which country is the 1804 dollar associated with?
The United States of America.
1804 dollar
The 1804 dollar or Bowed Liberty Dollar was a dollar coin struck by the Mint of the United States, of which fifteen specimens are currently known to exist. 
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1804 dollar [SEP] Introduction
The 1804 dollar or Bowed Liberty Dollar was a dollar coin struck by the Mint of the United States, of which fifteen specimens are currently known to exist. Though dated 1804, none were struck in that year; all were minted in the 1830s or later. They were first created for use in special proof coin sets used as diplomatic gifts during Edmund Roberts' trips to Siam and Muscat. Edmund Roberts distributed the coins in 1834 and 1835. Two additional sets were ordered for government officials in Japan and Cochinchina, but Roberts died in Macau before they could be delivered. Besides those 1804 dollars produced for inclusion in the diplomatic sets, the Mint struck some examples which were used to trade with collectors for pieces desired for the Mint's coin cabinet.