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We calculate the decay rates for $\eta_c \to \gamma \gamma$, $J/\psi \eta_c$ and \eta_c e^+e^-$ in lattice QCD with $u$, $d$, $s$ $c$ quarks sea first time. improve significantly on previous theory calculations to achieve accuracies of 1--2\%, giving results that are now more accurate than experimental values. In particular our transform theoretical picture $\eta_c\to\gamma\gamma$ decays. use gluon field configurations generated by MILC collaboration include $n_f=2+1+1$ flavours Highly Improved Staggered (HISQ) at four spacing values from 0.15 fm 0.06 u/d masses down their physical value. also implement valence using HISQ action. find ${\Gamma (\eta_c \gamma) = 6.788(45)_{\text{fit}}(41)_{\text{syst}} \: \mathrm{keV}}$, good agreement $\gamma\gamma K\overline{K}\pi$ but 4$\sigma$ tension Particle Data Group global fit result; we suggest this is revisited. $\Gamma (J/\psi \eta_c) 2.219(17)_{\text{fit}}(18)_{\text{syst}}(24)_{\text{expt}}(4)_{\text{QED}} \; \mathrm{keV}$, CLEO, predict Dalitz rate e^+ e^-) 0.01349(21)_{\text{latt}}(13)_{\text{QED}} \mathrm{keV}$. calibrate other approaches test simple relationships between form factors $J/\psi$ constant expected nonrelativistic limit.