We calculate the trispectrum T_zeta of the primordial curvature perturbation zeta, generated during a {it slow-roll} inflationary epoch by considering a two-field quadratic model of inflation with {it canonical} kinetic terms. We consider loop contributions as well as tree level terms, and show that it is possible to attain very high, {it including observable}, values for the level of non-gaussianity tau_{NL} if T_zeta is dominated by the one-loop contribution. Special attention is paid to the claim in JCAP {bf 0902}, 017 (2009) [arXiv:0812.0807 [astro-ph]] that, in the model studied in this paper and for the specific inflationary trajectory we choose, the quantum fluctuations of the fields overwhelm the classical evolution. We argue that such a claim actually does not apply to our model, although more research is needed in order to understand the role of quantum diffusion. We also consider the probability that an observer in an ensemble of realizations of the density field sees a non-gaussian distribution. In that respect, we show that the probability associated to the chosen inflationary trajectory is non-negligible. Finally, the levels of non-gaussianity f_{NL} and tau_{NL} in the bispectrum B_zeta and trispectrum T_zeta of zeta, respectively, are also studied for the case in which zeta is not generated during inflation.
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