UNCG, Department of Mathematics and Statistics, Sebastian Pauli

Number of totally ramified extensions of Q3 of degree 18

Introduction
Extensions of Q2 of degree 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 32
Extensions of Q2[x]/(x2+x+1) of degree 2, 4, 8 Extensions of Q2[x]/(x4+x+1) of degree 2, 4, 8
Extensions of Q3 of degree 3, 6, 9, 12, 15, 18, 21, 24, 27     Extensions of Q3[x]/(x2+2x+2) of degree 3, 6, 9
Extensions of Q5 of degree 5, 10, 15, 20, 25, 30, 35, 50, Extensions of Q7 of degree 7, 14, 21, 49
Extensions of Q11 of degree 11, 22, 33, Extensions of Q13 of degree 13, 26, 39,
Extensions of Q17 of degree 17, 34 Extensions of Q19 of degree 19,

Polynomial Invariants #Aut Splitting Field Number of
j Ramification Polygon Slopes Residual Polynomials eT fT #Gal Gal Polynomials Extensions
1{(1,1), (9,0), (18,0)}[ 1/8, 0 ](2z+1, z9+2)141664·3418T47612·324·324·32
(2z+2, z9+2)141664·3418T47612·32
2{(1,2), (9,0), (18,0)}[ 1/4, 0 ](2z2+1, z9+2)12816·32[144,182] = 18T7312·324·324·32
(2z2+2, z9+2)12816·32[144,182] = 18T7312·32
4{(1,4), (3,3), (9,0), (18,0)}[ 1/2, 0 ](2z4+z+1, z9+2)14416·34[1296,3508] = 18T29712·328·324·33
(2z4+2z+1, z9+2)14416·34[1296,3508] = 18T29712·32
(2z4+z+2, z9+2)14416·34[1296,3508] = 18T29712·32
(2z4+2z+2, z9+2)14416·34[1296,3508] = 18T29712·32
{(1,4), (9,0), (18,0)}[ 1/2, 0 ](2z4+1, z9+2)1248·32[72,40] = 18T3412·324·32
(2z4+2, z9+2)1248·32[72,40] = 18T3412·32
5{(1,5), (3,3), (9,0), (18,0)}[ 1, 1/2, 0 ](z2+1, 2z3+1, z9+2)12412·328·324·33
(2z2+1, 2z3+2, z9+2)32432·32
(z2+2, 2z3+1, z9+2)32432·32
(2z2+2, 2z3+2, z9+2)12412·32
{(1,5), (9,0), (18,0)}[ 5/8, 0 ](2z+1, z9+2)141664·3418T47612·324·32
(2z+2, z9+2)141664·3418T47612·32
7{(1,7), (3,3), (9,0), (18,0)}[ 2, 1/2, 0 ](z2+1, 2z3+1, z9+2)12432·338·334·34
(2z2+1, 2z3+2, z9+2)324322·33
(z2+2, 2z3+1, z9+2)324322·33
(2z2+2, 2z3+2, z9+2)12432·33
{(1,7), (3,6), (9,0), (18,0)}[ 1/2, 1, 0 ](z+1, 2z6+1, z9+2)141612·328·32
(2z+1, 2z6+2, z9+2)141612·32
(z+2, 2z6+1, z9+2)141612·32
(2z+2, 2z6+2, z9+2)141612·32
{(1,7), (9,0), (18,0)}[ 7/8, 0 ](2z+1, z9+2)141632·334·33
(2z+2, z9+2)141632·33
8{(1,8), (3,3), (9,0), (18,0)}[ 5/2, 1/2, 0 ](z+1, 2z3+1, z9+2)12432·338·334·34
(2z+1, 2z3+2, z9+2)12432·33
(z+2, 2z3+1, z9+2)12432·33
(2z+2, 2z3+2, z9+2)12432·33
{(1,8), (3,6), (9,0), (18,0)}[ 1, 0 ](2z8+z2+1, z9+2)18216·32[144,185] = 18T5912·328·32
(2z8+2z2+1, z9+2)18216·32[144,185] = 18T5912·32
