  In this article, we construct the
$[sc]_A[\bar{s}\bar{c}]_V-[sc]_V[\bar{s}\bar{c}]_A$ type tensor current to
study the mass and width of the $X(4274)$ with the QCD sum rules in details.
The predicted mass $M_{X}=(4.27\pm0.09) \, \rm{GeV}$ for the $J^{PC}=1^{++}$
tetraquark state is in excellent agreement with the experimental data $4273.3
\pm 8.3 ^{+17.2}_{-3.6} \mbox MeV}$ from the LHCb collaboration. The central
value of the width $\Gamma(X(4274)\to J/\psi \phi)=47.9\,{\rm{MeV}}$ is in
excellent agreement with the experimental data $56 \pm 11 ^{+8}_{-11} \mbox
MeV}}$ from the LHCb collaboration. The present work supports assigning the
$X(4274)$ to be the $J^{PC}=1^{++}$
$[sc]_A[\bar{s}\bar{c}]_V-[sc]_V[\bar{s}\bar{c}]_A$ tetraquark state with a
relative P-wave between the diquark and antidiquark constituents. Furthermore,
we obtain the mass of the $[sc]_A[\bar{s}\bar{c}]_V-[sc]_V[\bar{s}\bar{c}]_A$
type tetraquark state with $J^{PC}=1^{-+}$ as a byproduct.
