Rank 5 example: Y 2
= A(X)T2
+ B(X)T + C(X)
A(X) = X(X − 1) + k
2
,
2(b1 + b2)(m
2
3 − 1)
2
B(X) = 2 (m
4
3 − 2m
2
3 + 1)b
2
2 + (m
4
3 − 2m
2
3 + 1)b1b2


X
2
+
2 (m
4
3 − 2m
2
3 + 1)b1b2 + (m
4
3 − 2m
2
3 + 1)b
2
1


X +
− 4(m
4
3 + 2m
2
3 + 1)k
4
− 8(m
3
3 + m3)k
3
+
2 (m
4
3 − 2m
2
3 + 1)b
2
2 + (m
4
3 − 2m
2
3 + 1)b1b2 − 2m
2
3


k
2
+
− (m
4
3 − 2m
2
3 + 1)b
2
2 + 2(m
4
3 − 2m
2
3 + 1)b1b2 + (m
4
3 − 2m
2
3 + 1)b
2
1


,
C(X) =
B(X)2
+ 4X(X − 1)(X − r3)(X − r4)(X − r5)
4A(X)
,
where r3 = 2m3k+1
1−m2
3
, r4 = 1 − r3 and r5 = 2m5k+1
1−m2
5
.
With m5 = q−p
q+p and
p = 4(m
4
3 + 2m
2
3 + 1)k
4
+ 8(m
3
3 + m3)k
3
+ 4m
2
3k
2
q = −2 (m
4
3 − 2m
2
3 + 1)b
2
2 + 2(m
4
3 − 2m
2
3 + 1)b1b2 + (m
4
3 − 2m
2
3 + 1)b
2
1


k +
+ (m
4
3 − 2m
2
3 + 1)b
2
2 + 2(m
4
3 − 2m
2
3 + 1)b1b2 + (m
4
3 − 2m
2
3 + 1)b
2
1.
We have A(ri) =

(1 + m2
i )k + mi
1 − m2
i
2
.