Number Theory Tables, Department of Mathematics and Statistics, UNCG

Number of totally ramified extensions of Q5 of degree 10

Introduction

Polynomial Invariants #Aut Splitting Field Number of
j Ramification Polygon Slopes Residual Polynomials fT eT #Gal Gal Polynomials Extensions
1{(1,1), (5,0), (10,0)}[ 1/4, 0 ](2z+1, z5+2){1}28{ 400 }{ 10T28 }12·58·58·5
(2z+2, z5+2){1}28{ 400 }{ 10T28 }12·5
(2z+3, z5+2){1}28{ 400 }{ 10T28 }12·5
(2z+4, z5+2){1}28{ 400 }{ 10T28 }12·5
2{(1,2), (5,0), (10,0)}[ 1/2, 0 ](2z2+1, z5+2){2}24{ 40 }{ 10T5 }12·58·58·5
(2z2+2, z5+2){2}14{ 20 }{ 10T4 }12·5
(2z2+3, z5+2){2}14{ 20 }{ 10T4 }12·5
(2z2+4, z5+2){2}24{ 40 }{ 10T5 }12·5
3{(1,3), (5,0), (10,0)}[ 3/4, 0 ](2z+1, z5+2){1}28{ 400 }{ 10T28 }12·58·58·5
(2z+2, z5+2){1}28{ 400 }{ 10T28 }12·5
(2z+3, z5+2){1}28{ 400 }{ 10T28 }12·5
(2z+4, z5+2){1}28{ 400 }{ 10T28 }12·5
4{(1,4), (5,0), (10,0)}[ 1, 0 ](2z4+1, z5+2){2}42{ 40 }{ 10T5 }12·58·58·5
(2z4+2, z5+2){2}22{ 20 }{ 10T3 }12·5
(2z4+3, z5+2){5,10}12{ 10, 50 }{ 10T6, 10T2 }52·5
(2z4+4, z5+2){2}42{ 40 }{ 10T5 }12·5
6{(1,6), (5,0), (10,0)}[ 3/2, 0 ](2z2+1, z5+2){1,2}24{ 40, 200 }{ 10T5, 10T17 }52·528·528·52
(2z2+2, z5+2){1,2}14{ 20, 100 }{ 10T4, 10T10 }52·52
(2z2+3, z5+2){1,2}14{ 20, 100 }{ 10T4, 10T10 }52·52
(2z2+4, z5+2){1,2}24{ 40, 200 }{ 10T5, 10T17 }52·52
7{(1,7), (5,0), (10,0)}[ 7/4, 0 ](2z+1, z5+2){1}28{ 400 }{ 10T28 }52·528·528·52
(2z+2, z5+2){1}28{ 400 }{ 10T28 }52·52
(2z+3, z5+2){1}28{ 400 }{ 10T28 }52·52
(2z+4, z5+2){1}28{ 400 }{ 10T28 }52·52
8{(1,8), (5,0), (10,0)}[ 2, 0 ](2z4+1, z5+2){1,2}42{ 40, 200 }{ 10T5, 10T17 }52·528·528·52
(2z4+2, z5+2){1,2}22{ 20, 100 }{ 10T3, 10T9 }52·52
(2z4+3, z5+2){5,10}12{ 10, 50 }{ 10T1, 10T6 }522·52
(2z4+4, z5+2){1,2}42{ 40, 200 }{ 10T5, 10T17 }52·52
9{(1,9), (5,0), (10,0)}[ 9/4, 0 ](2z+1, z5+2){1}28{ 400 }{ 10T28 }52·528·528·52
(2z+2, z5+2){1}28{ 400 }{ 10T28 }52·52
(2z+3, z5+2){1}28{ 400 }{ 10T28 }52·52
(2z+4, z5+2){1}28{ 400 }{ 10T28 }52·52
10{(1,10), (5,0), (10,0)}[ 5/2, 0 ](2z2+3, z5+2){1,2}14{ 20, 100 }{ 10T4, 10T10 }522·532·532·53