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This work is a sequel of a previous work of one of the authors (Y.\^O), whichtreated certain congruence relation between an elliptic Gauss sum and acoefficient of power series expansion at the origin of the lemniscate sinefunction. We extend the previous result (in \cite{O}) which concerned only fornon-vanishing elliptic Gauss sums. We give new congruence relations betweenpower series coefficients of the lemniscate cosine function, which hold if andonly if the corresponding elliptic Gauss sum vanishes.

Arithmetic over the Gaussian Number Field on a Certain Family of Elliptic Curves with Complex Multiplication