  A Coulomb-excitation reorientation-effect measurement using the TIGRESS
$\gamma-$ray spectrometer at the TRIUMF/ISAC II facility has permitted the
first determination of the $\langle 2^+_1\mid\mid \hat{E2} \mid\mid
2^+_1\rangle$ diagonal matrix element in $^{12}$C from particle$-\gamma$
coincidence data. Required state-of-the-art no-core shell model calculations of
the nuclear polarizability for the ground and first-excited (2$^+_1$) states in
$^{12}$C using chiral NN N$^4$LO500 and NN+3NF350 interactions have been
performed. Consistent predictions show a larger polarizability than previously
anticipated. The polarizability of the 2$^+_1$ state is introduced into the
current and previous Coulomb-excitation reorientation-effect analysis of
$^{12}$C. Spectroscopic quadrupole moments of $Q_{_S}(2_1^+)= +0.053(44)$ eb
and $Q_{_S}(2_1^+)= +0.08(3)$ eb are determined, respectively, yielding a
weighted average of $Q_{_S}(2_1^+)= +0.071(25)$ eb, in agreement with recent ab
initio calculations. The present measurement confirms that the 2$^+_1$ state of
$^{12}$C is oblate and emphasizes the important role played by the nuclear
polarizability in Coulomb-excitation studies of light nuclei.