(2z8+z2+2, z9+2)18216·32[144,185] = 18T5912·32
(2z8+2z2+2, z9+2)38232·32
{(1,8), (9,0), (18,0)}[ 1, 0 ](2z8+1, z9+2)32232·324·32
(2z8+2, z9+2)1224·32[36,10] = 18T1112·32
10{(1,10), (3,3), (9,0), (18,0)}[ 7/2, 1/2, 0 ](z+1, 2z3+1, z9+2)124322·348·344·35
(2z+1, 2z3+2, z9+2)124322·34
(z+2, 2z3+1, z9+2)124322·34
(2z+2, 2z3+2, z9+2)124322·34
{(1,10), (3,6), (9,0), (18,0)}[ 2, 1, 0 ](z2+1, 2z6+1, z9+2)322322·338·33
(2z2+1, 2z6+2, z9+2)322322·33
(z2+2, 2z6+1, z9+2){3,6,9,18}2233 [!]2·33
(2z2+2, 2z6+2, z9+2)12232·33
{(1,10), (9,0), (18,0)}[ 5/4, 0 ](2z2+1, z9+2)12832·334·33
(2z2+2, z9+2)12832·33
11{(1,11), (3,3), (9,0), (18,0)}[ 4, 1/2, 0 ](z2+1, 2z3+1, z9+2)124322·348·344·35
(2z2+1, 2z3+2, z9+2)324332·34
(z2+2, 2z3+1, z9+2)324332·34
(2z2+2, 2z3+2, z9+2)124322·34
{(1,11), (3,6), (9,0), (18,0)}[ 5/2, 1, 0 ](z+1, 2z6+1, z9+2)324322·338·33
(2z+1, 2z6+2, z9+2)12432·33
(z+2, 2z6+1, z9+2)324322·33
(2z+2, 2z6+2, z9+2)12432·33
{(1,11), (9,0), (18,0)}[ 11/8, 0 ](2z+1, z9+2)141632·334·33
(2z+2, z9+2)141632·33
13{(1,13), (3,3), (9,0), (18,0)}[ 5, 1/2, 0 ](z2+1, 2z3+1, z9+2)124332·358·354·36
(2z2+1, 2z3+2, z9+2)324342·35
(z2+2, 2z3+1, z9+2)324342·35
(2z2+2, 2z3+2, z9+2)124332·35
{(1,13), (3,6), (9,0), (18,0)}[ 7/2, 1, 0 ](z+1, 2z6+1, z9+2)324332·348·34
(2z+1, 2z6+2, z9+2)124322·34
(z+2, 2z6+1, z9+2)324332·34
(2z+2, 2z6+2, z9+2)124322·34
{(1,13), (9,0), (18,0)}[ 13/8, 0 ](2z+1, z9+2)1416322·344·34
(2z+2, z9+2)1416322·34
14{(1,14), (3,3), (9,0), (18,0)}[ 11/2, 1/2, 0 ](z+1, 2z3+1, z9+2)124332·358·354·36
(2z+1, 2z3+2, z9+2)124332·35
(z+2, 2z3+1, z9+2)124332·35
(2z+2, 2z3+2, z9+2)124332·35
{(1,14), (3,6), (9,0), (18,0)}[ 4, 1, 0 ](z2+1, 2z6+1, z9+2)322332·348·34
(2z2+1, 2z6+2, z9+2)322332·34
(z2+2, 2z6+1, z9+2){3,6,9,18}2234 [!]2·34
(2z2+2, 2z6+2, z9+2)122322·34
{(1,14), (3,12), (9,0), (18,0)}[ 1, 2, 0 ](z2+1, 2z6+1, z9+2)12832·338·33
(2z2+1, 2z6+2, z9+2)328322·33
(z2+2, 2z6+1, z9+2)328322·33
(2z2+2, 2z6+2, z9+2)12832·33
{(1,14), (9,0), (18,0)}[ 7/4, 0 ](2z2+1, z9+2)128322·344·34
(2z2+2, z9+2)128322·34
16{(1,16), (3,3), (9,0), (18,0)}[ 13/2, 1/2, 0 ](z+1, 2z3+1, z9+2)124342·368·364·37
(2z+1, 2z3+2, z9+2)124342·36
(z+2, 2z3+1, z9+2)124342·36
(2z+2, 2z3+2, z9+2)124342·36
{(1,16), (3,6), (9,0), (18,0)}[ 5, 1, 0 ](z2+1, 2z6+1, z9+2)322342·358·35
(2z2+1, 2z6+2, z9+2)322342·35
(z2+2, 2z6+1, z9+2){3,6,9,18}2235 [!]2·35
(2z2+2, 2z6+2, z9+2)122332·35
