Number Theory Tables, Department of Mathematics and Statistics, UNCG

Number of totally ramified extensions of Q3 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), (3,0), (15,0)}[ 1/2, 0 ](2z+1, z12+2){1}410{ 3240 }{ 15T52 }15·310·310·3
(2z+2, z12+2){1}410{ 3240 }{ 15T52 }15·3
2{(1,2), (3,0), (15,0)}[ 1, 0 ](2z2+1, z12+2){3}45{ 1620, 4860 }{ 15T41, 15T56 }35·310·310·3
(2z2+2, z12+2){1}45{ 1620 }{ 15T42 }15·3
4{(1,4), (3,0), (15,0)}[ 2, 0 ](2z2+1, z12+2){3}45{ 1620, 4860 }{ 15T41, 15T56 }325·3210·3210·32
(2z2+2, z12+2){1}45{ 1620 }{ 15T42 }35·32
5{(1,5), (3,0), (15,0)}[ 5/2, 0 ](2z+1, z12+2){1}410{ 120, 9720 }{ 15T11, 15T64 }35·3210·3210·32
(2z+2, z12+2){1}410{ 120, 9720 }{ 15T11, 15T64 }35·32
7{(1,7), (3,0), (15,0)}[ 7/2, 0 ](2z+1, z12+2){1}410{ 3240, 9720 }{ 15T52, 15T64 }325·3310·3310·33
(2z+2, z12+2){1}410{ 3240, 9720 }{ 15T52, 15T64 }325·33
8{(1,8), (3,0), (15,0)}[ 4, 0 ](2z2+1, z12+2){3}45{ 1620, 4860 }{ 15T41, 15T56 }335·3310·3310·33
(2z2+2, z12+2){1}45{ 1620 }{ 15T42 }325·33
10{(1,10), (3,0), (15,0)}[ 5, 0 ](2z2+1, z12+2){3}45{ 60, 4860 }{ 15T56, 15T8 }345·3410·3410·34
(2z2+2, z12+2){1}45{ 60, 4860 }{ 15T54, 15T6 }335·34
11{(1,11), (3,0), (15,0)}[ 11/2, 0 ](2z+1, z12+2){1}410{ 3240, 9720 }{ 15T52, 15T64 }335·3410·3410·34
(2z+2, z12+2){1}410{ 3240, 9720 }{ 15T52, 15T64 }335·34
13{(1,13), (3,0), (15,0)}[ 13/2, 0 ](2z+1, z12+2){1}410{ 3240, 9720 }{ 15T52, 15T64 }345·3510·3510·35
(2z+2, z12+2){1}410{ 3240, 9720 }{ 15T52, 15T64 }345·35
14{(1,14), (3,0), (15,0)}[ 7, 0 ](2z2+1, z12+2){3}45{ 1620, 4860 }{ 15T41, 15T56 }355·3510·3510·35
(2z2+2, z12+2){1}45{ 1620, 4860 }{ 15T54, 15T42 }345·35
15{(1,15), (3,0), (15,0)}[ 15/2, 0 ](2z+1, z12+2){1}410{ 120, 9720 }{ 15T11, 15T64 }355·365·365·36