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∑ (b+ca) x^2 (c^2 y  b^2 z) = 0 or ∑ a^2 [(c+ab)y  (a+bc)z] yz = 0 

X(6), X(7), X(509), X(1486) A'B'C' = cevian triangle of X(7) 

K631 is the only Poncelet poristic cubic which is a pivotal cubic namely pK(X56, X7). See PseudoPivotal Cubics and Poristic Triangles, §2.3. 
