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http://functions.wolfram.com/09.19.17.0001.01
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\[CapitalDelta][\[Tau]]^6 == (1/16777216)
(\[CapitalDelta][\[Tau]/2]^2 \[CapitalDelta][(\[Tau] + 1)/2]^4 +
2 \[CapitalDelta][\[Tau]/2]^3 \[CapitalDelta][(\[Tau] + 1)/2]^3 +
\[CapitalDelta][\[Tau]/2]^4 \[CapitalDelta][(\[Tau] + 1)/2]^2 +
393216 \[CapitalDelta][\[Tau]/2] \[CapitalDelta][\[Tau]]^4
\[CapitalDelta][(\[Tau] + 1)/2] - 2304 \[CapitalDelta][\[Tau]]^2
\[CapitalDelta][\[Tau]/2]^2 \[CapitalDelta][(\[Tau] + 1)/2]^2) /;
\[CapitalDelta][\[Tau]] == Subscript[g, 2]^3 - 27 Subscript[g, 3]^2 &&
{Subscript[g, 2], Subscript[g, 3]} == WeierstrassInvariants[{1, \[Tau]}]
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Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["\[CapitalDelta]", "[", "\[Tau]", "]"]], "6"], "\[Equal]", RowBox[List[FractionBox["1", "16777216"], RowBox[List["(", " ", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["\[CapitalDelta]", "[", FractionBox["\[Tau]", "2"], "]"]], "2"], SuperscriptBox[RowBox[List["\[CapitalDelta]", "[", FractionBox[RowBox[List["\[Tau]", "+", "1"]], "2"], "]"]], "4"]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["\[CapitalDelta]", "[", FractionBox["\[Tau]", "2"], "]"]], "3"], SuperscriptBox[RowBox[List["\[CapitalDelta]", "[", FractionBox[RowBox[List["\[Tau]", "+", "1"]], "2"], "]"]], "3"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["\[CapitalDelta]", "[", FractionBox["\[Tau]", "2"], "]"]], "4"], SuperscriptBox[RowBox[List["\[CapitalDelta]", "[", FractionBox[RowBox[List["\[Tau]", "+", "1"]], "2"], "]"]], "2"]]], "+", RowBox[List["393216", " ", RowBox[List["\[CapitalDelta]", "[", FractionBox["\[Tau]", "2"], "]"]], SuperscriptBox[RowBox[List["\[CapitalDelta]", "[", "\[Tau]", "]"]], "4"], RowBox[List["\[CapitalDelta]", "[", FractionBox[RowBox[List["\[Tau]", "+", "1"]], "2"], "]"]]]], "-", RowBox[List["2304", " ", SuperscriptBox[RowBox[List["\[CapitalDelta]", "[", "\[Tau]", "]"]], "2"], " ", SuperscriptBox[RowBox[List["\[CapitalDelta]", "[", FractionBox["\[Tau]", "2"], "]"]], "2"], SuperscriptBox[RowBox[List["\[CapitalDelta]", "[", FractionBox[RowBox[List["\[Tau]", "+", "1"]], "2"], "]"]], "2"]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["\[CapitalDelta]", "[", "\[Tau]", "]"]], "\[Equal]", RowBox[List[SubsuperscriptBox["g", "2", "3"], "-", RowBox[List["27", SubsuperscriptBox["g", "3", "2"]]]]]]], "\[And]", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]], "\[Equal]", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List["1", ",", "\[Tau]"]], "}"]], "]"]]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mrow> <mi> Δ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> τ </mi> <mo> ) </mo> </mrow> <mn> 6 </mn> </msup> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 16777216 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mi> Δ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> τ </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> Δ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> τ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mrow> <mi> Δ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> τ </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> Δ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> τ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mi> Δ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> τ </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> Δ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> τ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2304 </mn> <mo> ⁢ </mo> <msup> <mrow> <mi> Δ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> τ </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> Δ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> τ </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> Δ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> τ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 393216 </mn> <mo> ⁢ </mo> <mrow> <mi> Δ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> τ </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> Δ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> τ </mi> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> ⁢ </mo> <mrow> <mi> Δ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> τ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Δ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> τ </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msubsup> <mi> g </mi> <mn> 2 </mn> <mn> 3 </mn> </msubsup> <mo> - </mo> <mrow> <mn> 27 </mn> <mo> ⁢ </mo> <msubsup> <mi> g </mi> <mn> 3 </mn> <mn> 2 </mn> </msubsup> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> ⩵ </mo> <mrow> <mo> { </mo> <mrow> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> τ </mi> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <msub> <mi> g </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> τ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <power /> <apply> <ci> Δ </ci> <ci> τ </ci> </apply> <cn type='integer'> 6 </cn> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 16777216 </cn> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <ci> Δ </ci> <apply> <times /> <ci> τ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Δ </ci> <apply> <times /> <apply> <plus /> <ci> τ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> Δ </ci> <apply> <times /> <ci> τ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <ci> Δ </ci> <apply> <times /> <apply> <plus /> <ci> τ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <ci> Δ </ci> <apply> <times /> <ci> τ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <ci> Δ </ci> <apply> <times /> <apply> <plus /> <ci> τ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2304 </cn> <apply> <power /> <apply> <ci> Δ </ci> <ci> τ </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Δ </ci> <apply> <times /> <ci> τ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Δ </ci> <apply> <times /> <apply> <plus /> <ci> τ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 393216 </cn> <apply> <ci> Δ </ci> <apply> <times /> <ci> τ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Δ </ci> <ci> τ </ci> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <ci> Δ </ci> <apply> <times /> <apply> <plus /> <ci> τ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> Δ </ci> <ci> τ </ci> </apply> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 27 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <list> <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> </list> <list> <apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> <ci> τ </ci> </apply> <apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 1 </cn> <ci> τ </ci> </apply> </list> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["\[CapitalDelta]", "[", "\[Tau]_", "]"]], "6"], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["\[CapitalDelta]", "[", FractionBox["\[Tau]", "2"], "]"]], "2"], " ", SuperscriptBox[RowBox[List["\[CapitalDelta]", "[", FractionBox[RowBox[List["\[Tau]", "+", "1"]], "2"], "]"]], "4"]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["\[CapitalDelta]", "[", FractionBox["\[Tau]", "2"], "]"]], "3"], " ", SuperscriptBox[RowBox[List["\[CapitalDelta]", "[", FractionBox[RowBox[List["\[Tau]", "+", "1"]], "2"], "]"]], "3"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["\[CapitalDelta]", "[", FractionBox["\[Tau]", "2"], "]"]], "4"], " ", SuperscriptBox[RowBox[List["\[CapitalDelta]", "[", FractionBox[RowBox[List["\[Tau]", "+", "1"]], "2"], "]"]], "2"]]], "+", RowBox[List["393216", " ", RowBox[List["\[CapitalDelta]", "[", FractionBox["\[Tau]", "2"], "]"]], " ", SuperscriptBox[RowBox[List["\[CapitalDelta]", "[", "\[Tau]", "]"]], "4"], " ", RowBox[List["\[CapitalDelta]", "[", FractionBox[RowBox[List["\[Tau]", "+", "1"]], "2"], "]"]]]], "-", RowBox[List["2304", " ", SuperscriptBox[RowBox[List["\[CapitalDelta]", "[", "\[Tau]", "]"]], "2"], " ", SuperscriptBox[RowBox[List["\[CapitalDelta]", "[", FractionBox["\[Tau]", "2"], "]"]], "2"], " ", SuperscriptBox[RowBox[List["\[CapitalDelta]", "[", FractionBox[RowBox[List["\[Tau]", "+", "1"]], "2"], "]"]], "2"]]]]], "16777216"], "/;", RowBox[List[RowBox[List[RowBox[List["\[CapitalDelta]", "[", "\[Tau]", "]"]], "\[Equal]", RowBox[List[SubsuperscriptBox["g", "2", "3"], "-", RowBox[List["27", " ", SubsuperscriptBox["g", "3", "2"]]]]]]], "&&", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]], "\[Equal]", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List["1", ",", "\[Tau]"]], "}"]], "]"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
