Polynomial Invariants | #Aut | Splitting Field | Number of | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
j | Ramification Polygon | Slopes | Residual Polynomials | fT | eT | #Gal | Gal | Polynomials | Extensions | |||
1 | {(1,1), (4,0)} | [ 1/3 ] | (z+1) | {1} | 1 | 3 | { 12 } | { 4T4 } | 1 | 22 | 3·22 | 3·22 |
(z+δ) | {1} | 1 | 3 | { 12 } | { 4T4 } | 1 | 22 | |||||
(z+(δ+1)) | {1} | 1 | 3 | { 12 } | { 4T4 } | 1 | 22 | |||||
3 | {(1,3), (2,2), (4,0)} | [ 1 ] | (z3+z+1) | {1} | 3 | 1 | { 12 } | { 4T4 } | 1 | 22 | 9·22 | 3·24 |
(z3+δz+1) | {1} | 3 | 1 | { 12 } | { 4T4 } | 1 | 22 | |||||
(z3+(δ+1)z+1) | {1} | 3 | 1 | { 12 } | { 4T4 } | 1 | 22 | |||||
(z3+z+δ) | {2} | 2 | 1 | { 8 } | { 4T3 } | 2 | 22 | |||||
(z3+δz+δ) | {2} | 2 | 1 | { 8 } | { 4T3 } | 2 | 22 | |||||
(z3+(δ+1)z+δ) | {2} | 2 | 1 | { 8 } | { 4T3 } | 2 | 22 | |||||
(z3+z+(δ+1)) | {2} | 2 | 1 | { 8 } | { 4T3 } | 2 | 22 | |||||
(z3+δz+(δ+1)) | {2} | 2 | 1 | { 8 } | { 4T3 } | 2 | 22 | |||||
(z3+(δ+1)z+(δ+1)) | {2} | 2 | 1 | { 8 } | { 4T3 } | 2 | 22 | |||||
{(1,3), (4,0)} | [ 1 ] | (z3+1) | {4} | 1 | 1 | { 4 } | { 4T2 } | 22 | 22 | 3·22 | ||
(z3+δ) | {1} | 3 | 1 | { 12 } | { 4T4 } | 1 | 22 | |||||
(z3+(δ+1)) | {1} | 3 | 1 | { 12 } | { 4T4 } | 1 | 22 | |||||
5 | {(1,5), (2,2), (4,0)} | [ 3, 1 ] | (z+1, z2+1) | {2,4} | 1 | 1 | 24 [3·22] | 24 | 9·24 | 3·26 | ||
(δz+1, z2+δ) | {2} | 1 | 1 | 24 [23] | 24 | |||||||
((δ+1)z+1, z2+(δ+1)) | {2} | 1 | 1 | 24 [23] | 24 | |||||||
(z+δ, z2+1) | {2,4} | 1 | 1 | 24 [3·22] | 24 | |||||||
(δz+δ, z2+δ) | {2} | 1 | 1 | 24 [23] | 24 | |||||||
((δ+1)z+δ, z2+(δ+1)) | {2,4} | 1 | 1 | 24 [3·22] | 24 | |||||||
(z+(δ+1), z2+1) | {2,4} | 1 | 1 | 24 [3·22] | 24 | |||||||
(δz+(δ+1), z2+δ) | {2,4} | 1 | 1 | 24 [3·22] | 24 | |||||||
((δ+1)z+(δ+1), z2+(δ+1)) | {2} | 1 | 1 | 24 [23] | 24 | |||||||
6 | {(1,6), (2,2), (4,0)} | [ 4, 1 ] | (z+1, z2+1) | {2} | 1 | 1 | 26 [25] | 26 | 3·26 | 3·26 | ||
(δz+δ, z2+δ) | {2} | 1 | 1 | 26 [25] | 26 | |||||||
((δ+1)z+(δ+1), z2+(δ+1)) | {2} | 1 | 1 | 26 [25] | 26 | |||||||
7 | {(1,7), (2,4), (4,0)} | [ 3, 2 ] | (z+1, z2+1) | {2} | 1 | 1 | 26 [25] | 26 | 3·26 | 3·26 | ||
(z+δ, z2+1) | {2} | 1 | 1 | 26 [25] | 26 | |||||||
(z+(δ+1), z2+1) | {2} | 1 | 1 | 26 [25] | 26 | |||||||
8 | {(1,8), (2,4), (4,0)} | [ 4, 2 ] | (z+1, z2+1) | {2,4} | 1 | 1 | 28 [9·24] | 28 | 28 | 28 |