We use our Ace-high's showdown value and excellent pot odds to catch a small river bet after the board double-pairs.
Checking is preferred here. While we have the range advantage, A3o has very little equity and no real draws, making it a poor candidate for a large c-bet.
Note: Betting A-high with no draws on this texture is unnecessary; checking back allows us to realize equity and catch bluffs later.
Checking back is the correct play. The Jack pairing the board is better for the SB's range, as they have more Jx than we do after we bet the flop.
Calling is mandatory given the price. The board double-pairing means our Ace-high has been upgraded to two pair, and we only need to be right ~20% of the time.