That pistol is a Remington New Model Police Revolver.
They were made in 1865- 1873 with a total production of around 18000. (That number includes some that were modified for cartridges).
The complete barrel markings should say,
" PATENTED SEPT. 14, 1858 MARCH 17, 1863
E.REMINGTON & SONS, ILION, NEW YORK
NEW MODEL"
This pistol was made to compete with the Colt 1862 Police Revolver and the Pocket Navy so it uses a smaller frame than the larger ARMY and NAVY revolver.
It is a .36 caliber gun with a 5 shot cylinder.
In 2007 when my "FLAYDERMAN'S GUIDE TO ANTIQUE AMERICAN FIREARMS" was printed it was valued at $475 in "good" condition and $1200 in "fine" condition.
Due to inflation that would be $658 and $1662 in 2019 US dollars.
I'm not sure about the pinkish colored frame. Flayderman's Guide says some of these pistols were nickle plated all over, some with a nickle plated frame and the rest were blued all over.
The color reminds me very much of the pump slide guide on my Remington 870 Wingmaster. It came from the factory with a pinkish tinge to it although the rest of the steel parts are blued.