Number Theory Tables, Department of Mathematics and Statistics, UNCG

Number of totally ramified extensions of Q5 of degree 15

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), (15,0)}[ 1/4, 0 ](3z+1, z10+3){1}212{ 600 }{ 15T27 }13·512·512·5
(3z+2, z10+3){1}212{ 600 }{ 15T27 }13·5
(3z+3, z10+3){1}212{ 600 }{ 15T27 }13·5
(3z+4, z10+3){1}212{ 600 }{ 15T27 }13·5
2{(1,2), (5,0), (15,0)}[ 1/2, 0 ](3z2+1, z10+3){1}26{ 300 }{ 15T17 }13·512·512·5
(3z2+2, z10+3){1}26{ 300 }{ 15T18 }13·5
(3z2+3, z10+3){1}26{ 300 }{ 15T18 }13·5
(3z2+4, z10+3){1}26{ 300 }{ 15T17 }13·5
3{(1,3), (5,0), (15,0)}[ 3/4, 0 ](3z+1, z10+3){1}212{ 120 }{ 15T11 }13·512·512·5
(3z+2, z10+3){1}212{ 120 }{ 15T11 }13·5
(3z+3, z10+3){1}212{ 120 }{ 15T11 }13·5
(3z+4, z10+3){1}212{ 120 }{ 15T11 }13·5
4{(1,4), (5,0), (15,0)}[ 1, 0 ](3z4+1, z10+3){1}43{ 300 }{ 15T17 }13·512·512·5
(3z4+2, z10+3){5}23{ 150, 750 }{ 15T32, 15T13 }53·5
(3z4+3, z10+3){1}23{ 150 }{ 15T14 }13·5
(3z4+4, z10+3){1}43{ 300 }{ 15T17 }13·5
6{(1,6), (5,0), (15,0)}[ 3/2, 0 ](3z2+1, z10+3){1}26{ 60, 1500 }{ 15T6, 15T37 }53·5212·5212·52
(3z2+2, z10+3){1}26{ 60, 1500 }{ 15T7, 15T40 }53·52
(3z2+3, z10+3){1}26{ 60, 1500 }{ 15T7, 15T40 }53·52
(3z2+4, z10+3){1}26{ 60, 1500 }{ 15T6, 15T37 }53·52
7{(1,7), (5,0), (15,0)}[ 7/4, 0 ](3z+1, z10+3){1}212{ 600, 3000 }{ 15T27, 15T49 }53·5212·5212·52
(3z+2, z10+3){1}212{ 600, 3000 }{ 15T27, 15T49 }53·52
(3z+3, z10+3){1}212{ 600, 3000 }{ 15T27, 15T49 }53·52
(3z+4, z10+3){1}212{ 600, 3000 }{ 15T27, 15T49 }53·52
8{(1,8), (5,0), (15,0)}[ 2, 0 ](3z4+1, z10+3){1}43{ 300 }{ 15T17 }53·5212·5212·52
(3z4+2, z10+3){5}23{ 150, 750 }{ 15T32, 15T13 }523·52
(3z4+3, z10+3){1}23{ 150 }{ 15T14 }53·52
(3z4+4, z10+3){1}43{ 300 }{ 15T17 }53·52
9{(1,9), (5,0), (15,0)}[ 9/4, 0 ](3z+1, z10+3){1}212{ 120, 3000 }{ 15T11, 15T49 }53·5212·5212·52
(3z+2, z10+3){1}212{ 120, 3000 }{ 15T11, 15T49 }53·52
(3z+3, z10+3){1}212{ 120, 3000 }{ 15T11, 15T49 }53·52
(3z+4, z10+3){1}212{ 120, 3000 }{ 15T11, 15T49 }53·52
11{(1,11), (5,0), (15,0)}[ 11/4, 0 ](3z+1, z10+3){1}212{ 600, 3000 }{ 15T27, 15T49 }523·5312·5312·53
(3z+2, z10+3){1}212{ 600, 3000 }{ 15T27, 15T49 }523·53
(3z+3, z10+3){1}212{ 600, 3000 }{ 15T27, 15T49 }523·53
(3z+4, z10+3){1}212{ 600, 3000 }{ 15T27, 15T49 }523·53
12{(1,12), (5,0), (15,0)}[ 3, 0 ](3z4+1, z10+3){1}43{ 60, 1500 }{ 15T6, 15T37 }523·5312·5312·53
(3z4+2, z10+3){5}23{ 30, 750 }{ 15T4, 15T32 }533·53
(3z4+3, z10+3){1}23{ 30, 750 }{ 15T2, 15T31 }523·53
(3z4+4, z10+3){1}43{ 60, 1500 }{ 15T6, 15T37 }523·53
13{(1,13), (5,0), (15,0)}[ 13/4, 0 ](3z+1, z10+3){1}212{ 600, 3000 }{ 15T27, 15T49 }523·5312·5312·53
(3z+2, z10+3){1}212{ 600, 3000 }{ 15T27, 15T49 }523·53
(3z+3, z10+3){1}212{ 600, 3000 }{ 15T27, 15T49 }523·53
(3z+4, z10+3){1}212{ 600, 3000 }{ 15T27, 15T49 }523·53
14{(1,14), (5,0), (15,0)}[ 7/2, 0 ](3z2+1, z10+3){1}26{ 300, 1500 }{ 15T17, 15T37 }523·5312·5312·53
(3z2+2, z10+3){1}26{ 300, 1500 }{ 15T40, 15T18 }523·53
(3z2+3, z10+3){1}26{ 300, 1500 }{ 15T40, 15T18 }523·53
(3z2+4, z10+3){1}26{ 300, 1500 }{ 15T17, 15T37 }523·53
15{(1,15), (5,0), (15,0)}[ 15/4, 0 ](3z+2, z10+3){1}212{ 120, 3000 }{ 15T11, 15T49 }533·543·543·54