Moral result
F : Y 2
= A(X)T2
+ B(X)T + C(X), withA, B, C ∈ Z[X].
This gives
Y 2
= A(X)

T +
B(X)
2A(X)
2
−
B2
(X) − 4A(X)C(X)
4A(X)
Observation
If ρ ∈ Q is such that:
(i) ρ is a root of ∆ := B2
(X) − 4A(X)C(X) ;
(ii) A(ρ) = L2
(ρ) ∈ (Q(ρ)∗
)2
;
Then
P(ρ) =

ρ, L(ρ)

T +
B(ρ)
2A(ρ)

∈ F(Q(ρ)(T)).
Trace of P(ρ) on F gives P[ρ] ∈ F(Q(T)).
If A(ρ) = B(ρ) = 0 and C(ρ) = L(ρ)2
, consider P(ρ) = (ρ, L(ρ)).