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* <math>x^2+y^2=1+dx^2y^2</math> with point <math>(a,b) </math> where <math>a=\frac{u^2-1}{u^2+1}, b=-\frac{(u-1)^2}{u^2+1}</math> and <math>d=\frac{(u^2+1)^3(u^2-4u+1)}{(u-1)^6(u+1)^2}, u\notin\{0,\pm1\}.</math>
Every Edwards curve with a point of order 3 can be written in the ways shown above. Curves with torsion group isomorphic to <math>\Z/2\Z\times \Z/8\Z</math> and <math>\Z/2\Z\times \Z/4\Z</math> may be more efficient at finding primes.<ref name=Bernstein2008>{{cite web|last1=Berstein|first1=Daniel J.|last2=Birkner|first2=Peter|last3=Lange|first3=Tanja|author3-link=Tanja Lange|last4=Peters|first4=Christiane|title=ECM Using Edwards Curves|url=https://eprint.iacr.org/2008/016.pdf|website=Cryptology ePrint Archive|date=January 9, 2008}} (see top of page 30 for examples of such curves)</ref>
==Stage 2==
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