Hi!
You are referring at original Nikon AF-S 12-24mm f/4
I guess; well, it takes 77mm thread.
I bought a great "Slim" genuine Nikon CPL II 77mm (Circular Polarizing Filter); I use it on other lenses, though.
Fitting Cir Pol on Wide, Filter's mounting-ring must be very thin ("slim") keeping its glass closest to the objectiv's frontal-lens, to avoid "vignetting" in picture.
Besides, an heavier mounting-ring would form an obstacle to the widest rays; by that, the corners of the picture would result as black-rounded at all, like if the pic is taken through (from inside) a tube (yet, this may be a desired effect, sometimes!).
In this "slim" version, the first mounting-ring is 3-4 mm only, in thickness.
But I am not aware using it on a such 12mm wide Focal-length; I successfully used it up to 20mm widing.
However, I do not think either Nikon or other brands make any better type of Cir Pol, at now.
Nikon makes some special ones for COOLPIX Digital (not-SLR/single-lens) Cameras, like FF-CP10 CPL Filter; but it is 48 mm thread (size) only, allowed to COOLPIX 8400.
The largest size required in COOLPIX System, would be 72mm by the WC-E63 Wide Angle Adapter; so, it takes filters among any 72mm common ones, yet; no special product worked out.
Instead, they declare regarding
Using Filters with Nikon 10.5 mm DX Lens
(a shorter than Your 12mm Focal-length):
"Due to the extreme wide angle of view of the Nikon 10.5mm DX lens, filters cannot be attached to the front of the lens as per a regular lens. Instead Nikon has provided a gelatin filter holder at the rear of the lens allowing the use of filters without causing vignetting."
Since they did not constructed any rear Filter-holder on 12-24mm Lens (is is so?), I think this Objectiv works well with the Cir Pol NIKON CPL II 77mm ("II"="Slim Version").
Sorry, I did not find any pic of the filter to do attachment here.
Greetings.
