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\begin{abstract}
For $P\in \mathbb{F}_2[z]$ with $P(0)=1$ and $\deg(P)\geq 1$, let
${\cal A}={\cal A}(P)$ be the unique
subset of $\mathbb{N}$ such that $\sum_{n\geq 0}p({\cal
A},n)z^n\equiv P(z)$ (mod $2$), where $p({\cal A},n)$ is the
number of partitions of $n$ with parts in ${\cal A}$. Let $p$ be
an odd prime number, and let $P$ be irreducible of order $p$ ; i.e.,
$p$ is the smallest positive integer such that $P$ divides $1+z^p$
in $\F_2[z]$.  N. Baccar proved that the elements of
${\cal A}(P)$ of the form $2^km$, where $k\geq 0$ and $m$ is odd,
are given by the $2$-adic expansion of a zero  of some polynomial
$R_m$ with integer coefficients. Let $s_p$ be the order of $2$
modulo $p$, i.e., the smallest positive integer such that
$2^{s_p}\equiv 1$ (mod $p$). Improving on the method  with which $R_m$ was obtained explicitly only when
$s_p=\frac{p-1}{2}$, here we make explicit $R_m$ when
$s_p=\frac{p-1}{3}$. For that, we have used the number of points of the elliptic curve $x^3+ay^3 =1 $ modulo $p$.
\end{abstract}


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