Motivations and notations
F : Y 2
= X3
+ α2(T)X2
+ α4(T)x + α6(T), with αi(T) ∈ Z[T].
• If deg αi(T) ≤ 2 then F is a rational elliptic surface.
Shioda-Tate’s formula gives:
rF/Q(T ) = 8 −
X
v
(mv − 1),
where mv is the number of irreducible components in v.
Here, we consider
F : Y 2
= X3
+ α2(T)X2
+ α4(T)x + α6(T), with deg αi(T) ≤ 2.
• We write
F : Y 2
= A(X)T2
+ B(X)T + C(X), with A, B, C ∈ Z[X],
et deg A, B ≤ 2 et deg C − X3
≤ 2.
Using Nagao’s formula, we obtain a close formula for r in function
of A, B and C. And we find r points naturally.