Number Theory Tables, Department of Mathematics and Statistics, UNCG

Number of totally ramified extensions of Q2 of degree 8

j Ramification Polygon Slopes Residual Polynomials #A Polynomials Extensions
1{(1,1), (8,0)}[ 1/7 ](z+1)11888
3{(1,3), (8,0)}[ 3/7 ](z+1)118816
{(1,3), (2,2), (8,0)}[ 1, 1/3 ](z+1, z2+1)1288
5{(1,5), (8,0)}[ 5/7 ](z+1)12161632
{(1,5), (2,2), (8,0)}[ 3, 1/3 ](z+1, z2+1)141616
7{(1,7), (8,0)}[ 1 ](z7+1)128864
{(1,7), (2,2), (8,0)}[ 5, 1/3 ](z+1, z2+1)183232
{(1,7), (2,6), (8,0)}[ 1 ](z7+z+1)1188
{(1,7), (4,4), (8,0)}[ 1 ](z7+z3+1)1188
9{(1,9), (8,0)}[ 9/7 ](z+1)121616128
{(1,9), (2,2), (8,0)}[ 7, 1/3 ](z+1, z2+1)116 [!]6464
{(1,9), (2,6), (8,0)}[ 3, 1 ](z+1, z6+1)18 [!]1616
{(1,9), (4,4), (8,0)}[ 5/3, 1 ](z+1, z4+1)14 [!]1616
10{(1,10), (2,2), (8,0)}[ 8, 1/3 ](z+1, z2+1)132 [!]128128128
11{(1,11), (8,0)}[ 11/7 ](z+1)143232128
{(1,11), (2,6), (8,0)}[ 5, 1 ](z+1, z6+1)116 [!]3232
{(1,11), (4,4), (8,0)}[ 7/3, 1 ](z+1, z4+1)18 [!]3232
13{(1,13), (8,0)}[ 13/7 ](z+1)143232256
{(1,13), (2,6), (8,0)}[ 7, 1 ](z+1, z6+1)132 [!]6464
{(1,13), (2,10), (8,0)}[ 3, 5/3 ](z+1, z2+1)18 [!]3232
{(1,13), (4,4), (8,0)}[ 3, 1 ](z3+1, z4+1)116 [!]3232
{(1,13), (2,10), (4,4), (8,0)}[ 3, 1 ](z3+z+1, z4+1)18 [!]3232
14{(1,14), (2,6), (8,0)}[ 8, 1 ](z+1, z6+1)164 [!]128128256
15{(1,15), (2,10), (8,0)}[ 5, 5/3 ](z+1, z2+1)1166464256
{(1,15), (4,4), (8,0)}[ 11/3, 1 ](z+1, z4+1)116 [!]6464
{(1,15), (2,10), (4,4), (8,0)}[ 5, 3, 1 ](z+1, z2+1, z4+1)164 [!]6464
{(1,15), (4,8), (8,0)}[ 7/3, 2 ](z+1, z4+1)116 [!]6464
17{(1,17), (2,10), (8,0)}[ 7, 5/3 ](z+1, z2+1)132 [!]128128512
{(1,17), (2,10), (4,4), (8,0)}[ 7, 3, 1 ](z+1, z2+1, z4+1)1128 [!]128128
{(1,17), (2,12), (4,4), (8,0)}[ 5, 4, 1 ](z+1, z2+1, z4+1)1128 [!]128128
{(1,17), (4,8), (8,0)}[ 3, 2 ](z3+1, z4+1)132 [!]6464
{(1,17), (2,14), (4,8), (8,0)}[ 3, 2 ](z3+z+1, z4+1)116 [!]6464
18{(1,18), (2,10), (8,0)}[ 8, 5/3 ](z+1, z2+1)164 [!]256256512
{(1,18), (2,10), (4,4), (8,0)}[ 8, 3, 1 ](z+1, z2+1, z4+1)1256 [!]256256
19{(1,19), (2,12), (4,4), (8,0)}[ 7, 4, 1 ](z+1, z2+1, z4+1)1256 [!]256256512
{(1,19), (4,8), (8,0)}[ 11/3, 2 ](z+1, z4+1)132 [!]128128
{(1,19), (2,14), (4,8), (8,0)}[ 5, 3, 2 ](z+1, z2+1, z4+1)1128 [!]128128
20{(1,20), (2,12), (4,4), (8,0)}[ 8, 4, 1 ](z+1, z2+1, z4+1)1512 [!]512512512
21{(1,21), (2,14), (4,8), (8,0)}[ 7, 3, 2 ](z+1, z2+1, z4+1)1256 [!]256256512
{(1,21), (2,16), (4,8), (8,0)}[ 5, 4, 2 ](z+1, z2+1, z4+1)1256 [!]256256
22{(1,22), (2,14), (4,8), (8,0)}[ 8, 3, 2 ](z+1, z2+1, z4+1)1512 [!]512512512
23{(1,23), (2,16), (4,8), (8,0)}[ 7, 4, 2 ](z+1, z2+1, z4+1)1512 [!]512512512
24{(1,24), (2,16), (4,8), (8,0)}[ 8, 4, 2 ](z+1, z2+1, z4+1)11024 [!]102410241024