We extract a subfamily 𝑬(𝒌): 𝒚𝟐 = 𝒙𝟑 + (𝒌𝟐− 𝟏)𝒙 of elliptic curves from the curve 𝑬(𝒕): 𝒚𝟐 = 𝒙𝟑− (𝒂 + 𝒃𝒕)𝒙 .

Then, by imposing, successively, points on the obtained curves 𝑬(𝒌), we increase the rank. 

At the end, we show that its rank is at least 3 over ℚ(𝑘).