Number Theory Tables, Department of Mathematics and Statistics, UNCG

Number of totally ramified extensions of Q7 of degree 7

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)}[ 1/6 ](z+1){1}16{ 42 }{ 7T4 }176·76·7
(z+2){1}16{ 42 }{ 7T4 }17
(z+3){1}16{ 42 }{ 7T4 }17
(z+4){1}16{ 42 }{ 7T4 }17
(z+5){1}16{ 42 }{ 7T4 }17
(z+6){1}16{ 42 }{ 7T4 }17
2{(1,2), (7,0)}[ 1/3 ](z2+1){1}23{ 42 }{ 7T4 }176·76·7
(z2+2){1}23{ 42 }{ 7T4 }17
(z2+3){1}13{ 21 }{ 7T3 }17
(z2+4){1}23{ 42 }{ 7T4 }17
(z2+5){1}13{ 21 }{ 7T3 }17
(z2+6){1}13{ 21 }{ 7T3 }17
3{(1,3), (7,0)}[ 1/2 ](z3+1){1}12{ 14 }{ 7T2 }176·76·7
(z3+2){1}32{ 42 }{ 7T4 }17
(z3+3){1}32{ 42 }{ 7T4 }17
(z3+4){1}32{ 42 }{ 7T4 }17
(z3+5){1}32{ 42 }{ 7T4 }17
(z3+6){1}12{ 14 }{ 7T2 }17
4{(1,4), (7,0)}[ 2/3 ](z2+1){1}23{ 42 }{ 7T4 }176·76·7
(z2+2){1}23{ 42 }{ 7T4 }17
(z2+3){1}13{ 21 }{ 7T3 }17
(z2+4){1}23{ 42 }{ 7T4 }17
(z2+5){1}13{ 21 }{ 7T3 }17
(z2+6){1}13{ 21 }{ 7T3 }17
5{(1,5), (7,0)}[ 5/6 ](z+1){1}16{ 42 }{ 7T4 }176·76·7
(z+2){1}16{ 42 }{ 7T4 }17
(z+3){1}16{ 42 }{ 7T4 }17
(z+4){1}16{ 42 }{ 7T4 }17
(z+5){1}16{ 42 }{ 7T4 }17
(z+6){1}16{ 42 }{ 7T4 }17
6{(1,6), (7,0)}[ 1 ](z6+1){1}21{ 14 }{ 7T2 }176·76·7
(z6+2){1}61{ 42 }{ 7T4 }17
(z6+3){1}31{ 21 }{ 7T3 }17
(z6+4){1}61{ 42 }{ 7T4 }17
(z6+5){1}31{ 21 }{ 7T3 }17
(z6+6){7}11{ 7 }{ 7T1 }77
7{(1,7), (7,0)}[ 7/6 ](z+6){1}16{ 42 }{ 7T4 }7727272