  We present a study of the intermittent properties of a shell model of
turbulence with unprecedented statistics, about $\sim 10^7$ eddy turn over
time, achieved thanks to an implementation on a large-scale parallel GPU
factory. This allows us to quantify the inertial range anomalous scaling
properties of the velocity fluctuations up to the 24th order moment. Through a
careful assessment of the statistical and systematic uncertainties, we show
that none of the phenomenological and theoretical models previously proposed in
the literature to predict the anomalous power-law exponents in the inertial
range is in agreement with our high-precision numerical measurements. We find
that at asymptotically high order moments, the anomalous exponents tend towards
a linear scaling, suggesting that extreme turbulent events are dominated by one
leading singularity. We found that systematic corrections to scaling induced by
the infrared and ultraviolet (viscous) cut-offs are the main limitations to
precision for low-order moments, while high orders are mainly affected by the
finite statistical samples. The unprecedentedly high fidelity numerical results
reported in this work offer an ideal benchmark for the development of future
theoretical models of intermittency in dynamical systems for either extreme
events (high-order moments) or typical fluctuations (low-order moments). For
the latter, we show that we achieve a precision in the determination of the
inertial range scaling exponents of the order of one part over ten thousand
(5th significant digit), which must be considered a record for
out-of-equilibrium fluid-mechanics systems and models.
