Entry 8

The previous entry ended with \(\cos\big(\tfrac{2\pi}{29}\big)\). This is not by Ramanujan, but I got reminded about something elegant regarding \(p=29\) which he would have appreciated. Does anyone know why the octic found by Igor Schein in 1999 $$x^8-x^7+29x^2+29 = 0$$is solvable in radicals, specifically by the \(29\)th root of unity? Any other octic or higher with a similarly simple form?

Incidentally, the discriminant of \(x^8-x^7+ax^2+a\) is $$F(a)=186624 a^3 - 3561092 a^2 + 29511140 a - 7^7$$ and the only integer solution to the elliptic curve \(F(a) = y^2\) is \(a = 29\).