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X(2), X(5), X(6), X(13), X(14), X(15), X(16), X(17), X(18), X(140)

X(42488) → X(42497)

other points below

Geometric properties :

K1194 is a KHO-cubic. See K1191 for explanations and also CL075.

Its KHO-equation is : x^2 (4y - 3z) - (2y - z) (3y - z) (y - z) = 0.

Other points on K1194

Q1 : (3,4,7) : 3 a^4-7 a^2 b^2+4 b^4-7 a^2 c^2-8 b^2 c^2+4 c^4-2 Sqrt[3] a^2 S : : , SEARCH = 1.84597594037453

Q2 : (-3,4,7) : 3 a^4-7 a^2 b^2+4 b^4-7 a^2 c^2-8 b^2 c^2+4 c^4+2 Sqrt[3] a^2 S : : , SEARCH = 3.394278959915633

Q3 : (3,2,8) : 6 a^4-8 a^2 b^2+2 b^4-8 a^2 c^2-4 b^2 c^2+2 c^4-2 Sqrt[3] a^2 S : : , SEARCH = 3.15768283859599

Q4 : (-3,2,8) : 6 a^4-8 a^2 b^2+2 b^4-8 a^2 c^2-4 b^2 c^2+2 c^4+2 Sqrt[3] a^2 S : : , SEARCH = 7.910971580119713

Q5 : (12,5,7) : -2 a^4+7 a^2 b^2-5 b^4+7 a^2 c^2+10 b^2 c^2-5 c^4+8 Sqrt[3] a^2 S : : , SEARCH = 1.210882227623435

Q6 : (-12,5,7) : -2 a^4+7 a^2 b^2-5 b^4+7 a^2 c^2+10 b^2 c^2-5 c^4-8 Sqrt[3] a^2 S : : , SEARCH = 0.1893155465687874

Q7 : (6,7,6) : -a^4-6 a^2 b^2+7 b^4-6 a^2 c^2-14 b^2 c^2+7 c^4-4 Sqrt[3] a^2 S : : , SEARCH = 0.4836738608084304

Q8 : (-6,7,6) : -a^4-6 a^2 b^2+7 b^4-6 a^2 c^2-14 b^2 c^2+7 c^4+4 Sqrt[3] a^2 S : : , SEARCH = -3.72392685147395

Triples of collinear points on K1194

X2, X5, X140 – X2, X13, X16 – X2, X14, X15 – X2, X17, Q1 – X2, X18, Q2 – X2, Q3, Q8 – X2, Q4, Q7 – X5, X13, X18 – X5, X14, X17 – X5, X15, Q1 – X5, X16, Q2 – X5, Q5, Q8 – X5, Q6, Q7 – X6, X13, X14 – X6, X15, X16 – X6, X17, X18 – X6, Q1, Q2 – X6, Q3, Q4 – X6, Q5, Q6 – X6, Q7, Q8 – X13, X15, Q5 – X13, X17, Q3 – X13, X140, Q1 – X13, Q2, Q7 – X14, X16, Q6 – X14, X18, Q4 – X14, X140, Q2 – X14, Q1, Q8 – X15, X17, Q7 – X15, X18, X140 – X15, Q2, Q3 – X16, X17, X140 – X16, X18, Q8 – X16, Q1, Q4 – X17, Q2, Q5 – X18, Q1, Q6 – X140, Q3, Q6 – X140, Q4, Q5.

Q1 lies on the lines (X2,X17), (X5,X15), (X13,X140), (X14,X3090), (X18,X396), (X61,X1656).

Q2 lies on the lines (X2,X18), (X5,X16), (X13,X3090), (X14,X140), (X17,X395),(X62,X1656).

***

More generally, a cubic with KHO-equation below passes through X(2), X(6), X(13), X(14), X(15), X(16), X(17), X(18).

x^2(v y + w z) - (2y - z)[3 v y z - w (3 y^2 - 8 y z + z^2)] = 0, where v, w are two real numbers.

These cubics are in a same pencil which contains K1191, K1194, K1196 and two proper nodal cubics. They are those passing through X(14813) (an acnodal cubic) and X(14814) (a crunodal cubic).