Number Theory Tables, Department of Mathematics and Statistics, UNCG

Number of totally ramified extensions of Q11 of degree 22

Introduction
Extensions of Q2 of degree 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 32
Extensions of Q2[x]/(x2+x+1) of degree 2, 4, 8 Extensions of Q2[x]/(x4+x+1) of degree 2, 4, 8
Extensions of Q3 of degree 3, 6, 9, 12, 15, 18, 21, 24, 27     Extensions of Q3[x]/(x2+2x+2) of degree 3, 6, 9
Extensions of Q5 of degree 5, 10, 15, 20, 25, 30, 35, 50, Extensions of Q7 of degree 7, 14, 21, 49
Extensions of Q11 of degree 11, 22, 33, Extensions of Q13 of degree 13, 26, 39,
Extensions of Q17 of degree 17, 34 Extensions of Q19 of degree 19,

Polynomial Invariants #Aut Splitting Field Number of
j Ramification Polygon Slopes Residual Polynomials fT eT #Gal Gal Polynomials Extensions
1{(1,1), (11,0), (22,0)}[ 1/10, 0 ](2z+1, z11+2){1}220{ 4840 }{ 22T20 }12·1120·1120·11
(2z+2, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+3, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+4, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+5, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+6, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+7, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+8, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+9, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+10, z11+2){1}220{ 4840 }{ 22T20 }12·11
2{(1,2), (11,0), (22,0)}[ 1/5, 0 ](2z2+1, z11+2){2}110{ 110 }{ 22T4 }12·1120·1120·11
(2z2+2, z11+2){2}210{ 220 }{ 22T6 }12·11
(2z2+3, z11+2){2}110{ 110 }{ 22T4 }12·11
(2z2+4, z11+2){2}110{ 110 }{ 22T4 }12·11
(2z2+5, z11+2){2}110{ 110 }{ 22T4 }12·11
(2z2+6, z11+2){2}210{ 220 }{ 22T6 }12·11
(2z2+7, z11+2){2}210{ 220 }{ 22T6 }12·11
(2z2+8, z11+2){2}210{ 220 }{ 22T6 }12·11
(2z2+9, z11+2){2}110{ 110 }{ 22T4 }12·11
(2z2+10, z11+2){2}210{ 220 }{ 22T6 }12·11
3{(1,3), (11,0), (22,0)}[ 3/10, 0 ](2z+1, z11+2){1}220{ 4840 }{ 22T20 }12·1120·1120·11
(2z+2, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+3, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+4, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+5, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+6, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+7, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+8, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+9, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+10, z11+2){1}220{ 4840 }{ 22T20 }12·11
4{(1,4), (11,0), (22,0)}[ 2/5, 0 ](2z2+1, z11+2){2}110{ 110 }{ 22T5 }12·1120·1120·11
(2z2+2, z11+2){2}210{ 220 }{ 22T6 }12·11
(2z2+3, z11+2){2}110{ 110 }{ 22T5 }12·11
(2z2+4, z11+2){2}110{ 110 }{ 22T5 }12·11
(2z2+5, z11+2){2}110{ 110 }{ 22T5 }12·11
(2z2+6, z11+2){2}210{ 220 }{ 22T6 }12·11
(2z2+7, z11+2){2}210{ 220 }{ 22T6 }12·11
(2z2+8, z11+2){2}210{ 220 }{ 22T6 }12·11
(2z2+9, z11+2){2}110{ 110 }{ 22T5 }12·11
(2z2+10, z11+2){2}210{ 220 }{ 22T6 }12·11
