We consider the family of elliptic curves \(E_{a,b}:y^2=x^3+a(x-b)^2\) with \(a,b \in {\mathbb {Z}}\) . These elliptic curves have a rational 3-isogeny, say \(\varphi \) . We give an upper and a lower bound on the rank of the \(\varphi \) -Selmer group of \(E_{a,b}\) over \(K:={\mathbb {Q}}(\zeta _3)\) in terms of the 3-part of the ideal class group of a certain quadratic extension of K. Using our bounds on the Selmer groups, we construct infinitely many curves in this family with arbitrarily large 3-Selmer rank over K and no non-trivial K-rational point of order 3. We also show that for a positive proportion of natural numbers n, the curve \(E_{n,n}/{\mathbb {Q}}\) has root number \(-1\) and 3-Selmer rank equal to 1.
