Polynomial Invariants | #Aut | Splitting Field | Number of | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
j | Ramification Polygon | Slopes | Residual Polynomials | fT | eT | #Gal | Gal | Polynomials | Extensions | |||
1 | {(1,1), (3,0), (12,0)} | [ 1/2, 0 ] | (z+1, z9+1) | {1} | 2 | 8 | { 144 } | { 12T84 } | 1 | 4·3 | 8·3 | 8·3 |
(z+2, z9+1) | {1} | 2 | 8 | { 144 } | { 12T84 } | 1 | 4·3 | |||||
2 | {(1,2), (3,0), (12,0)} | [ 1, 0 ] | (z2+1, z9+1) | {2} | 2 | 4 | { 72 } | { 12T36 } | 1 | 4·3 | 8·3 | 8·3 |
(z2+2, z9+1) | {3,6} | 2 | 4 | { 72, 216 } | { 12T121, 12T35 } | 3 | 4·3 | |||||
4 | {(1,4), (3,0), (12,0)} | [ 2, 0 ] | (z2+1, z9+1) | {1,2} | 2 | 4 | { 24, 216 } | { 12T13, 12T120 } | 3 | 4·32 | 8·32 | 8·32 |
(z2+2, z9+1) | {3,6} | 2 | 4 | { 24, 72, 216, 648 } | { 12T42, 12T116, 12T167, 12T15 } | 32 | 4·32 | |||||
5 | {(1,5), (3,0), (12,0)} | [ 5/2, 0 ] | (z+1, z9+1) | {1} | 2 | 8 | { 144, 1296 } | { 12T84, 12T212 } | 3 | 4·32 | 8·32 | 8·32 |
(z+2, z9+1) | {1} | 2 | 8 | { 144, 1296 } | { 12T84, 12T212 } | 3 | 4·32 | |||||
7 | {(1,7), (3,0), (12,0)} | [ 7/2, 0 ] | (z+1, z9+1) | {1} | 2 | 8 | { 144, 1296 } | { 12T84, 12T212 } | 32 | 4·33 | 8·33 | 8·33 |
(z+2, z9+1) | {1} | 2 | 8 | { 144, 1296 } | { 12T84, 12T212 } | 32 | 4·33 | |||||
8 | {(1,8), (3,0), (12,0)} | [ 4, 0 ] | (z2+1, z9+1) | {1,2} | 2 | 4 | { 24, 72, 216, 648 } | { 12T118, 12T12, 12T38, 12T169 } | 32 | 4·33 | 8·33 | 8·33 |
(z2+2, z9+1) | {3,6} | 2 | 4 | { 24, 72, 216, 648 } | { 12T121, 12T42, 12T167, 12T14 } | 33 | 4·33 | |||||
10 | {(1,10), (3,0), (12,0)} | [ 5, 0 ] | (z2+1, z9+1) | {1,2} | 2 | 4 | { 72, 216, 648 } | { 12T118, 12T120, 12T36, 12T169 } | 33 | 4·34 | 8·34 | 8·34 |
(z2+2, z9+1) | {3,6} | 2 | 4 | { 72, 216, 648 } | { 12T121, 12T167, 12T116, 12T35 } | 34 | 4·34 | |||||
11 | {(1,11), (3,0), (12,0)} | [ 11/2, 0 ] | (z+1, z9+1) | {1} | 2 | 8 | { 144, 1296 } | { 12T84, 12T212 } | 33 | 4·34 | 8·34 | 8·34 |
(z+2, z9+1) | {1} | 2 | 8 | { 144, 1296 } | { 12T84, 12T212 } | 33 | 4·34 | |||||
12 | {(1,12), (3,0), (12,0)} | [ 6, 0 ] | (z2+2, z9+1) | {3} | 2 | 4 | 35 | 4·35 | 4·35 | 4·35 |