{(1,16), (3,12), (9,0), (18,0)}[ 2, 0 ](2z8+z2+1, z9+2)182322·348·34
(2z8+2z2+1, z9+2)182322·34
(2z8+z2+2, z9+2)182322·34
(2z8+2z2+2, z9+2)382332·34
{(1,16), (9,0), (18,0)}[ 2, 0 ](2z8+1, z9+2)322332·344·34
(2z8+2, z9+2)122322·34
17{(1,17), (3,3), (9,0), (18,0)}[ 7, 1/2, 0 ](z2+1, 2z3+1, z9+2)124342·368·364·37
(2z2+1, 2z3+2, z9+2)324352·36
(z2+2, 2z3+1, z9+2)324352·36
(2z2+2, 2z3+2, z9+2)124342·36
{(1,17), (3,6), (9,0), (18,0)}[ 11/2, 1, 0 ](z+1, 2z6+1, z9+2)324342·358·35
(2z+1, 2z6+2, z9+2)124332·35
(z+2, 2z6+1, z9+2)324342·35
(2z+2, 2z6+2, z9+2)124332·35
{(1,17), (3,12), (9,0), (18,0)}[ 5/2, 2, 0 ](z+1, 2z6+1, z9+2){1,2,3,6,9}2433 [!]2·348·34
(2z+1, 2z6+2, z9+2)124322·34
(z+2, 2z6+1, z9+2){1,2,3,6,9}2433 [!]2·34
(2z+2, 2z6+2, z9+2)124322·34
{(1,17), (3,15), (9,0), (18,0)}[ 1, 5/2, 0 ](z2+1, 2z3+1, z9+2)141632·338·33
(2z2+1, 2z3+2, z9+2)3416322·33
(z2+2, 2z3+1, z9+2)3416322·33
(2z2+2, 2z3+2, z9+2)141632·33
{(1,17), (9,0), (18,0)}[ 17/8, 0 ](2z+1, z9+2)1416322·344·34
(2z+2, z9+2)1416322·34
19{(1,19), (3,3), (9,0), (18,0)}[ 8, 1/2, 0 ](z2+1, 2z3+1, z9+2)124352·378·374·38
(2z2+1, 2z3+2, z9+2)324362·37
(z2+2, 2z3+1, z9+2)324362·37
(2z2+2, 2z3+2, z9+2)124352·37
{(1,19), (3,6), (9,0), (18,0)}[ 13/2, 1, 0 ](z+1, 2z6+1, z9+2)324352·368·36
(2z+1, 2z6+2, z9+2)124342·36
(z+2, 2z6+1, z9+2)324352·36
(2z+2, 2z6+2, z9+2)124342·36
{(1,19), (3,12), (9,0), (18,0)}[ 7/2, 2, 0 ](z+1, 2z6+1, z9+2){1,2,3,6,9}2434 [!]2·358·35
(2z+1, 2z6+2, z9+2)124332·35
(z+2, 2z6+1, z9+2){1,2,3,6,9}2434 [!]2·35
(2z+2, 2z6+2, z9+2)124332·35
{(1,19), (3,15), (9,0), (18,0)}[ 2, 5/2, 0 ](z2+1, 2z3+1, z9+2)1416322·348·34
(2z2+1, 2z3+2, z9+2)3416332·34
(z2+2, 2z3+1, z9+2)3416332·34
(2z2+2, 2z3+2, z9+2)1416322·34
20{(1,20), (3,3), (9,0), (18,0)}[ 17/2, 1/2, 0 ](z+1, 2z3+1, z9+2)124352·378·374·38
(2z+1, 2z3+2, z9+2)124352·37
(z+2, 2z3+1, z9+2)124352·37
(2z+2, 2z3+2, z9+2)124352·37
{(1,20), (3,6), (9,0), (18,0)}[ 7, 1, 0 ](z2+1, 2z6+1, z9+2)322352·368·36
(2z2+1, 2z6+2, z9+2)322352·36
(z2+2, 2z6+1, z9+2){3,6,9,18}2236 [!]2·36
(2z2+2, 2z6+2, z9+2)122342·36
{(1,20), (3,12), (9,0), (18,0)}[ 4, 2, 0 ](z2+1, 2z6+1, z9+2){1,2,3,6,9}2234 [!]2·358·35
(2z2+1, 2z6+2, z9+2)322342·35
(z2+2, 2z6+1, z9+2){3,6,9,18}2235 [!]2·35
(2z2+2, 2z6+2, z9+2)122332·35
{(1,20), (3,15), (9,0), (18,0)}[ 5/2, 0 ](2z4+z+1, z9+2)144322·348·34
(2z4+2z+1, z9+2)144322·34
(2z4+z+2, z9+2)144322·34
