HYACINTHOS 29469

[Antreas P. Hatzipolakis]:

Update of Hyacinthos 26850

[Peter Moses]:

1).

(name pending) =

= (a^8 - 3*a^6*b^2 + 4*a^4*b^4 - 3*a^2*b^6 + b^8 - 4*a^6*c^2 - 6*a^4*b^2*c^2 - 6*a^2*b^4*c^2 - 4*b^6*c^2 + 6*a^4*c^4 + 13*a^2*b^2*c^4 + 6*b^4*c^4 - 4*a^2*c^6 - 4*b^2*c^6 + c^8)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 3*a^6*c^2 - 6*a^4*b^2*c^2 + 13*a^2*b^4*c^2 - 4*b^6*c^2 + 4*a^4*c^4 - 6*a^2*b^2*c^4 + 6*b^4*c^4 - 3*a^2*c^6 - 4*b^2*c^6 + c^8) : :  

= 1/((a^2*SA*(S^2 + 5*SA^2) - (3*S^2 - SA^2)*SB*SC)) : :

= lies on the circumconic {A,B,C,X(4),X(1656)}(*)  and this line: {1656,18451}

(*) ABCH X{1656, 2963, 4994, 14938}

 

 

2). 
X(15424) =

= 1/(a^2*SA*(S^2 + 5*SA^2)) : :

=lies on the circumconic {A,B,C,X(4),X(93)}(*) and these lines: {4, 3521}, {93, 403}, {186, 1105}, {235, 6344}, {264, 16868}, {1217, 7505}, {1300, 3518}, {3147, 18852}, {8884, 11815}, {14249, 16263}, {18808, 23290}, {26863, 32085}

= X(i)-isoconjugate of X(j) for these (i,j): {255, 3520}, {2169, 11591}
= barycentric product X(i)*X(j) for these {i,j}: {324, 11815}, {2052, 3521}, {23290, 30527}
= barycentric quotient X(i)/X(j) for these {i,j}: {53, 11591}, {393, 3520}, {3521, 394}, {11815, 97}

(*) ABCH X{93, 225, 254, 264, 393, 847, 1093, 1105, 1179, 1217, 1300, 1826, 6344, 6526, 6531, 8737, 8738, 8741, 8742, 8801, 8884...}.

3)  lies on  the circumconic: {A,B,C,X(4),X(1657)} 
4). lies on  the circumconic: {A,B,C,X(4),X(550)}  

Best regards,
Peter Moses.