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The BL Lac object S5~0716+714, a highly variable blazar, underwent animpressive outburst in January 2015 (Phase A), followed by minor activity inFebruary (Phase B). The MAGIC observations were triggered by the optical fluxobserved in Phase A, corresponding to the brightest ever reported state of thesource in the R-band. The comprehensive dataset collected is investigated inorder to shed light on the mechanism of the broadband emission.Multi-wavelength light curves have been studied together with the broadbandSpectral Energy Distributions (SEDs). The data set collected spans from radio,optical photometry and polarimetry, X-ray, high-energy (HE, 0.1 GeV < E < 100GeV) with \textit{Fermi}-LAT to the very-high-energy (VHE, E>100 GeV) withMAGIC. The flaring state of Phase A was detected in all the energy bands,providing for the first time a multi-wavelength sample of simultaneous datafrom the radio band to the VHE. In the constructed SED the\textit{Swift}-XRT+\textit{NuSTAR} data constrain the transition between thesynchrotron and inverse Compton components very accurately, while the secondpeak is constrained from 0.1~GeV to 600~GeV by \textit{Fermi}+MAGIC data. Thebroadband SED cannot be described with a one-zone synchrotron self-Comptonmodel as it severely underestimates the optical flux in order to reproduce theX-ray to $\gamma$-ray data. Instead we use a two-zone model. The EVPA shows anunprecedented fast rotation. An estimation of the redshift of the source bycombined HE and VHE data provides a value of $z = 0.31 \pm 0.02_{stats} \pm0.05_{sys}$, confirming the literature value. The data show the VHE emissionoriginating in the entrance and exit of a superluminal knot in and out arecollimation shock in the inner jet. A shock-shock interaction in the jetseems responsible for the observed flares and EVPA swing. This scenario is alsoconsistent with the SED modelling.

Multi-wavelength characterization of the blazar S5~0716+714 during an unprecedented outburst phase