Number Theory Tables, Department of Mathematics and Statistics, UNCG

Number of totally ramified extensions of Q3 of degree 21

Introduction

Polynomial Invariants #Aut Splitting Field Number of
j Ramification Polygon Slopes Residual Polynomials eT fT #Gal Gal Polynomials Extensions
1{(1,1), (3,0), (21,0)}[ 1/2, 0 ](z+1, z18+1){1}614{ 28·37 }{ 21T92 }17·314·314·3
(z+2, z18+1){1}614{ 28·37 }{ 21T92 }17·3
2{(1,2), (3,0), (21,0)}[ 1, 0 ](z2+1, z18+1){1}67{ 14·37 }{ 21T78 }17·314·314·3
(z2+2, z18+1){3}6737·3
4{(1,4), (3,0), (21,0)}[ 2, 0 ](z2+1, z18+1){1}6737·3214·3214·32
(z2+2, z18+1){3}67327·32
5{(1,5), (3,0), (21,0)}[ 5/2, 0 ](z+1, z18+1){1}61437·3214·3214·32
(z+2, z18+1){1}61437·32
7{(1,7), (3,0), (21,0)}[ 7/2, 0 ](z+1, z18+1){1}614327·3314·3314·33
(z+2, z18+1){1}614327·33
8{(1,8), (3,0), (21,0)}[ 4, 0 ](z2+1, z18+1){1}67327·3314·3314·33
(z2+2, z18+1){3}67337·33
10{(1,10), (3,0), (21,0)}[ 5, 0 ](z2+1, z18+1){1}67337·3414·3414·34
(z2+2, z18+1){3}67347·34
11{(1,11), (3,0), (21,0)}[ 11/2, 0 ](z+1, z18+1){1}614337·3414·3414·34
(z+2, z18+1){1}614337·34
13{(1,13), (3,0), (21,0)}[ 13/2, 0 ](z+1, z18+1){1}614347·3514·3514·35
(z+2, z18+1){1}614347·35
14{(1,14), (3,0), (21,0)}[ 7, 0 ](z2+1, z18+1){1}67347·3514·3514·35
(z2+2, z18+1){3}67357·35
16{(1,16), (3,0), (21,0)}[ 8, 0 ](z2+1, z18+1){1}67357·3614·3614·36
(z2+2, z18+1){3}67367·36
17{(1,17), (3,0), (21,0)}[ 17/2, 0 ](z+1, z18+1){1}614357·3614·3614·36
(z+2, z18+1){1}614357·36
19{(1,19), (3,0), (21,0)}[ 19/2, 0 ](z+1, z18+1){1}614367·3714·3714·37
(z+2, z18+1){1}614367·37
20{(1,20), (3,0), (21,0)}[ 10, 0 ](z2+1, z18+1){1}67367·3714·3714·37
(z2+2, z18+1){3}67377·37
21{(1,21), (3,0), (21,0)}[ 21/2, 0 ](z+2, z18+1){1}614377·387·387·38