5{(1,5), (11,0), (22,0)}[ 1/2, 0 ](2z5+1, z11+2){1}10412·1120·1120·11
(2z5+2, z11+2){1}24{ 968 }{ 22T10 }12·11
(2z5+3, z11+2){1}10412·11
(2z5+4, z11+2){1}10412·11
(2z5+5, z11+2){1}10412·11
(2z5+6, z11+2){1}10412·11
(2z5+7, z11+2){1}10412·11
(2z5+8, z11+2){1}10412·11
(2z5+9, z11+2){1}24{ 968 }{ 22T10 }12·11
(2z5+10, z11+2){1}10412·11
6{(1,6), (11,0), (22,0)}[ 3/5, 0 ](2z2+1, z11+2){2}110{ 110 }{ 22T4 }12·1120·1120·11
(2z2+2, z11+2){2}210{ 220 }{ 22T6 }12·11
(2z2+3, z11+2){2}110{ 110 }{ 22T4 }12·11
(2z2+4, z11+2){2}110{ 110 }{ 22T4 }12·11
(2z2+5, z11+2){2}110{ 110 }{ 22T4 }12·11
(2z2+6, z11+2){2}210{ 220 }{ 22T6 }12·11
(2z2+7, z11+2){2}210{ 220 }{ 22T6 }12·11
(2z2+8, z11+2){2}210{ 220 }{ 22T6 }12·11
(2z2+9, z11+2){2}110{ 110 }{ 22T4 }12·11
(2z2+10, z11+2){2}210{ 220 }{ 22T6 }12·11
7{(1,7), (11,0), (22,0)}[ 7/10, 0 ](2z+1, z11+2){1}220{ 4840 }{ 22T20 }12·1120·1120·11
(2z+2, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+3, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+4, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+5, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+6, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+7, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+8, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+9, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+10, z11+2){1}220{ 4840 }{ 22T20 }12·11
8{(1,8), (11,0), (22,0)}[ 4/5, 0 ](2z2+1, z11+2){2}110{ 110 }{ 22T5 }12·1120·1120·11
(2z2+2, z11+2){2}210{ 220 }{ 22T6 }12·11
(2z2+3, z11+2){2}110{ 110 }{ 22T5 }12·11
(2z2+4, z11+2){2}110{ 110 }{ 22T5 }12·11
(2z2+5, z11+2){2}110{ 110 }{ 22T5 }12·11
(2z2+6, z11+2){2}210{ 220 }{ 22T6 }12·11
(2z2+7, z11+2){2}210{ 220 }{ 22T6 }12·11
(2z2+8, z11+2){2}210{ 220 }{ 22T6 }12·11
(2z2+9, z11+2){2}110{ 110 }{ 22T5 }12·11
(2z2+10, z11+2){2}210{ 220 }{ 22T6 }12·11
9{(1,9), (11,0), (22,0)}[ 9/10, 0 ](2z+1, z11+2){1}220{ 4840 }{ 22T20 }12·1120·1120·11
(2z+2, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+3, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+4, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+5, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+6, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+7, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+8, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+9, z11+2){1}220{ 4840 }{ 22T20 }12·11
(2z+10, z11+2){1}220{ 4840 }{ 22T20 }12·11
10{(1,10), (11,0), (22,0)}[ 1, 0 ](2z10+1, z11+2){2}52{ 110 }{ 22T4 }12·1120·1120·11
(2z10+2, z11+2){2}22{ 44 }{ 22T3 }12·11
(2z10+3, z11+2){2}52{ 110 }{ 22T4 }12·11
(2z10+4, z11+2){2}52{ 110 }{ 22T4 }12·11
(2z10+5, z11+2){2}52{ 110 }{ 22T4 }12·11
(2z10+6, z11+2){2}102{ 220 }{ 22T6 }12·11
(2z10+7, z11+2){2}102{ 220 }{ 22T6 }12·11
(2z10+8, z11+2){2}102{ 220 }{ 22T6 }12·11
(2z10+9, z11+2){11,22}12{ 22, 242 }{ 22T7, 22T2 }112·11
(2z10+10, z11+2){2}102{ 220 }{ 22T6 }12·11
12{(1,12), (11,0), (22,0)}[ 6/5, 0 ](2z2+1, z11+2){1,2}110{ 110, 1210 }{ 22T5, 22T11 }112·11220·11220·112
