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Its rank is $q$ when $q \\equiv 1 \\bmod 3$ and its rank is $q-2$ when $q \\equiv 2 \\bmod 3$. We describe an explicit method for producing points on this elliptic curve. In case $q \\not\\equiv 11 \\bmod 12$, our method produces points which generate a full-rank subgroup. 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