Number Theory Tables, Department of Mathematics and Statistics, UNCG

Number of totally ramified extensions of Q3 of degree 9

j Ramification Polygon Slopes Residual Polynomials #A Polynomials Extensions
1{(1,1), (9,0)}[ 1/8 ](z+1)22181818
2{(1,2), (9,0)}[ 1/4 ](z2+1)1191818
(z2+2)119
4{(1,4), (9,0)}[ 1/2 ](z4+1)1191854
(z4+2)119
{(1,4), (3,3), (9,0)}[ 1/2 ](z4+z+1)221836
(z4+z+2)2218
5{(1,5), (9,0)}[ 5/8 ](z+1)22181854
{(1,5), (3,3), (9,0)}[ 1, 1/2 ](z2+1, z3+1)11936
(2z2+1, z3+2)139
(z2+2, z3+1)139
(2z2+2, z3+2)119
7{(1,7), (9,0)}[ 7/8 ](z+1)265454162
{(1,7), (3,3), (9,0)}[ 2, 1/2 ](z2+1, z3+1)1327108
(2z2+1, z3+2)1927
(z2+2, z3+1)1927
(2z2+2, z3+2)1327
8{(1,8), (9,0)}[ 1 ](z8+1)11918162
(z8+2)139
{(1,8), (3,3), (9,0)}[ 5/2, 1/2 ](z+1, z3+1)1327108
(2z+1, z3+2)1327
(z+2, z3+1)1327
(2z+2, z3+2)1327
{(1,8), (3,6), (9,0)}[ 1 ](z8+z2+1)13936
(z8+2z2+1)119
(z8+z2+2)119
(z8+2z2+2)119
10{(1,10), (9,0)}[ 5/4 ](z2+1)132754486
(z2+2)1327
{(1,10), (3,3), (9,0)}[ 7/2, 1/2 ](z+1, z3+1)1981324
(2z+1, z3+2)1981
(z+2, z3+1)1981
(2z+2, z3+2)1981
{(1,10), (3,6), (9,0)}[ 2, 1 ](z2+1, z6+1)1327108
(2z2+1, z6+2)127 [!]27
(z2+2, z6+1)1927
(2z2+2, z6+2)19 [!]27
11{(1,11), (9,0)}[ 11/8 ](z+1)265454486
{(1,11), (3,3), (9,0)}[ 4, 1/2 ](z2+1, z3+1)1981324
(2z2+1, z3+2)12781
(z2+2, z3+1)12781
(2z2+2, z3+2)1981
{(1,11), (3,6), (9,0)}[ 5/2, 1 ](z+1, z6+1)2654108
(2z+1, z6+2)218 [!]54
12{(1,12), (3,3), (9,0)}[ 9/2, 1/2 ](2z+1, z3+2)127243486486
(z+2, z3+1)127243
13{(1,13), (3,6), (9,0)}[ 7/2, 1 ](z+1, z6+1)218162324486
(2z+1, z6+2)254 [!]162
{(1,13), (3,9), (9,0)}[ 2, 3/2 ](2z2+1, z3+2)12781162
(2z2+2, z3+2)1981
14{(1,14), (3,6), (9,0)}[ 4, 1 ](z2+1, z6+1)1981324486
(2z2+1, z6+2)181 [!]81
(z2+2, z6+1)127 [!]81
(2z2+2, z6+2)127 [!]81
{(1,14), (3,9), (9,0)}[ 5/2, 3/2 ](2z+1, z3+2)1981162
(2z+2, z3+2)1981
15{(1,15), (3,6), (9,0)}[ 9/2, 1 ](2z+1, z6+2)181 [!]243486486
(z+2, z6+1)127243
16{(1,16), (3,9), (9,0)}[ 7/2, 3/2 ](2z+1, z3+2)127243486486
(2z+2, z3+2)127243
17{(1,17), (3,9), (9,0)}[ 4, 3/2 ](2z2+1, z3+2)181243486486
(2z2+2, z3+2)127243
18{(1,18), (3,9), (9,0)}[ 9/2, 3/2 ](2z+1, z3+2)181729729729