  We study the qualitative properties of groundstates of the time-independent
magnetic semilinear Schr\"odinger equation \[ - (\nabla + i A)^2 u + u =
|u|^{p-2} u, \qquad \text{ in } \mathbb{R}^N, \] where the magnetic potential
$A$ induces a constant magnetic field. When the latter magnetic field is small
enough, we show that the groundstate solution is unique up to magnetic
translations and rotations in the complex phase space, that groundstate
solutions share the rotational invariance of the magnetic field and that the
presence of a magnetic field induces a Gaussian decay. In this small magnetic
field r\'egime, the corresponding ground-energy is a convex differentiable
function of the magnetic field.
