This file presents the choices of y that demonstrate the truth of Lemma 3.4.

Each quadruple is [q,a,b,y], where (a,b) is in Sigma.
Note that wlog we may assume that b=a^2 by part (i),
and hence a and b are both squares.
Thus wlog we may assume that 1-a and 1-b are not squares, by part (ii).

[11,4,5,3],

[13,3,9,7],
[13,9,3,5],

[17,8,13,6],
[17,15,4,7],

[19,5,6,4],
[19,6,17,6],

[23,2,4,2],
[23,3,9,2],

Write GF(25) as Z_5[t]/(t^2+t+2)
[25,2t+3,3t+1,2t],
[25,3t+1,2t+3,t+3],

Write GF(27) as Z_3[t]/(t^3+2t+1)
[27,2t,t^2,2t],
[27,2t+1,t^2+t+1,2t],
[27,2t+2,t^2+2t+1,2t+1],

[29,4,16,3],
[29,22,20,2],

[31,8,2,14],
[31,9,19,2],
[31,19,20,5],

[37,3,9,2],
[37,9,7,2],
[37,25,33,14],
[37,33,16,5],

[41,4,16,12],
[41,8,23,3],
[41,20,31,11],
[41,31,18,11],
[41,36,25,11],
[41,39,4,6],

[43,10,14,9],
[43,14,24,4],
[43,15,10,4],
[43,16,41,6],
[43,24,17,10],