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Let $S_g$ be a closed orientable surface of genus $g > 1$. Consider theminimal asymptotic translation length $L_{\mathcal{T}}(k, g)$ on theTeichm\"uller space of $S_g$, among pseudo-Anosov mapping classes of $S_g$acting trivially on a $k$-dimensional subspace of $H_1(S_g)$, $0 \le k \le 2g$.The asymptotics of $L_{\mathcal{T}}(k, g)$ for extreme cases $k = 0, 2g$ havebeen shown by several authors. Jordan Ellenberg asked whether there is a lowerbound for $L_{\mathcal{T}}(k, g)$ interpolating the known results on$L_{\mathcal{T}}(0, g)$ and $L_{\mathcal{T}}(2g, g)$, which was affirmativelyanswered by Agol, Leininger, and Margalit.  In this paper, we study an analogue of Ellenberg's question, replacingTeichm\"uller spaces with curve complexes. We provide lower and upper bound onthe minimal asymptotic translation length $L_{\mathcal{C}}(k, g)$ on the curvecomplex, whose lower bound interpolates the known results on$L_{\mathcal{C}}(0, g)$ and $L_{\mathcal{C}}(2g, g)$.  Finally, for each $g$, we construct a non-Torelli pseudo-Anosov $f_g \in\operatorname{Mod}(S_g)$ which does not normally generates$\operatorname{Mod}(S_g)$ and so that the asymptotic translation length of$f_g$ on curve complexes decays more quickly than a constant multiple of $1/g$as $g \to \infty$. From this, we provide a restriction on how small theasymptotic translation lengths on curve complexes should be if the similarphenomenon as in the work of Lanier and Margalit on Teichm\"uller spaces holdsfor curve complexes.

Minimal asymptotic translation lengths on curve complexes and homology of mapping tori