(2z4+2z+2, z9+2)144322·34
21{(1,21), (3,3), (9,0), (18,0)}[ 9, 1/2, 0 ](2z2+1, 2z3+2, z9+2)324372·384·384·38
(z2+2, 2z3+1, z9+2)324372·38
22{(1,22), (3,6), (9,0), (18,0)}[ 8, 1, 0 ](z2+1, 2z6+1, z9+2){1,2,3,6,9}2236 [!]2·378·374·38
(2z2+1, 2z6+2, z9+2)322362·37
(z2+2, 2z6+1, z9+2){3,6,9,18}2237 [!]2·37
(2z2+2, 2z6+2, z9+2)122352·37
{(1,22), (3,12), (9,0), (18,0)}[ 5, 2, 0 ](z2+1, 2z6+1, z9+2){1,2,3,6,9}2235 [!]2·368·36
(2z2+1, 2z6+2, z9+2)322352·36
(z2+2, 2z6+1, z9+2){3,6,9,18}2236 [!]2·36
(2z2+2, 2z6+2, z9+2)122342·36
{(1,22), (3,15), (9,0), (18,0)}[ 7/2, 5/2, 0 ](z+1, 2z3+1, z9+2)124332·358·35
(2z+1, 2z3+2, z9+2)124332·35
(z+2, 2z3+1, z9+2)124332·35
(2z+2, 2z3+2, z9+2)124332·35
{(1,22), (3,18), (9,0), (18,0)}[ 2, 3, 0 ](z2+1, 2z6+1, z9+2)128332·354·35
(z2+2, 2z6+1, z9+2)328342·35
23{(1,23), (3,6), (9,0), (18,0)}[ 17/2, 1, 0 ](z+1, 2z6+1, z9+2){1,2,3,6,9}2436 [!]2·378·374·38
(2z+1, 2z6+2, z9+2)124352·37
(z+2, 2z6+1, z9+2){1,2,3,6,9}2436 [!]2·37
(2z+2, 2z6+2, z9+2)124352·37
{(1,23), (3,12), (9,0), (18,0)}[ 11/2, 2, 0 ](z+1, 2z6+1, z9+2){1,2,3,6,9}2435 [!]2·368·36
(2z+1, 2z6+2, z9+2)124342·36
(z+2, 2z6+1, z9+2){1,2,3,6,9}2435 [!]2·36
(2z+2, 2z6+2, z9+2)124342·36
{(1,23), (3,15), (9,0), (18,0)}[ 4, 5/2, 0 ](z2+1, 2z3+1, z9+2)124332·358·35
(2z2+1, 2z3+2, z9+2)324342·35
(z2+2, 2z3+1, z9+2)324342·35
(2z2+2, 2z3+2, z9+2)124332·35
{(1,23), (3,18), (9,0), (18,0)}[ 5/2, 3, 0 ](z+1, 2z6+1, z9+2)1416332·354·35
(z+2, 2z6+1, z9+2)1416332·35
24{(1,24), (3,6), (9,0), (18,0)}[ 9, 1, 0 ](2z2+1, 2z6+2, z9+2)322372·384·384·38
(z2+2, 2z6+1, z9+2){3,6,9,18}2238 [!]2·38
25{(1,25), (3,12), (9,0), (18,0)}[ 13/2, 2, 0 ](z+1, 2z6+1, z9+2){1,2,3,6,9}2436 [!]2·378·374·38
(2z+1, 2z6+2, z9+2)124352·37
(z+2, 2z6+1, z9+2){1,2,3,6,9}2436 [!]2·37
(2z+2, 2z6+2, z9+2)124352·37
{(1,25), (3,15), (9,0), (18,0)}[ 5, 5/2, 0 ](z2+1, 2z3+1, z9+2)124342·368·36
(2z2+1, 2z3+2, z9+2)324352·36
(z2+2, 2z3+1, z9+2)324352·36
(2z2+2, 2z3+2, z9+2)124342·36
{(1,25), (3,18), (9,0), (18,0)}[ 7/2, 3, 0 ](z+1, 2z6+1, z9+2){1,2,3,6,9}2435 [!]2·364·36
(z+2, 2z6+1, z9+2){1,2,3,6,9}2435 [!]2·36
26{(1,26), (3,12), (9,0), (18,0)}[ 7, 2, 0 ](z2+1, 2z6+1, z9+2){1,2,3,6,9}2236 [!]2·378·374·38
(2z2+1, 2z6+2, z9+2)322362·37
(z2+2, 2z6+1, z9+2){3,6,9,18}2237 [!]2·37
(2z2+2, 2z6+2, z9+2)122352·37
{(1,26), (3,15), (9,0), (18,0)}[ 11/2, 5/2, 0 ](z+1, 2z3+1, z9+2)124342·368·36
(2z+1, 2z3+2, z9+2)124342·36
