mi> g </mi> <mn> 2 </mn> <mn> 3 </mn> </msubsup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 108 </mn> <mo> ⁢ </mo> <msubsup> <mi> g </mi> <mn> 3 </mn> <mn> 2 </mn> </msubsup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> m </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ⩵ </mo> <mn> 10 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <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> <mrow> <mi> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mfrac> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> ⩵ </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mstyle scriptlevel='0'> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> <mstyle scriptlevel='0'> <mo> , </mo> </mstyle> <mrow> <mstyle scriptlevel='0'> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> </mrow> <mstyle scriptlevel='0'> <mo> } </mo> </mstyle> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 108 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> 10 </cn> </apply> <apply> <and /> <apply> <eq /> <ci> m </ci> <apply> <ci> InverseEllipticNomeQ </ci> <apply> <exp /> <apply> <times /> <imaginaryi /> <pi /> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <list> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </list> <list> <apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </apply> </list> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[SubsuperscriptBox["g_", "2", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["m_", "3"]]], "-", RowBox[List["3", " ", SuperscriptBox["m_", "2"]]], "-", RowBox[List["3", " ", "m_"]], "+", "2"]], ")"]], "2"]]], "-", RowBox[List["108", " ", SubsuperscriptBox["g_", "3", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["m_", "2"], "-", "m_", "+", "1"]], ")"]], "3"]]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["m", "\[Equal]", RowBox[List["InverseEllipticNomeQ", "[", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", SubscriptBox["\[Omega]", "3"]]], SubscriptBox["\[Omega]", "1"]]], "]"]]]], "&&", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "\[Equal]", RowBox[List["WeierstrassHalfPeriods", "[", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]], "]"]]]]]]]]]]]]

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

2001-10-29