  PbCuTe$_2$O$_6$ is considered as one of the rare candidate materials for a
three-dimensional quantum spin liquid (QSL). This assessment was based on the
results of various magnetic experiments, performed mainly on polycrystalline
material. More recent measurements on single crystals revealed an even more
exotic behavior, yielding ferroelectric order below $T_{\text{FE}}\approx
1\,\text{K}$, accompanied by distinct lattice distortions, and a somewhat
modified magnetic response which is still consistent with a QSL. Here we report
on low-temperature measurements of various thermodynamic, magnetic and
dielectric properties of single crystalline PbCuTe$_2$O$_6$ in magnetic fields
$B\leq 14.5\,\text{T}$. The combination of these various probes allows us to
construct a detailed $B$-$T$ phase diagram including a ferroelectric phase for
$B \leq$ $8\,\text{T}$ and a $B$-induced magnetic phase at $B \geq$
$11\,\text{T}$. These phases are preceded by or coincide with a structural
transition from a cubic high-temperature phase into a distorted non-cubic
low-temperature state. The phase diagram discloses two quantum critical points
(QCPs) in the accessible field range, a ferroelectric QCP at $B_{c1}$ =
$7.9\,\text{T}$ and a magnetic QCP at $B_{c2}$ = $11\,\text{T}$. Field-induced
lattice distortions, observed in the state at $T>$ $1\,\text{K}$ and which are
assigned to the effect of spin-orbit interaction of the Cu$^{2+}$-ions, are
considered as the key mechanism by which the magnetic field couples to the
dielectric degrees of freedom in this material.
