Polynomial Invariants |
#Aut |
Splitting Field |
Number of
|
j |
Ramification Polygon |
Slopes |
Residual Polynomials |
fT |
eT |
#Gal |
Gal |
Polynomials |
Extensions
|
1 | {(1,1), (4,0), (12,0)} | [ 1/3, 0 ] | (z+1, z8+1) | {1} | 6 | 9 | { 3456 } | { 12T254 } | 1 | 3·22 | 3·22 | 3·22 |
3 | {(1,3), (2,2), (4,0), (12,0)} | [ 1, 0 ] | (z3+z+1, z8+1) | {1} | 6 | 3 | | | 1 | 3·22 | 3·22 | 3·23 |
{(1,3), (4,0), (12,0)} | [ 1, 0 ] | (z3+1, z8+1) | {2} | 2 | 3 | { 24, 48 } | { 12T8, 12T27 } | 2 | 3·22 | 3·22 |
5 | {(1,5), (2,2), (4,0), (12,0)} | [ 3, 1, 0 ] | (z+1, z2+1, z8+1) | {2,4} | 2 | 3 | { 96, 192, ...} | { 12T64, ..., 12T98, 12T62, 12T65 } | 23 [3·2] | 3·23 | 3·23 | 3·24 |
{(1,5), (4,0), (12,0)} | [ 5/3, 0 ] | (z+1, z8+1) | {1} | 6 | 9 | { 3456 } | { 12T254 } | 2 | 3·23 | 3·23 |
7 | {(1,7), (2,2), (4,0), (12,0)} | [ 5, 1, 0 ] | (z+1, z2+1, z8+1) | {2,4} | 2 | 3 | { 48, 192, ...} | { ..., 12T24, 12T97, 12T27, 12T98, 12T23, 12T96, 12T22 } | 24 [3·22] | 3·24 | 3·24 | 3·25 |
{(1,7), (2,6), (4,0), (12,0)} | [ 1, 3, 0 ] | (z+1, z2+1, z8+1) | {1} | 6 | 9 | { 3456 } | { 12T254 } | 22 | 3·23 | 3·23 |
{(1,7), (4,0), (12,0)} | [ 7/3, 0 ] | (z+1, z8+1) | {1} | 6 | 9 | { 3456 } | { 12T254 } | 22 | 3·24 | 3·24 |
9 | {(1,9), (2,2), (4,0), (12,0)} | [ 7, 1, 0 ] | (z+1, z2+1, z8+1) | {2} | 2 | 3 | | | 25 [24] | 3·25 | 3·25 | 3·26 |
{(1,9), (2,6), (4,0), (12,0)} | [ 3, 0 ] | (z3+z+1, z8+1) | {1} | 6 | 3 | { 72, 1152 } | { 12T43, 12T206 } | 22 | 3·24 | 3·24 |
{(1,9), (4,0), (12,0)} | [ 3, 0 ] | (z3+1, z8+1) | {2} | 2 | 3 | { 24, 96 } | { 12T13, 12T52, 12T49 } | 23 | 3·24 | 3·24 |
11 | {(1,11), (2,2), (4,0), (12,0)} | [ 9, 1, 0 ] | (z+1, z2+1, z8+1) | {2,4} | 2 | 3 | { 96, 192, ...} | { 12T100, 12T96, 12T67, ..., 12T65, 12T103, 12T98, 12T68, 12T97, 12T66, 12T64, 12T62, 12T102 } | 26 [5·23] | 3·26 | 3·26 | 3·27 |
{(1,11), (2,6), (4,0), (12,0)} | [ 5, 3, 0 ] | (z+1, z2+1, z8+1) | {2} | 2 | 3 | { 192, ...} | { 12T113, ..., 12T109 } | 25 [24] | 3·25 | 3·25 |
{(1,11), (2,10), (4,0), (12,0)} | [ 1, 5, 0 ] | (z+1, z2+1, z8+1) | {1} | 6 | 9 | { 3456 } | { 12T254 } | 23 | 3·24 | 3·24 |
{(1,11), (4,0), (12,0)} | [ 11/3, 0 ] | (z+1, z8+1) | {1} | 6 | 9 | { 3456 } | { 12T254 } | 23 | 3·25 | 3·25 |
13 | {(1,13), (2,2), (4,0), (12,0)} | [ 11, 1, 0 ] | (z+1, z2+1, z8+1) | {2,4} | 2 | 3 | { 48, 192, ...} | { 12T100, 12T96, ..., 12T27, 12T23, 12T103, 12T98, 12T97, 12T24, 12T22, 12T102 } | 27 [5·24] | 3·27 | 3·27 | 3·28 |
