Number Theory Tables, Department of Mathematics and Statistics, UNCG

Number of totally ramified extensions of Q7 of degree 14

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), (14,0)}[ 1/6, 0 ](2z+1, z7+2){1}212{ 1176 }{ 14T32 }12·712·712·7
(2z+2, z7+2){1}212{ 1176 }{ 14T32 }12·7
(2z+3, z7+2){1}212{ 1176 }{ 14T32 }12·7
(2z+4, z7+2){1}212{ 1176 }{ 14T32 }12·7
(2z+5, z7+2){1}212{ 1176 }{ 14T32 }12·7
(2z+6, z7+2){1}212{ 1176 }{ 14T32 }12·7
2{(1,2), (7,0), (14,0)}[ 1/3, 0 ](2z2+1, z7+2){2}26{ 84 }{ 14T7 }12·712·712·7
(2z2+2, z7+2){2}26{ 84 }{ 14T7 }12·7
(2z2+3, z7+2){2}16{ 42 }{ 14T4 }12·7
(2z2+4, z7+2){2}26{ 84 }{ 14T7 }12·7
(2z2+5, z7+2){2}16{ 42 }{ 14T4 }12·7
(2z2+6, z7+2){2}16{ 42 }{ 14T4 }12·7
3{(1,3), (7,0), (14,0)}[ 1/2, 0 ](2z3+1, z7+2){1}64{ 1176 }{ 14T32 }12·712·712·7
(2z3+2, z7+2){1}24{ 392 }{ 14T20 }12·7
(2z3+3, z7+2){1}64{ 1176 }{ 14T32 }12·7
(2z3+4, z7+2){1}64{ 1176 }{ 14T32 }12·7
(2z3+5, z7+2){1}24{ 392 }{ 14T20 }12·7
(2z3+6, z7+2){1}64{ 1176 }{ 14T32 }12·7
4{(1,4), (7,0), (14,0)}[ 2/3, 0 ](2z2+1, z7+2){2}26{ 84 }{ 14T7 }12·712·712·7
(2z2+2, z7+2){2}26{ 84 }{ 14T7 }12·7
(2z2+3, z7+2){2}16{ 42 }{ 14T5 }12·7
(2z2+4, z7+2){2}26{ 84 }{ 14T7 }12·7
(2z2+5, z7+2){2}16{ 42 }{ 14T5 }12·7
(2z2+6, z7+2){2}16{ 42 }{ 14T5 }12·7
5{(1,5), (7,0), (14,0)}[ 5/6, 0 ](2z+1, z7+2){1}212{ 1176 }{ 14T32 }12·712·712·7
(2z+2, z7+2){1}212{ 1176 }{ 14T32 }12·7
(2z+3, z7+2){1}212{ 1176 }{ 14T32 }12·7
(2z+4, z7+2){1}212{ 1176 }{ 14T32 }12·7
(2z+5, z7+2){1}212{ 1176 }{ 14T32 }12·7
(2z+6, z7+2){1}212{ 1176 }{ 14T32 }12·7
6{(1,6), (7,0), (14,0)}[ 1, 0 ](2z6+1, z7+2){2}62{ 84 }{ 14T7 }12·712·712·7
(2z6+2, z7+2){2}22{ 28 }{ 14T3 }12·7
(2z6+3, z7+2){2}32{ 42 }{ 14T4 }12·7
(2z6+4, z7+2){2}62{ 84 }{ 14T7 }12·7
(2z6+5, z7+2){7,14}12{ 14, 98 }{ 14T2, 14T8 }72·7
(2z6+6, z7+2){2}32{ 42 }{ 14T4 }12·7
8{(1,8), (7,0), (14,0)}[ 4/3, 0 ](2z2+1, z7+2){1,2}26{ 84, 588 }{ 14T24, 14T7 }72·7212·7212·72
