Application: A = k2
, F : Y 2
= k2
T2
+ BT + C
• r = Ω((B2
− 4k2
C)∗
) − 1.
Corollary
Let C : Y 2
= C(X) an elliptic curve defined over Q such that
C(Q) = {O}. Then, for all B of degree ≤ 2 and all k ∈ Z∗
, the
polynomial B2
− 4k2
C is a power of an irreducible polynomial.
• C : Y 2
= X3
+ X + 5 has rank 0 and no nonzero torsion point, so
P(X) = 4k2
X3
− b2
1X2
− (2b0b1 − 4k2
)X − (b2
0 − 20k2
)
is irreducible for all k ∈ Z∗
et b0, b1 ∈ Z.