{(1,13), (2,6), (4,0), (12,0)} | [ 7, 3, 0 ] | (z+1, z2+1, z8+1) | {2,4} | 2 | 3 | { 48, 192, 384, ...} | { 12T113, ..., 12T24, 12T23, 12T109, 12T148, 12T101 } | 26 [5·23] | 3·26 | 3·26 |
{(1,13), (2,10), (4,0), (12,0)} | [ 3, 5, 0 ] | (z+1, z2+1, z8+1) | {1} | 6 | 9 | { 3456 } | { 12T254 } | 24 [.] | 3·25 | 3·25 |
14 | {(1,14), (2,2), (4,0), (12,0)} | [ 12, 1, 0 ] | (z+1, z2+1, z8+1) | {2} | 2 | 3 | | | 28 [27] | 3·28 | 3·28 | 3·28 |
15 | {(1,15), (2,6), (4,0), (12,0)} | [ 9, 3, 0 ] | (z+1, z2+1, z8+1) | {2,4} | 2 | 3 | { 24, 96, 384, ...} | { 12T13, ..., 12T49, 12T48, 12T148, 12T10 } | 27 [5·24] | 3·27 | 3·27 | 3·28 |
{(1,15), (2,10), (4,0), (12,0)} | [ 5, 0 ] | (z3+z+1, z8+1) | {1} | 6 | 3 | { 288, 1152 } | { 12T128, 12T206 } | 24 | 3·26 | 3·26 |
{(1,15), (2,12), (4,0), (12,0)} | [ 3, 6, 0 ] | (z+1, z2+1, z8+1) | {2} | 2 | 3 | { 24, 48, 96, 192, 384 } | { 12T8, 12T100, 12T49, 12T66, 12T148, 12T22 } | 25 [.] | 3·26 | 3·26 |
17 | {(1,17), (2,6), (4,0), (12,0)} | [ 11, 3, 0 ] | (z+1, z2+1, z8+1) | {2} | 2 | 3 | { 192, ...} | { ..., 12T109 } | 28 [27] | 3·28 | 3·28 | 3·29 |
{(1,17), (2,10), (4,0), (12,0)} | [ 7, 5, 0 ] | (z+1, z2+1, z8+1) | {2,4} | 2 | 3 | { 48, 96, 192, 768, ...} | { ..., 12T24, 12T103, 12T184, 12T23, 12T48 } | 27 [3·25] | 3·27 | 3·27 |
{(1,17), (2,12), (4,0), (12,0)} | [ 5, 6, 0 ] | (z+1, z2+1, z8+1) | {1} | 6 | 9 | | | 26 [.] | 3·27 | 3·27 |
18 | {(1,18), (2,6), (4,0), (12,0)} | [ 12, 3, 0 ] | (z+1, z2+1, z8+1) | {2,3,4,6,12} | 2 | 3 | | | 29 [?] | 3·29 | 3·29 | 3·29 |
19 | {(1,19), (2,10), (4,0), (12,0)} | [ 9, 5, 0 ] | (z+1, z2+1, z8+1) | {2} | 2 | 3 | | | 28 [27] | 3·28 | 3·28 | 3·29 |
{(1,19), (2,12), (4,0), (12,0)} | [ 7, 6, 0 ] | (z+1, z2+1, z8+1) | {2} | 2 | 3 | { 192, ...} | { 12T113, ..., 12T109 } | 28 [27] | 3·28 | 3·28 |
21 | {(1,21), (2,10), (4,0), (12,0)} | [ 11, 5, 0 ] | (z+1, z2+1, z8+1) | {2,3,4,6,12} | 2 | 3 | | | 29 [?] | 3·29 | 3·29 | 3·210 |
{(1,21), (2,12), (4,0), (12,0)} | [ 9, 6, 0 ] | (z+1, z2+1, z8+1) | {2,3,4,6,12} | 2 | 3 | | | 29 [?] | 3·29 | 3·29 |
22 | {(1,22), (2,10), (4,0), (12,0)} | [ 12, 5, 0 ] | (z+1, z2+1, z8+1) | {2,3,4,6,12} | 2 | 3 | | | 210 [?] | 3·210 | 3·210 | 3·210 |
23 | {(1,23), (2,12), (4,0), (12,0)} | [ 11, 6, 0 ] | (z+1, z2+1, z8+1) | {2,3,4,6,12} | 2 | 3 | | | 210 [?] | 3·210 | 3·210 | 3·210 |
24 | {(1,24), (2,12), (4,0), (12,0)} | [ 12, 6, 0 ] | (z+1, z2+1, z8+1) | {2,3,4,6,12} | 2 | 3 | | | 211 [?] | 3·211 | 3·211 | 3·211 |