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X(2), X(17), X(18), X(20), X(61), X(62), X(1131), X(1132), X(3068), X(3069), X(3070), X(3071), X(39641), X(39642)

Q167 is a KHO-curve, see K1191 and CL075 for information. See also Q073, another KHO-quartic.

The KHO-equation is : 2x^4 - x^2 ( 94y^2 - 59yz + 12z^2) + (2y - z) (2y + z) (66y^2 - 101yz + 54z^2) = 0.

Other KHO-centers on the curve : (±88 √3, 45, 86), (±80 √3, 99, 38), (240, -29, 22), (240, 29, -22).

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Peter Moses also found several other similar KHO-quartics listed in the table below.

KHO-equation

X(i) on the curve for these i

2x^4 + 3x^2 y (10y - 9z) - 9 (y -z) (2y - z) (6y^2 - 3yz + 2z^2) = 0

see Q168

2x^4 - 3x^2 y (10y - z) + 3 (2y -z) (2y + z) (6y^2 - 7yz + 6z^2) = 0

2, 17, 18, 20, 371, 372, 1131, 1132, 1587, 1588, 3068, 3069, 3070, 3071, 39641, 39642

2x^4 + 3x^2 y (46y - 27z) - 9 (2y -z) (2y + z) (12y^2 - 3yz - 2z^2) = 0

2, 17, 18, 20, 371, 372, 485, 486, 3068, 3069, 3070, 3071, 39641, 39642