(2z2+2, z11+2){1,2}210{ 220, 2420 }{ 22T6, 22T15 }112·112
(2z2+3, z11+2){1,2}110{ 110, 1210 }{ 22T5, 22T11 }112·112
(2z2+4, z11+2){1,2}110{ 110, 1210 }{ 22T5, 22T11 }112·112
(2z2+5, z11+2){1,2}110{ 110, 1210 }{ 22T5, 22T11 }112·112
(2z2+6, z11+2){1,2}210{ 220, 2420 }{ 22T6, 22T15 }112·112
(2z2+7, z11+2){1,2}210{ 220, 2420 }{ 22T6, 22T15 }112·112
(2z2+8, z11+2){1,2}210{ 220, 2420 }{ 22T6, 22T15 }112·112
(2z2+9, z11+2){1,2}110{ 110, 1210 }{ 22T5, 22T11 }112·112
(2z2+10, z11+2){1,2}210{ 220, 2420 }{ 22T6, 22T15 }112·112
13{(1,13), (11,0), (22,0)}[ 13/10, 0 ](2z+1, z11+2){1}220{ 4840 }{ 22T20 }112·11220·11220·112
(2z+2, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+3, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+4, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+5, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+6, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+7, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+8, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+9, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+10, z11+2){1}220{ 4840 }{ 22T20 }112·112
14{(1,14), (11,0), (22,0)}[ 7/5, 0 ](2z2+1, z11+2){1,2}110{ 110, 1210 }{ 22T11, 22T4 }112·11220·11220·112
(2z2+2, z11+2){1,2}210{ 220, 2420 }{ 22T6, 22T15 }112·112
(2z2+3, z11+2){1,2}110{ 110, 1210 }{ 22T11, 22T4 }112·112
(2z2+4, z11+2){1,2}110{ 110, 1210 }{ 22T11, 22T4 }112·112
(2z2+5, z11+2){1,2}110{ 110, 1210 }{ 22T11, 22T4 }112·112
(2z2+6, z11+2){1,2}210{ 220, 2420 }{ 22T6, 22T15 }112·112
(2z2+7, z11+2){1,2}210{ 220, 2420 }{ 22T6, 22T15 }112·112
(2z2+8, z11+2){1,2}210{ 220, 2420 }{ 22T6, 22T15 }112·112
(2z2+9, z11+2){1,2}110{ 110, 1210 }{ 22T11, 22T4 }112·112
(2z2+10, z11+2){1,2}210{ 220, 2420 }{ 22T6, 22T15 }112·112
15{(1,15), (11,0), (22,0)}[ 3/2, 0 ](2z5+1, z11+2){1}104112·11220·11220·112
(2z5+2, z11+2){1}24{ 968 }{ 22T10 }112·112
(2z5+3, z11+2){1}104112·112
(2z5+4, z11+2){1}104112·112
(2z5+5, z11+2){1}104112·112
(2z5+6, z11+2){1}104112·112
(2z5+7, z11+2){1}104112·112
(2z5+8, z11+2){1}104112·112
(2z5+9, z11+2){1}24{ 968 }{ 22T10 }112·112
(2z5+10, z11+2){1}104112·112
16{(1,16), (11,0), (22,0)}[ 8/5, 0 ](2z2+1, z11+2){1,2}110{ 110, 1210 }{ 22T5, 22T11 }112·11220·11220·112
(2z2+2, z11+2){1,2}210{ 220, 2420 }{ 22T6, 22T15 }112·112
(2z2+3, z11+2){1,2}110{ 110, 1210 }{ 22T5, 22T11 }112·112
(2z2+4, z11+2){1,2}110{ 110, 1210 }{ 22T5, 22T11 }112·112
(2z2+5, z11+2){1,2}110{ 110, 1210 }{ 22T5, 22T11 }112·112
(2z2+6, z11+2){1,2}210{ 220, 2420 }{ 22T6, 22T15 }112·112
(2z2+7, z11+2){1,2}210{ 220, 2420 }{ 22T6, 22T15 }112·112
(2z2+8, z11+2){1,2}210{ 220, 2420 }{ 22T6, 22T15 }112·112
(2z2+9, z11+2){1,2}110{ 110, 1210 }{ 22T5, 22T11 }112·112
(2z2+10, z11+2){1,2}210{ 220, 2420 }{ 22T6, 22T15 }112·112
17{(1,17), (11,0), (22,0)}[ 17/10, 0 ](2z+1, z11+2){1}220{ 4840 }{ 22T20 }112·11220·11220·112
