Thermodynamic and shadow radius analysis of the charged Einstein-Euler-Heisenberg black hole

We perform the thermodynamic and shadow radius analysis of an electrically charged black hole (EC) with electric charge $q$ and coupling constant $\mu$ obtained from the Einstein-Euler-Heisenberg nonlinear electodynamics. For $\mu=0.03$, we have four solution branches of the horizon including low, hot, negative, and cold ones, while for $\mu=0.3,3$ there exist single branches without limitation on $q$. The shadow radius for the low branch is the nearly same as that for the Reissner-Norstr"om black hole for $q<1$ case, while one finds the $q>1$ negative branch which is constrained by the EHT observation.