(z+2, 2z3+1, z9+2)124342·36
(2z+2, 2z3+2, z9+2)124342·36
{(1,26), (3,18), (9,0), (18,0)}[ 4, 3, 0 ](z2+1, 2z6+1, z9+2){1,2,3,6,9}2235 [!]2·364·36
(z2+2, 2z6+1, z9+2){3,6,9,18}2236 [!]2·36
28{(1,28), (3,12), (9,0), (18,0)}[ 8, 2, 0 ](z2+1, 2z6+1, z9+2){1,2,3,6,9}2237 [!]2·388·384·39
(2z2+1, 2z6+2, z9+2)322372·38
(z2+2, 2z6+1, z9+2){3,6,9,18}2238 [!]2·38
(2z2+2, 2z6+2, z9+2)122362·38
{(1,28), (3,15), (9,0), (18,0)}[ 13/2, 5/2, 0 ](z+1, 2z3+1, z9+2)124352·378·37
(2z+1, 2z3+2, z9+2)124352·37
(z+2, 2z3+1, z9+2)124352·37
(2z+2, 2z3+2, z9+2)124352·37
{(1,28), (3,18), (9,0), (18,0)}[ 5, 3, 0 ](z2+1, 2z6+1, z9+2){1,2,3,6,9}2236 [!]2·374·37
(z2+2, 2z6+1, z9+2){3,6,9,18}2237 [!]2·37
29{(1,29), (3,12), (9,0), (18,0)}[ 17/2, 2, 0 ](z+1, 2z6+1, z9+2){1,2,3,6,9}2437 [!]2·388·384·39
(2z+1, 2z6+2, z9+2)124362·38
(z+2, 2z6+1, z9+2){1,2,3,6,9}2437 [!]2·38
(2z+2, 2z6+2, z9+2)124362·38
{(1,29), (3,15), (9,0), (18,0)}[ 7, 5/2, 0 ](z2+1, 2z3+1, z9+2)124352·378·37
(2z2+1, 2z3+2, z9+2)324362·37
(z2+2, 2z3+1, z9+2)324362·37
(2z2+2, 2z3+2, z9+2)124352·37
{(1,29), (3,18), (9,0), (18,0)}[ 11/2, 3, 0 ](z+1, 2z6+1, z9+2){1,2,3,6,9}2436 [!]2·374·37
(z+2, 2z6+1, z9+2){1,2,3,6,9}2436 [!]2·37
30{(1,30), (3,12), (9,0), (18,0)}[ 9, 2, 0 ](2z2+1, 2z6+2, z9+2)322382·394·394·39
(z2+2, 2z6+1, z9+2){3,6,9,18}2239 [!]2·39
31{(1,31), (3,15), (9,0), (18,0)}[ 8, 5/2, 0 ](z2+1, 2z3+1, z9+2)124362·388·384·39
(2z2+1, 2z3+2, z9+2)324372·38
(z2+2, 2z3+1, z9+2)324372·38
(2z2+2, 2z3+2, z9+2)124362·38
{(1,31), (3,18), (9,0), (18,0)}[ 13/2, 3, 0 ](z+1, 2z6+1, z9+2){1,2,3,6,9}2437 [!]2·384·38
(z+2, 2z6+1, z9+2){1,2,3,6,9}2437 [!]2·38
32{(1,32), (3,15), (9,0), (18,0)}[ 17/2, 5/2, 0 ](z+1, 2z3+1, z9+2)124362·388·384·39
(2z+1, 2z3+2, z9+2)124362·38
(z+2, 2z3+1, z9+2)124362·38
(2z+2, 2z3+2, z9+2)124362·38
{(1,32), (3,18), (9,0), (18,0)}[ 7, 3, 0 ](z2+1, 2z6+1, z9+2){1,2,3,6,9}2237 [!]2·384·38
(z2+2, 2z6+1, z9+2){3,6,9,18}2238 [!]2·38
33{(1,33), (3,15), (9,0), (18,0)}[ 9, 5/2, 0 ](2z2+1, 2z3+2, z9+2)324382·394·394·39
(z2+2, 2z3+1, z9+2)324382·39
34{(1,34), (3,18), (9,0), (18,0)}[ 8, 3, 0 ](z2+1, 2z6+1, z9+2){1,2,3,6,9}2238 [!]2·394·394·39
(z2+2, 2z6+1, z9+2){3,6,9,18}2239 [!]2·39
35{(1,35), (3,18), (9,0), (18,0)}[ 17/2, 3, 0 ](z+1, 2z6+1, z9+2)324382·394·394·39
(z+2, 2z6+1, z9+2)324382·39
36{(1,36), (3,18), (9,0), (18,0)}[ 9, 3, 0 ](z2+2, 2z6+1, z9+2){3,6,9,18}12310 [!]2·3102·3102·310