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We present a lattice calculation of the Hadronic Vacuum Polarization (HVP)contribution of the strange and charm quarks to the anomalous magnetic momentof the muon including leading-order electromagnetic corrections. We employ thegauge configurations generated by the European Twisted Mass Collaboration(ETMC) with $N_f = 2+1+1$ dynamical quarks at three values of the latticespacing ($a \simeq 0.062, 0.082, 0.089$ fm) with pion masses in the range$M_\pi \simeq 210 - 450$ MeV. The strange and charm quark masses are tuned attheir physical values. Neglecting disconnected diagrams and after theextrapolations to the physical pion mass and to the continuum limit we obtain:$a_\mu^s(\alpha_{em}^2) = (53.1 \pm 2.5) \cdot 10^{-10}$,$a_\mu^s(\alpha_{em}^3) = (-0.018 \pm 0.011) \cdot 10^{-10}$ and$a_\mu^c(\alpha_{em}^2) = (14.75 \pm 0.56) \cdot 10^{-10}$,$a_\mu^c(\alpha_{em}^3) = (-0.030 \pm 0.013) \cdot 10^{-10}$ for the strangeand charm contributions, respectively.

Strange and charm HVP contributions to the muon ($g - 2)$ including QED corrections with twisted-mass fermions