Polynomial Invariants | #Aut | Splitting Field | Number of | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
j | Ramification Polygon | Slopes | Residual Polynomials | fT | eT | #Gal | Gal | Polynomials | Extensions | |||
1 | {(1,1), (3,0), (15,0)} | [ 1/2, 0 ] | (2z+1, z12+2) | {1} | 4 | 10 | { 3240 } | { 15T52 } | 1 | 5·3 | 10·3 | 10·3 |
(2z+2, z12+2) | {1} | 4 | 10 | { 3240 } | { 15T52 } | 1 | 5·3 | |||||
2 | {(1,2), (3,0), (15,0)} | [ 1, 0 ] | (2z2+1, z12+2) | {3} | 4 | 5 | { 1620, 4860 } | { 15T41, 15T56 } | 3 | 5·3 | 10·3 | 10·3 |
(2z2+2, z12+2) | {1} | 4 | 5 | { 1620 } | { 15T42 } | 1 | 5·3 | |||||
4 | {(1,4), (3,0), (15,0)} | [ 2, 0 ] | (2z2+1, z12+2) | {3} | 4 | 5 | { 1620, 4860 } | { 15T41, 15T56 } | 32 | 5·32 | 10·32 | 10·32 |
(2z2+2, z12+2) | {1} | 4 | 5 | { 1620 } | { 15T42 } | 3 | 5·32 | |||||
5 | {(1,5), (3,0), (15,0)} | [ 5/2, 0 ] | (2z+1, z12+2) | {1} | 4 | 10 | { 120, 9720 } | { 15T11, 15T64 } | 3 | 5·32 | 10·32 | 10·32 |
(2z+2, z12+2) | {1} | 4 | 10 | { 120, 9720 } | { 15T11, 15T64 } | 3 | 5·32 | |||||
7 | {(1,7), (3,0), (15,0)} | [ 7/2, 0 ] | (2z+1, z12+2) | {1} | 4 | 10 | { 3240, 9720 } | { 15T52, 15T64 } | 32 | 5·33 | 10·33 | 10·33 |
(2z+2, z12+2) | {1} | 4 | 10 | { 3240, 9720 } | { 15T52, 15T64 } | 32 | 5·33 | |||||
8 | {(1,8), (3,0), (15,0)} | [ 4, 0 ] | (2z2+1, z12+2) | {3} | 4 | 5 | { 1620, 4860 } | { 15T41, 15T56 } | 33 | 5·33 | 10·33 | 10·33 |
(2z2+2, z12+2) | {1} | 4 | 5 | { 1620 } | { 15T42 } | 32 | 5·33 | |||||
10 | {(1,10), (3,0), (15,0)} | [ 5, 0 ] | (2z2+1, z12+2) | {3} | 4 | 5 | { 60, 4860 } | { 15T56, 15T8 } | 34 | 5·34 | 10·34 | 10·34 |
(2z2+2, z12+2) | {1} | 4 | 5 | { 60, 4860 } | { 15T54, 15T6 } | 33 | 5·34 | |||||
11 | {(1,11), (3,0), (15,0)} | [ 11/2, 0 ] | (2z+1, z12+2) | {1} | 4 | 10 | { 3240, 9720 } | { 15T52, 15T64 } | 33 | 5·34 | 10·34 | 10·34 |
(2z+2, z12+2) | {1} | 4 | 10 | { 3240, 9720 } | { 15T52, 15T64 } | 33 | 5·34 | |||||
13 | {(1,13), (3,0), (15,0)} | [ 13/2, 0 ] | (2z+1, z12+2) | {1} | 4 | 10 | { 3240, 9720 } | { 15T52, 15T64 } | 34 | 5·35 | 10·35 | 10·35 |
(2z+2, z12+2) | {1} | 4 | 10 | { 3240, 9720 } | { 15T52, 15T64 } | 34 | 5·35 | |||||
14 | {(1,14), (3,0), (15,0)} | [ 7, 0 ] | (2z2+1, z12+2) | {3} | 4 | 5 | { 1620, 4860 } | { 15T41, 15T56 } | 35 | 5·35 | 10·35 | 10·35 |
(2z2+2, z12+2) | {1} | 4 | 5 | { 1620, 4860 } | { 15T54, 15T42 } | 34 | 5·35 | |||||
15 | {(1,15), (3,0), (15,0)} | [ 15/2, 0 ] | (2z+1, z12+2) | {1} | 4 | 10 | { 120, 9720 } | { 15T11, 15T64 } | 35 | 5·36 | 5·36 | 5·36 |