(2z+2, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+3, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+4, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+5, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+6, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+7, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+8, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+9, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+10, z11+2){1}220{ 4840 }{ 22T20 }112·112
18{(1,18), (11,0), (22,0)}[ 9/5, 0 ](2z2+1, z11+2){1,2}110{ 110, 1210 }{ 22T11, 22T4 }112·11220·11220·112
(2z2+2, z11+2){1,2}210{ 220, 2420 }{ 22T6, 22T15 }112·112
(2z2+3, z11+2){1,2}110{ 110, 1210 }{ 22T11, 22T4 }112·112
(2z2+4, z11+2){1,2}110{ 110, 1210 }{ 22T11, 22T4 }112·112
(2z2+5, z11+2){1,2}110{ 110, 1210 }{ 22T11, 22T4 }112·112
(2z2+6, z11+2){1,2}210{ 220, 2420 }{ 22T6, 22T15 }112·112
(2z2+7, z11+2){1,2}210{ 220, 2420 }{ 22T6, 22T15 }112·112
(2z2+8, z11+2){1,2}210{ 220, 2420 }{ 22T6, 22T15 }112·112
(2z2+9, z11+2){1,2}110{ 110, 1210 }{ 22T11, 22T4 }112·112
(2z2+10, z11+2){1,2}210{ 220, 2420 }{ 22T6, 22T15 }112·112
19{(1,19), (11,0), (22,0)}[ 19/10, 0 ](2z+1, z11+2){1}220{ 4840 }{ 22T20 }112·11220·11220·112
(2z+2, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+3, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+4, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+5, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+6, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+7, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+8, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+9, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+10, z11+2){1}220{ 4840 }{ 22T20 }112·112
20{(1,20), (11,0), (22,0)}[ 2, 0 ](2z10+1, z11+2){1,2}52{ 110, 1210 }{ 22T5, 22T11 }112·11220·11220·112
(2z10+2, z11+2){1,2}22{ 44, 484 }{ 22T3, 22T9 }112·112
(2z10+3, z11+2){1,2}52{ 110, 1210 }{ 22T5, 22T11 }112·112
(2z10+4, z11+2){1,2}52{ 110, 1210 }{ 22T5, 22T11 }112·112
(2z10+5, z11+2){1,2}52{ 110, 1210 }{ 22T5, 22T11 }112·112
(2z10+6, z11+2){1,2}102{ 220, ...}{ 22T6, ... }112·112
(2z10+7, z11+2){1,2}102{ 220, ...}{ 22T6, ... }112·112
(2z10+8, z11+2){1,2}102{ 220, ...}{ 22T6, ... }112·112
(2z10+9, z11+2){11,22}12{ 22, 242 }{ 22T7, 22T1 }1122·112
(2z10+10, z11+2){1,2}102{ 220, ...}{ 22T6, ... }112·112
21{(1,21), (11,0), (22,0)}[ 21/10, 0 ](2z+1, z11+2){1}220{ 4840 }{ 22T20 }112·11220·11220·112
(2z+2, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+3, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+4, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+5, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+6, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+7, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+8, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+9, z11+2){1}220{ 4840 }{ 22T20 }112·112
(2z+10, z11+2){1}220{ 4840 }{ 22T20 }112·112
22{(1,22), (11,0), (22,0)}[ 11/5, 0 ](2z2+9, z11+2){1,2}110{ 110, 1210 }{ 22T11, 22T4 }1122·1132·1132·113