EllipticK[1 - z^2]/EllipticK[z^2] == Sqrt[13] /; z == 1/Sqrt[2 (649 + 180 Sqrt[13] + 6 Sqrt[23382 + 6485 Sqrt[13]])]

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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> <mfrac> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> </mfrac> <mo> &#10869; </mo> <msqrt> <mn> 13 </mn> </msqrt> </mrow> <mo> /; </mo> <mrow> <mi> z </mi> <mo> &#10869; </mo> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 649 </mn> <mo> + </mo> <mrow> <mn> 180 </mn> <mo> &#8290; </mo> <msqrt> <mn> 13 </mn> </msqrt> </mrow> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 23382 </mn> <mo> + </mo> <mrow> <mn> 6485 </mn> <mo> &#8290; </mo> <msqrt> <mn> 13 </mn> </msqrt> </mrow> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 13 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <eq /> <ci> z </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 649 </cn> <apply> <times /> <cn type='integer'> 180 </cn> <apply> <power /> <cn type='integer'> 13 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 23382 </cn> <apply> <times /> <cn type='integer'> 6485 </cn> <apply> <power /> <cn type='integer'> 13 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2002-12-18