v^6 - u^6 + 5 (v^2 - u^2) v^2 u^2 - 4 u v (1 - u^4 v^4) == 0 /; u == InverseEllipticNomeQ[z^5]^(1/8) && v == InverseEllipticNomeQ[z]^(1/8) && 0 <= z <= 1

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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> <mrow> <msup> <mi> v </mi> <mn> 6 </mn> </msup> <mo> - </mo> <msup> <mi> u </mi> <mn> 6 </mn> </msup> <mo> + </mo> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> v </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> u </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> v </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> u </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> v </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msup> <mi> u </mi> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> v </mi> <mn> 4 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> u </mi> </mrow> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> u </mi> <mo> &#10869; </mo> <mroot> <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> z </mi> <mn> 5 </mn> </msup> <mo> ) </mo> </mrow> <mn> 8 </mn> </mroot> </mrow> <mo> &#8743; </mo> <mrow> <mi> v </mi> <mo> &#10869; </mo> <mroot> <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> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 8 </mn> </mroot> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &#8804; </mo> <mi> z </mi> <mo> &#8804; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <power /> <ci> v </ci> <cn type='integer'> 6 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> u </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <plus /> <apply> <power /> <ci> v </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> u </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> v </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> u </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> v </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> u </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <ci> v </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> <ci> u </ci> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <and /> <apply> <eq /> <ci> u </ci> <apply> <power /> <apply> <ci> InverseEllipticNomeQ </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 8 </cn> </apply> </apply> <apply> <eq /> <ci> v </ci> <apply> <power /> <apply> <ci> InverseEllipticNomeQ </ci> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 8 </cn> </apply> </apply> <apply> <leq /> <cn type='integer'> 0 </cn> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29