Number Theory Tables, Department of Mathematics and Statistics, UNCG

Number of totally ramified extensions of Q3 of degree 12

Introduction

Polynomial Invariants #Aut Splitting Field Number of
j Ramification Polygon Slopes Residual Polynomials fT eT #Gal Gal Polynomials Extensions
1{(1,1), (3,0), (12,0)}[ 1/2, 0 ](z+1, z9+1){1}28{ 144 }{ 12T84 }14·38·38·3
(z+2, z9+1){1}28{ 144 }{ 12T84 }14·3
2{(1,2), (3,0), (12,0)}[ 1, 0 ](z2+1, z9+1){2}24{ 72 }{ 12T36 }14·38·38·3
(z2+2, z9+1){3,6}24{ 72, 216 }{ 12T121, 12T35 }34·3
4{(1,4), (3,0), (12,0)}[ 2, 0 ](z2+1, z9+1){1,2}24{ 24, 216 }{ 12T13, 12T120 }34·328·328·32
(z2+2, z9+1){3,6}24{ 24, 72, 216, 648 }{ 12T42, 12T116, 12T167, 12T15 }324·32
5{(1,5), (3,0), (12,0)}[ 5/2, 0 ](z+1, z9+1){1}28{ 144, 1296 }{ 12T84, 12T212 }34·328·328·32
(z+2, z9+1){1}28{ 144, 1296 }{ 12T84, 12T212 }34·32
7{(1,7), (3,0), (12,0)}[ 7/2, 0 ](z+1, z9+1){1}28{ 144, 1296 }{ 12T84, 12T212 }324·338·338·33
(z+2, z9+1){1}28{ 144, 1296 }{ 12T84, 12T212 }324·33
8{(1,8), (3,0), (12,0)}[ 4, 0 ](z2+1, z9+1){1,2}24{ 24, 72, 216, 648 }{ 12T118, 12T12, 12T38, 12T169 }324·338·338·33
(z2+2, z9+1){3,6}24{ 24, 72, 216, 648 }{ 12T121, 12T42, 12T167, 12T14 }334·33
10{(1,10), (3,0), (12,0)}[ 5, 0 ](z2+1, z9+1){1,2}24{ 72, 216, 648 }{ 12T118, 12T120, 12T36, 12T169 }334·348·348·34
(z2+2, z9+1){3,6}24{ 72, 216, 648 }{ 12T121, 12T167, 12T116, 12T35 }344·34
11{(1,11), (3,0), (12,0)}[ 11/2, 0 ](z+1, z9+1){1}28{ 144, 1296 }{ 12T84, 12T212 }334·348·348·34
(z+2, z9+1){1}28{ 144, 1296 }{ 12T84, 12T212 }334·34
12{(1,12), (3,0), (12,0)}[ 6, 0 ](z2+2, z9+1){3}24354·354·354·35