(2z2+2, z7+2){1,2}26{ 84, 588 }{ 14T24, 14T7 }72·72
(2z2+3, z7+2){1,2}16{ 42, 294 }{ 14T14, 14T5 }72·72
(2z2+4, z7+2){1,2}26{ 84, 588 }{ 14T24, 14T7 }72·72
(2z2+5, z7+2){1,2}16{ 42, 294 }{ 14T14, 14T5 }72·72
(2z2+6, z7+2){1,2}16{ 42, 294 }{ 14T14, 14T5 }72·72
9{(1,9), (7,0), (14,0)}[ 3/2, 0 ](2z3+1, z7+2){1}64{ 1176 }{ 14T32 }72·7212·7212·72
(2z3+2, z7+2){1}24{ 392 }{ 14T20 }72·72
(2z3+3, z7+2){1}64{ 1176 }{ 14T32 }72·72
(2z3+4, z7+2){1}64{ 1176 }{ 14T32 }72·72
(2z3+5, z7+2){1}24{ 392 }{ 14T20 }72·72
(2z3+6, z7+2){1}64{ 1176 }{ 14T32 }72·72
10{(1,10), (7,0), (14,0)}[ 5/3, 0 ](2z2+1, z7+2){1,2}26{ 84, 588 }{ 14T24, 14T7 }72·7212·7212·72
(2z2+2, z7+2){1,2}26{ 84, 588 }{ 14T24, 14T7 }72·72
(2z2+3, z7+2){1,2}16{ 42, 294 }{ 14T14, 14T4 }72·72
(2z2+4, z7+2){1,2}26{ 84, 588 }{ 14T24, 14T7 }72·72
(2z2+5, z7+2){1,2}16{ 42, 294 }{ 14T14, 14T4 }72·72
(2z2+6, z7+2){1,2}16{ 42, 294 }{ 14T14, 14T4 }72·72
11{(1,11), (7,0), (14,0)}[ 11/6, 0 ](2z+1, z7+2){1}212{ 1176 }{ 14T32 }72·7212·7212·72
(2z+2, z7+2){1}212{ 1176 }{ 14T32 }72·72
(2z+3, z7+2){1}212{ 1176 }{ 14T32 }72·72
(2z+4, z7+2){1}212{ 1176 }{ 14T32 }72·72
(2z+5, z7+2){1}212{ 1176 }{ 14T32 }72·72
(2z+6, z7+2){1}212{ 1176 }{ 14T32 }72·72
12{(1,12), (7,0), (14,0)}[ 2, 0 ](2z6+1, z7+2){1,2}62{ 84, 588 }{ 14T24, 14T7 }72·7212·7212·72
(2z6+2, z7+2){1,2}22{ 28, 196 }{ 14T13, 14T3 }72·72
(2z6+3, z7+2){1,2}32{ 42, 294 }{ 14T14, 14T5 }72·72
(2z6+4, z7+2){1,2}62{ 84, 588 }{ 14T24, 14T7 }72·72
(2z6+5, z7+2){7,14}12{ 14, 98 }{ 14T1, 14T8 }722·72
(2z6+6, z7+2){1,2}32{ 42, 294 }{ 14T14, 14T5 }72·72
13{(1,13), (7,0), (14,0)}[ 13/6, 0 ](2z+1, z7+2){1}212{ 1176 }{ 14T32 }72·7212·7212·72
(2z+2, z7+2){1}212{ 1176 }{ 14T32 }72·72
(2z+3, z7+2){1}212{ 1176 }{ 14T32 }72·72
(2z+4, z7+2){1}212{ 1176 }{ 14T32 }72·72
(2z+5, z7+2){1}212{ 1176 }{ 14T32 }72·72
(2z+6, z7+2){1}212{ 1176 }{ 14T32 }72·72
14{(1,14), (7,0), (14,0)}[ 7/3, 0 ](2z2+5, z7+2){1,2}16{ 42, 294 }{ 14T14, 14T4 }722·732·732·73