Number Theory Tables, Department of Mathematics and Statistics, UNCG

Number of totally ramified extensions of Q2 of degree 18

Introduction

Polynomial Invariants #Aut Splitting Field Number of
j Ramification Polygon Slopes Residual Polynomials fT eT #Gal Gal Polynomials Extensions
1{(1,1), (2,0), (18,0)}[ 1, 0 ](z+1, z16+1){2}69{ 3456 }{ 18T433 }29·29·29·2
3{(1,3), (2,0), (18,0)}[ 3, 0 ](z+1, z16+1){2}69{ 216, 13824 }{ 18T101, 18T98, 18T588 }229·229·229·22
5{(1,5), (2,0), (18,0)}[ 5, 0 ](z+1, z16+1){2}69{ 3456, 13824, ...}{ ..., 18T588, 18T433 }239·239·239·23
7{(1,7), (2,0), (18,0)}[ 7, 0 ](z+1, z16+1){2}69{ 3456, 13824 }{ 18T588, 18T433 }249·249·249·24
9{(1,9), (2,0), (18,0)}[ 9, 0 ](z+1, z16+1){2}69{ 108, 432, 6912, 27648 }{ 18T45, 18T147, 18T656, 18T512 }259·259·259·25
11{(1,11), (2,0), (18,0)}[ 11, 0 ](z+1, z16+1){2}69{ 3456, 13824, ...}{ ..., 18T588, 18T434, 18T433 }269·269·269·26
13{(1,13), (2,0), (18,0)}[ 13, 0 ](z+1, z16+1){2}69{ 3456, 6912, 13824, 27648, ...}{ 18T588, 18T512, 18T656, 18T434, 18T592, 18T433 }279·279·279·27
15{(1,15), (2,0), (18,0)}[ 15, 0 ](z+1, z16+1){2}69{ 13824, 27648, ...}{ 18T656, 18T588 }289·289·289·28
17{(1,17), (2,0), (18,0)}[ 17, 0 ](z+1, z16+1){2}69{ 3456, 6912, 13824, 27648, ...}{ 18T588, 18T656, 18T512, 18T434, 18T592, 18T433 }299·299·299·29
18{(1,18), (2,0), (18,0)}[ 18, 0 ](z+1, z16+1){2}69{ 6912, 27648, ...}{ 18T656, 18T512 }210 [.]9·2109·2109·210