Sol on the coinage is one of the commonest images of the third and early fourth centuries: a radiate figure with the right hand raised and a globe or whip in the left, over legends such as SOLI INVICTO COMITI, to the unconquered Sun, the companion. Constantine went on striking that legend for well over a decade after 312, which is the single fact that most complicates any tidy account of his conversion. Sun and emperor share attributes in this period, the radiate crown above all, and the message is less theological than political: here is a ruler who cannot be overcome.
The December date rests on one document. The Chronography of 354, a Roman almanac compiled by Furius Dionysius Filocalus, records under the eighth day before the kalends of January a natalis Invicti with thirty chariot races. The same compilation, in its list of martyrs, gives the birth of Christ on the same day. Both entries are there, and they are the earliest surviving evidence for either.
What follows from that is disputed. The older view holds that the Christian feast was placed on the pagan one to displace it. The calculation hypothesis holds that the date was reached independently, from a reckoning that put the conception at the spring equinox, and points to Christian writers using the date before Aurelian founded anything. Neither reading has carried the field, and the atlas leaves the question open rather than repeating the version that is easier to tell.