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Nov 24, 2016 at 13:10 comment added Tito Piezas III @GerryMyerson:The corrected Lehmer quintic does satisfy all three Diophantine relations. (I made a small typo when I was checking it earlier.)
Nov 24, 2016 at 11:46 comment added Gerry Myerson I checked Storer's book, he doesn't do the quintic.
Nov 24, 2016 at 7:51 comment added Tito Piezas III Thanks. Courtesy of the answers by Myerson and Elkies, the depressed forms of these satisfy the three Diophantine relations in the post. And I edited the post to include a second simple cyclic family which, incidentally, contains $p=151$.
Nov 24, 2016 at 4:57 history edited Will Jagy CC BY-SA 3.0
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Nov 24, 2016 at 3:25 history edited Will Jagy CC BY-SA 3.0
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Nov 24, 2016 at 3:17 history edited Will Jagy CC BY-SA 3.0
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Nov 24, 2016 at 2:02 history answered Will Jagy CC BY-SA 3.0