Number Theory Tables, Department of Mathematics and Statistics, UNCG

Number of totally ramified extensions of Q7 of degree 21

Introduction

Polynomial Invariants #Aut Splitting Field Number of
j Ramification Polygon Slopes Residual Polynomials fT eT #Gal Gal Polynomials Extensions
1{(1,1), (7,0), (21,0)}[ 1/6, 0 ](3z+1, z14+3){1}318{ 18522 }{ 21T66 }13·718·718·7
(3z+2, z14+3){1}318{ 18522 }{ 21T66 }13·7
(3z+3, z14+3){1}318{ 18522 }{ 21T66 }13·7
(3z+4, z14+3){1}318{ 18522 }{ 21T66 }13·7
(3z+5, z14+3){1}318{ 18522 }{ 21T66 }13·7
(3z+6, z14+3){1}318{ 18522 }{ 21T66 }13·7
2{(1,2), (7,0), (21,0)}[ 1/3, 0 ](3z2+1, z14+3){1}39{ 9261 }{ 21T49 }13·718·718·7
(3z2+2, z14+3){1}39{ 9261 }{ 21T49 }13·7
(3z2+3, z14+3){1}69{ 18522 }{ 21T66 }13·7
(3z2+4, z14+3){1}39{ 9261 }{ 21T49 }13·7
(3z2+5, z14+3){1}69{ 18522 }{ 21T66 }13·7
(3z2+6, z14+3){1}69{ 18522 }{ 21T66 }13·7
3{(1,3), (7,0), (21,0)}[ 1/2, 0 ](3z3+1, z14+3){3}36{ 126 }{ 21T9 }13·718·718·7
(3z3+2, z14+3){3}36{ 126 }{ 21T9 }13·7
(3z3+3, z14+3){3}16{ 42 }{ 21T4 }13·7
(3z3+4, z14+3){3}16{ 42 }{ 21T4 }13·7
(3z3+5, z14+3){3}36{ 126 }{ 21T9 }13·7
(3z3+6, z14+3){3}36{ 126 }{ 21T9 }13·7
4{(1,4), (7,0), (21,0)}[ 2/3, 0 ](3z2+1, z14+3){1}39{ 9261 }{ 21T49 }13·718·718·7
(3z2+2, z14+3){1}39{ 9261 }{ 21T49 }13·7
(3z2+3, z14+3){1}69{ 18522 }{ 21T66 }13·7
(3z2+4, z14+3){1}39{ 9261 }{ 21T49 }13·7
(3z2+5, z14+3){1}69{ 18522 }{ 21T66 }13·7
(3z2+6, z14+3){1}69{ 18522 }{ 21T66 }13·7
5{(1,5), (7,0), (21,0)}[ 5/6, 0 ](3z+1, z14+3){1}318{ 18522 }{ 21T66 }13·718·718·7
(3z+2, z14+3){1}318{ 18522 }{ 21T66 }13·7
(3z+3, z14+3){1}318{ 18522 }{ 21T66 }13·7
(3z+4, z14+3){1}318{ 18522 }{ 21T66 }13·7
(3z+5, z14+3){1}318{ 18522 }{ 21T66 }13·7
(3z+6, z14+3){1}318{ 18522 }{ 21T66 }13·7
6{(1,6), (7,0), (21,0)}[ 1, 0 ](3z6+1, z14+3){3}33{ 63 }{ 21T7 }13·718·718·7
(3z6+2, z14+3){3}33{ 63 }{ 21T7 }13·7
(3z6+3, z14+3){3}23{ 42 }{ 21T4 }13·7
(3z6+4, z14+3){7}1373·7
(3z6+5, z14+3){3}63{ 126 }{ 21T9 }13·7
(3z6+6, z14+3){3}63{ 126 }{ 21T9 }13·7
8{(1,8), (7,0), (21,0)}[ 4/3, 0 ](3z2+1, z14+3){1}3973·7218·7218·72
(3z2+2, z14+3){1}3973·72
(3z2+3, z14+3){1}6973·72
(3z2+4, z14+3){1}3973·72
(3z2+5, z14+3){1}6973·72
(3z2+6, z14+3){1}6973·72
9{(1,9), (7,0), (21,0)}[ 3/2, 0 ](3z3+1, z14+3){1}3673·7218·7218·72
