X(13), X(14), X(15), X(16), X(39), X(76), X(538), X(755)
X(14901) = X13-X14 /\ X39-X755
X(14902) = X13-X15 /\ X476-X755
X(14903) = X14-X16 /\ X476-X755
X(14904) = X13-X16 /\ X99-X755
X(14905) = X14-X15 /\ X99-X755
K290 is the orthopivotal cubic with orthopivot X(39), the Brocard midpoint. See the FG paper "Orthocorrespondence and orthopivotal cubics" in the Downloads page and Orthopivotal cubics in the glossary.
Philippe Deléham has found a characterization of K290 which he describes as follows. Let M be a point and M' the point with barycentric coordinates (AM)^2 : (BM)^2 : (CM)^2. For a given point P, the points M, M', P are collinear if and only if M lies on a cubic K_P. K_P is a circum-cubic if and only if P = X(76), the isotomic conjugate of the Lemoine point, and this circum-cubic is K290.