Number Theory Tables, Department of Mathematics and Statistics, UNCG

Number of totally ramified extensions of Q3 of degree 9

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), (9,0)}[ 1/8 ](z+1){1}28{ 144 }{ 9T19 }1322·322·32
(z+2){1}28{ 144 }{ 9T19 }132
2{(1,2), (9,0)}[ 1/4 ](z2+1){1}24{ 72 }{ 9T14 }1322·322·32
(z2+2){1}24{ 72 }{ 9T16 }132
4{(1,4), (3,3), (9,0)}[ 1/2 ](z4+z+1){1}32{ 54 }{ 9T11 }1324·322·33
(z4+2z+1){1}32{ 54 }{ 9T11 }132
(z4+z+2){1}42{ 72 }{ 9T15 }132
(z4+2z+2){1}42{ 72 }{ 9T15 }132
{(1,4), (9,0)}[ 1/2 ](z4+1){1}22{ 36 }{ 9T9 }1322·32
(z4+2){1}22{ 36 }{ 9T8 }132
5{(1,5), (3,3), (9,0)}[ 1, 1/2 ](z2+1, z3+1){1}22{ 108 }{ 9T18 }1324·322·33
(2z2+1, z3+2){3}12{ 162 }{ 9T20 }332
(z2+2, z3+1){3}12{ 162 }{ 9T20 }332
(2z2+2, z3+2){1}22{ 108 }{ 9T18 }132
{(1,5), (9,0)}[ 5/8 ](z+1){1}28{ 144 }{ 9T19 }1322·32
(z+2){1}28{ 144 }{ 9T19 }132
7{(1,7), (3,3), (9,0)}[ 2, 1/2 ](z2+1, z3+1){1}22{ 36, 108 }{ 9T8, 9T18 }3334·332·34
(2z2+1, z3+2){3}12{ 18, 162 }{ 9T4, 9T20 }3233
(z2+2, z3+1){3}12{ 18, 162 }{ 9T4, 9T20 }3233
(2z2+2, z3+2){1}22{ 36, 108 }{ 9T8, 9T18 }333
{(1,7), (9,0)}[ 7/8 ](z+1){1}28{ 144 }{ 9T19 }3332·33
(z+2){1}28{ 144 }{ 9T19 }333
8{(1,8), (3,3), (9,0)}[ 5/2, 1/2 ](z+1, z3+1){1}12{ 54 }{ 9T10 }3334·332·34
(2z+1, z3+2){1}12{ 54 }{ 9T10 }333
(z+2, z3+1){1}22{ 324 }{ 9T24 }333
(2z+2, z3+2){1}22{ 324 }{ 9T24 }333
{(1,8), (3,6), (9,0)}[ 1 ](z8+z2+1){3}31{ 27 }{ 9T7 }3324·32
(z8+2z2+1){1}61{ 54 }{ 9T11 }132
(z8+z2+2){1}81{ 72 }{ 9T15 }132
(z8+2z2+2){1}81{ 72 }{ 9T15 }132
{(1,8), (9,0)}[ 1 ](z8+1){1}41{ 36 }{ 9T9 }1322·32
(z8+2){3}21{ 18 }{ 9T4 }332
10{(1,10), (3,3), (9,0)}[ 7/2, 1/2 ](z+1, z3+1){1}12{ 18, 54 }{ 9T11, 9T10, 9T3 }32344·342·35
(2z+1, z3+2){1}12{ 18, 54 }{ 9T5, 9T11, 9T10 }3234
(z+2, z3+1){1}22{ 36, 324 }{ 9T8, 9T24 }3234
(2z+2, z3+2){1}22{ 108, 324 }{ 9T18, 9T24 }3234
{(1,10), (3,6), (9,0)}[ 2, 1 ](z2+1, z6+1){1}21{ 54 }{ 9T11 }3334·33
(2z2+1, z6+2){3}11{ 81 }{ 9T17 }33 [32]33
(z2+2, z6+1){3}21{ 162 }{ 9T20 }3233
(2z2+2, z6+2){1}21{ 54 }{ 9T13 }32 [3]33
11{(1,11), (3,3), (9,0)}[ 4, 1/2 ](z2+1, z3+1){1}22{ 108 }{ 9T18 }32344·342·35
(2z2+1, z3+2){3}12{ 54, 162 }{ 9T12, 9T20 }3334
(z2+2, z3+1){3}12{ 162 }{ 9T20 }3334
(2z2+2, z3+2){1}22{ 324 }{ 9T24 }3234
{(1,11), (3,6), (9,0)}[ 5/2, 1 ](z+1, z6+1){1}22{ 108 }{ 9T18 }3334·33
(2z+1, z6+2){3}12{ 18 }{ 9T4 }32 [32]33
(z+2, z6+1){1}22{ 36 }{ 9T8 }333
(2z+2, z6+2){1}12{ 54 }{ 9T13 }32 [3]33
12{(1,12), (3,3), (9,0)}[ 9/2, 1/2 ](2z+1, z3+2){1}12{ 162 }{ 9T21 }33352·352·35
(z+2, z3+1){1}22{ 324 }{ 9T24 }3335
13{(1,13), (3,6), (9,0)}[ 7/2, 1 ](z+1, z6+1){1}22{ 108 }{ 9T18 }32344·342·35
(2z+1, z6+2){1}12{ 54 }{ 9T13 }33 [32]34
(z+2, z6+1){1}22{ 324 }{ 9T24 }3234
(2z+2, z6+2){1}12{ 162 }{ 9T22 }33 [32]34
{(1,13), (3,9), (9,0)}[ 2, 3/2 ](2z2+1, z3+2){3}12{ 54 }{ 9T12 }33342·34
(2z2+2, z3+2){1}22{ 108 }{ 9T18 }3234
14{(1,14), (3,6), (9,0)}[ 4, 1 ](z2+1, z6+1){1}21{ 18, 54 }{ 9T10, 9T3 }32344·342·35
(2z2+1, z6+2){3,9}11{ 9, 27, 81 }{ 9T1, 9T17, 9T6 }34 [11·3]34
(z2+2, z6+1){3}21{ 162 }{ 9T20 }33 [.]34
(2z2+2, z6+2){1}21{ 162 }{ 9T22 }33 [32]34
{(1,14), (3,9), (9,0)}[ 5/2, 3/2 ](2z+1, z3+2){1}12{ 54 }{ 9T11 }32342·34
(2z+2, z3+2){1}22{ 108 }{ 9T18 }3234
15{(1,15), (3,6), (9,0)}[ 9/2, 1 ](2z+1, z6+2){1}12{ 162 }{ 9T22 }34 [33]352·352·35
(z+2, z6+1){1}22{ 324 }{ 9T24 }3335
16{(1,16), (3,9), (9,0)}[ 7/2, 3/2 ](2z+1, z3+2){1}12{ 162 }{ 9T21 }33352·352·35
(2z+2, z3+2){1}22{ 324 }{ 9T24 }3335
17{(1,17), (3,9), (9,0)}[ 4, 3/2 ](2z2+1, z3+2){3}12{ 162 }{ 9T20 }34352·352·35
(2z2+2, z3+2){1}22{ 324 }{ 9T24 }3335
18{(1,18), (3,9), (9,0)}[ 9/2, 3/2 ](2z+1, z3+2){1}12{ 18, 54, 162 }{ 9T10, 9T21, 9T3 }34363636