It doesn't really look quite right to be original black. The black paint is too shiney. Black lenses in Nikon mount had a chrome front ring and chrome rear mount. Also, most black Nikkors of that era have some brassing. But I haven't got much experience in LTM lenses. According to Bob Rotoloni's Nikon RF handbook, the lenses switched from chrome to black around serial number 714000, whereas your lens is 712XXX. My Black Nikon-S version is 717XXX. My black lens also doesn't have the W-Nikkor C that yours has, just a W-Nikkor with no C.
Two possible sources for a black lens out of series such as yours:
1. A later black repaint, not uncommon in Japan, either for a collector or a user.
2. A factory-black special paint job from 1956-ish, its approximate year of construction, for a special professional order, such as Life magazine or some other photojournalism group. Several special runs of black Nikon cameras were made by request for professional photographers.
In any case, black LTM lenses are extremely rare because by the time Nikon switched to black-barrel lenses, Leica had introduced the M mount.