Fix $A,B\in\Bbb{Z}$ with $4A^3+27B^2\neq 0$. The author considers the elliptic curve defined over the rationals $$ E\:y^2=x^3+Ax+B $$ and its quadratic twist $$ E_d\:dy^2=x^3+Ax+B $$ for a square-free integer $d$. Let $h\:\Bbb{P}^1(\Bbb{Q})\to \Bbb{R}_{\geq 0}$ denote the logarithmic Weil height. Given a point $P\in E_d(\Bbb{Q})$, let $$ \widehat{h}_{E_d}(P)=\frac{1}{2}\lim_{n\to\infty}\frac{1}{4^n}h_x(2^nP), $$ where $h_x\:\Bbb{P}^2(\Bbb{Q})\to\Bbb{R}_{\geq 0}$ is defined by $h_x(0:1:0)=0$ and $$ h_x(x:y:z)=h(x:z)\quad\text{for}\quad(x:y:z)\neq (0:1:0) . $$ Further, for $E_d(\Bbb{Q})$, $\eta_d(A,B)$ is defined by $$ \log\eta_d(A,B)=\min\{\widehat{h}_{E_d}(P)\,:\,P \in E_d(\Bbb{Q})\setminus E_d(\Bbb{Q})_{\rm tors}\}. $$ It is known from [P. Le~Boudec, Bull. Lond. Math. Soc. {\bf 48} (2016), no.~1, 99--108; MR3455753] that $$ \eta_d(A,B)\gg d^{1/8} $$ and the bound is sharp. The main result of this paper is about the analytic ranks of the curves $E_d$ with $$ d\in\Cal{D}_{A,B}(\alpha;X)\coloneq\left\{d\,:\,d\leq X,d\ \text{square-free},\eta_d(A,B)\leq d^{1/8+\alpha}\right\}, $$ where $\alpha>0$. Assume that $0<\alpha<1/120$ and $x^3+Ax+B$ is irreducible over $\Bbb{Z}$. The author proves that $$ \liminf_{X\to\infty}\frac{1}{\#\Cal{D}_{A,B} (\alpha;X)}\sum_{d\in\Cal{D}_{A,B}(\alpha ; X)}{\rm rank}_{\rm an}(E_d)>1, $$ where the analytic rank of $E_d$ is denoted by ${\rm rank}_{\rm an}(E_d)$. From this, the author deduces two corollaries on the $p^{\infty}$-Selmer group of $E_d$ and the algebraic rank of $E_d$ under the parity conjecture.