# Mordell-Weil rank bound

Given an elliptic curve $y^2 = x(x^2 + bx + c)$ is a non-singular curve, say $c > 0$ and $b^2 - 4c > 0$. Can we show the bound on the rank $r$ in terms of $\nu(c)$ and $\nu(b^2 - 4c)$ without quoting the Selmer groups? Well, a classical approach on $2$-descent says $$r \leqslant \nu(c) + \nu(b^2 - 4c) - 1.$$ I suspect that we can just show $$r \leqslant \nu(c) + \nu(b^2 - 4c)$$ by using a simpler argument, however I stuck here. Could you please give any hints?