Number Theory Tables, Department of Mathematics and Statistics, UNCG

Number of totally ramified extensions of Q(δ)≅Q3[x]/(x2+2 + O(3^30)x+2 + O(3^30)) of degree 6

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), (6,0)}[ 1/2, 0 ](2z+1, z3+2){1}1412·316·316·3
(2z+δ, z3+2){1}1412·3
(2z+(δ+1), z3+2){1}1412·3
(2z+(2δ+1), z3+2){1}1412·3
(2z+2, z3+2){1}1412·3
(2z+2δ, z3+2){1}1412·3
(2z+(2δ+2), z3+2){1}1412·3
(2z+(δ+2), z3+2){1}1412·3
2{(1,2), (3,0), (6,0)}[ 1, 0 ](2z2+1, z3+2){3,6}12{ 6, ...}{ ..., 6T2 }32·316·316·3
(2z2+δ, z3+2){2}2212·3
(2z2+(δ+1), z3+2){3,6}12{ 6, ...}{ ..., 6T2 }32·3
(2z2+(2δ+1), z3+2){2}2212·3
(2z2+2, z3+2){3,6}12{ 6, ...}{ ..., 6T2 }32·3
(2z2+2δ, z3+2){2}2212·3
(2z2+(2δ+2), z3+2){3,6}12{ 6, ...}{ ..., 6T2 }32·3
(2z2+(δ+2), z3+2){2}2212·3
4{(1,4), (3,0), (6,0)}[ 2, 0 ](2z2+1, z3+2){3,6}12{ 6, ...}{ 6T1, ... }332·3316·3316·33
(2z2+δ, z3+2){1,2}22322·33
(2z2+(δ+1), z3+2){3,6}12{ 6, ...}{ 6T1, ... }332·33
(2z2+(2δ+1), z3+2){1,2}22322·33
(2z2+2, z3+2){3,6}12{ 6, ...}{ 6T1, ... }332·33
(2z2+2δ, z3+2){1,2}22322·33
(2z2+(2δ+2), z3+2){3,6}12{ 6, ...}{ 6T1, ... }332·33
(2z2+(δ+2), z3+2){1,2}22322·33
5{(1,5), (3,0), (6,0)}[ 5/2, 0 ](2z+1, z3+2){1}14322·3316·3316·33
(2z+δ, z3+2){1}14322·33
(2z+(δ+1), z3+2){1}14322·33
(2z+(2δ+1), z3+2){1}14322·33
(2z+2, z3+2){1}14322·33
(2z+2δ, z3+2){1}14322·33
(2z+(2δ+2), z3+2){1}14322·33
(2z+(δ+2), z3+2){1}14322·33
6{(1,6), (3,0), (6,0)}[ 3, 0 ](2z2+1, z3+2){3,6}12{ 6, ...}{ ..., 6T2 }352·352·352·35