  The Weisfeiler-Leman (WL) dimension of a graph parameter $f$ is the minimum
$k$ such that, if $G_1$ and $G_2$ are indistinguishable by the $k$-dimensional
WL-algorithm then $f(G_1)=f(G_2)$. The WL-dimension of $f$ is $\infty$ if no
such $k$ exists. We study the WL-dimension of graph parameters characterised by
the number of answers from a fixed conjunctive query to the graph. Given a
conjunctive query $\varphi$, we quantify the WL-dimension of the function that
maps every graph $G$ to the number of answers of $\varphi$ in $G$.
  The works of Dvor\'ak (J. Graph Theory 2010), Dell, Grohe, and Rattan (ICALP
2018), and Neuen (ArXiv 2023) have answered this question for full conjunctive
queries, which are conjunctive queries without existentially quantified
variables. For such queries $\varphi$, the WL-dimension is equal to the
treewidth of the Gaifman graph of $\varphi$.
  In this work, we give a characterisation that applies to all conjunctive
qureies. Given any conjunctive query $\varphi$, we prove that its WL-dimension
is equal to the semantic extension width $\mathsf{sew}(\varphi)$, a novel width
measure that can be thought of as a combination of the treewidth of $\varphi$
and its quantified star size, an invariant introduced by Durand and Mengel
(ICDT 2013) describing how the existentially quantified variables of $\varphi$
are connected with the free variables. Using the recently established
equivalence between the WL-algorithm and higher-order Graph Neural Networks
(GNNs) due to Morris et al. (AAAI 2019), we obtain as a consequence that the
function counting answers to a conjunctive query $\varphi$ cannot be computed
by GNNs of order smaller than $\mathsf{sew}(\varphi)$.