(3z3+2, z14+3){1}3673·72
(3z3+3, z14+3){1}1673·72
(3z3+4, z14+3){1}1673·72
(3z3+5, z14+3){1}3673·72
(3z3+6, z14+3){1}3673·72
10{(1,10), (7,0), (21,0)}[ 5/3, 0 ](3z2+1, z14+3){1}3973·7218·7218·72
(3z2+2, z14+3){1}3973·72
(3z2+3, z14+3){1}6973·72
(3z2+4, z14+3){1}3973·72
(3z2+5, z14+3){1}6973·72
(3z2+6, z14+3){1}6973·72
11{(1,11), (7,0), (21,0)}[ 11/6, 0 ](3z+1, z14+3){1}31873·7218·7218·72
(3z+2, z14+3){1}31873·72
(3z+3, z14+3){1}31873·72
(3z+4, z14+3){1}31873·72
(3z+5, z14+3){1}31873·72
(3z+6, z14+3){1}31873·72
12{(1,12), (7,0), (21,0)}[ 2, 0 ](3z6+1, z14+3){1}3373·7218·7218·72
(3z6+2, z14+3){1}3373·72
(3z6+3, z14+3){1}2373·72
(3z6+4, z14+3){7}13723·72
(3z6+5, z14+3){1}6373·72
(3z6+6, z14+3){1}6373·72
13{(1,13), (7,0), (21,0)}[ 13/6, 0 ](3z+1, z14+3){1}31873·7218·7218·72
(3z+2, z14+3){1}31873·72
(3z+3, z14+3){1}31873·72
(3z+4, z14+3){1}31873·72
(3z+5, z14+3){1}31873·72
(3z+6, z14+3){1}31873·72
15{(1,15), (7,0), (21,0)}[ 5/2, 0 ](3z3+1, z14+3){1}36723·7318·7318·73
(3z3+2, z14+3){1}36723·73
(3z3+3, z14+3){1}16723·73
(3z3+4, z14+3){1}16723·73
(3z3+5, z14+3){1}36723·73
(3z3+6, z14+3){1}36723·73
16{(1,16), (7,0), (21,0)}[ 8/3, 0 ](3z2+1, z14+3){1}39723·7318·7318·73
(3z2+2, z14+3){1}39723·73
(3z2+3, z14+3){1}69723·73
(3z2+4, z14+3){1}39723·73
(3z2+5, z14+3){1}69723·73
(3z2+6, z14+3){1}69723·73
17{(1,17), (7,0), (21,0)}[ 17/6, 0 ](3z+1, z14+3){1}318723·7318·7318·73
(3z+2, z14+3){1}318723·73
(3z+3, z14+3){1}318723·73
(3z+4, z14+3){1}318723·73
(3z+5, z14+3){1}318723·73
(3z+6, z14+3){1}318723·73
18{(1,18), (7,0), (21,0)}[ 3, 0 ](3z6+1, z14+3){1}33723·7318·7318·73
(3z6+2, z14+3){1}33723·73
(3z6+3, z14+3){1}23723·73
(3z6+4, z14+3){7}13733·73
(3z6+5, z14+3){1}63723·73
(3z6+6, z14+3){1}63723·73
19{(1,19), (7,0), (21,0)}[ 19/6, 0 ](3z+1, z14+3){1}318723·7318·7318·73
(3z+2, z14+3){1}318723·73
(3z+3, z14+3){1}318723·73
(3z+4, z14+3){1}318723·73
(3z+5, z14+3){1}318723·73
(3z+6, z14+3){1}318723·73
20{(1,20), (7,0), (21,0)}[ 10/3, 0 ](3z2+1, z14+3){1}39723·7318·7318·73
(3z2+2, z14+3){1}39723·73
(3z2+3, z14+3){1}69723·73
(3z2+4, z14+3){1}39723·73
(3z2+5, z14+3){1}69723·73
(3z2+6, z14+3){1}69723·73
21{(1,21), (7,0), (21,0)}[ 7/2, 0 ](3z3+4, z14+3){1}16733·743·743·74