UNCG, Department of Mathematics and Statistics, Sebastian Pauli

Number of totally ramified extensions of Q17 of degree 17

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), (17,0)}[ 1/16 ](z+1){1}116{ 272 }{ 17T5 }11716·1716·17
(z+2){1}116{ 272 }{ 17T5 }117
(z+3){1}116{ 272 }{ 17T5 }117
(z+4){1}116{ 272 }{ 17T5 }117
(z+5){1}116{ 272 }{ 17T5 }117
(z+6){1}116{ 272 }{ 17T5 }117
(z+7){1}116{ 272 }{ 17T5 }117
(z+8){1}116{ 272 }{ 17T5 }117
(z+9){1}116{ 272 }{ 17T5 }117
(z+10){1}116{ 272 }{ 17T5 }117
(z+11){1}116{ 272 }{ 17T5 }117
(z+12){1}116{ 272 }{ 17T5 }117
(z+13){1}116{ 272 }{ 17T5 }117
(z+14){1}116{ 272 }{ 17T5 }117
(z+15){1}116{ 272 }{ 17T5 }117
(z+16){1}116{ 272 }{ 17T5 }117
2{(1,2), (17,0)}[ 1/8 ](z2+1){1}18{ 136 }{ 17T4 }11716·1716·17
(z2+2){1}18{ 136 }{ 17T4 }117
(z2+3){1}28{ 272 }{ 17T5 }117
(z2+4){1}18{ 136 }{ 17T4 }117
(z2+5){1}28{ 272 }{ 17T5 }117
(z2+6){1}28{ 272 }{ 17T5 }117
(z2+7){1}28{ 272 }{ 17T5 }117
(z2+8){1}18{ 136 }{ 17T4 }117
(z2+9){1}18{ 136 }{ 17T4 }117
(z2+10){1}28{ 272 }{ 17T5 }117
(z2+11){1}28{ 272 }{ 17T5 }117
(z2+12){1}28{ 272 }{ 17T5 }117
(z2+13){1}18{ 136 }{ 17T4 }117
(z2+14){1}28{ 272 }{ 17T5 }117
(z2+15){1}18{ 136 }{ 17T4 }117
(z2+16){1}18{ 136 }{ 17T4 }117
3{(1,3), (17,0)}[ 3/16 ](z+1){1}116{ 272 }{ 17T5 }11716·1716·17
(z+2){1}116{ 272 }{ 17T5 }117
(z+3){1}116{ 272 }{ 17T5 }117
(z+4){1}116{ 272 }{ 17T5 }117
(z+5){1}116{ 272 }{ 17T5 }117
(z+6){1}116{ 272 }{ 17T5 }117
(z+7){1}116{ 272 }{ 17T5 }117
(z+8){1}116{ 272 }{ 17T5 }117
(z+9){1}116{ 272 }{ 17T5 }117
(z+10){1}116{ 272 }{ 17T5 }117
(z+11){1}116{ 272 }{ 17T5 }117
(z+12){1}116{ 272 }{ 17T5 }117
(z+13){1}116{ 272 }{ 17T5 }117
(z+14){1}116{ 272 }{ 17T5 }117
(z+15){1}116{ 272 }{ 17T5 }117
(z+16){1}116{ 272 }{ 17T5 }117
4{(1,4), (17,0)}[ 1/4 ](z4+1){1}14{ 68 }{ 17T3 }11716·1716·17
(z4+2){1}24{ 136 }{ 17T4 }117
(z4+3){1}44{ 272 }{ 17T5 }117
(z4+4){1}14{ 68 }{ 17T3 }117
(z4+5){1}44{ 272 }{ 17T5 }117
(z4+6){1}44{ 272 }{ 17T5 }117
(z4+7){1}44{ 272 }{ 17T5 }117
(z4+8){1}24{ 136 }{ 17T4 }117
(z4+9){1}24{ 136 }{ 17T4 }117
(z4+10){1}44{ 272 }{ 17T5 }117
(z4+11){1}44{ 272 }{ 17T5 }117
(z4+12){1}44{ 272 }{ 17T5 }117
(z4+13){1}14{ 68 }{ 17T3 }117
(z4+14){1}44{ 272 }{ 17T5 }117
(z4+15){1}24{ 136 }{ 17T4 }117
(z4+16){1}14{ 68 }{ 17T3 }117
5{(1,5), (17,0)}[ 5/16 ](z+1){1}116{ 272 }{ 17T5 }11716·1716·17
(z+2){1}116{ 272 }{ 17T5 }117
(z+3){1}116{ 272 }{ 17T5 }117
(z+4){1}116{ 272 }{ 17T5 }117
(z+5){1}116{ 272 }{ 17T5 }117
(z+6){1}116{ 272 }{ 17T5 }117
(z+7){1}116{ 272 }{ 17T5 }117
(z+8){1}116{ 272 }{ 17T5 }117
(z+9){1}116{ 272 }{ 17T5 }117
(z+10){1}116{ 272 }{ 17T5 }117
(z+11){1}116{ 272 }{ 17T5 }117
(z+12){1}116{ 272 }{ 17T5 }117
(z+13){1}116{ 272 }{ 17T5 }117
(z+14){1}116{ 272 }{ 17T5 }117
(z+15){1}116{ 272 }{ 17T5 }117
(z+16){1}116{ 272 }{ 17T5 }117
6{(1,6), (17,0)}[ 3/8 ](z2+1){1}18{ 136 }{ 17T4 }11716·1716·17
(z2+2){1}18{ 136 }{ 17T4 }117
(z2+3){1}28{ 272 }{ 17T5 }117
(z2+4){1}18{ 136 }{ 17T4 }117
(z2+5){1}28{ 272 }{ 17T5 }117
(z2+6){1}28{ 272 }{ 17T5 }117
(z2+7){1}28{ 272 }{ 17T5 }117
(z2+8){1}18{ 136 }{ 17T4 }117
(z2+9){1}18{ 136 }{ 17T4 }117
(z2+10){1}28{ 272 }{ 17T5 }117
(z2+11){1}28{ 272 }{ 17T5 }117
(z2+12){1}28{ 272 }{ 17T5 }117
(z2+13){1}18{ 136 }{ 17T4 }117
(z2+14){1}28{ 272 }{ 17T5 }117
(z2+15){1}18{ 136 }{ 17T4 }117
(z2+16){1}18{ 136 }{ 17T4 }117
7{(1,7), (17,0)}[ 7/16 ](z+1){1}116{ 272 }{ 17T5 }11716·1716·17
(z+2){1}116{ 272 }{ 17T5 }117
(z+3){1}116{ 272 }{ 17T5 }117
(z+4){1}116{ 272 }{ 17T5 }117
(z+5){1}116{ 272 }{ 17T5 }117
(z+6){1}116{ 272 }{ 17T5 }117
(z+7){1}116{ 272 }{ 17T5 }117
(z+8){1}116{ 272 }{ 17T5 }117
(z+9){1}116{ 272 }{ 17T5 }117
(z+10){1}116{ 272 }{ 17T5 }117
(z+11){1}116{ 272 }{ 17T5 }117
(z+12){1}116{ 272 }{ 17T5 }117
(z+13){1}116{ 272 }{ 17T5 }117
(z+14){1}116{ 272 }{ 17T5 }117
(z+15){1}116{ 272 }{ 17T5 }117
(z+16){1}116{ 272 }{ 17T5 }117
8{(1,8), (17,0)}[ 1/2 ](z8+1){1}12{ 34 }{ 17T2 }11716·1716·17
(z8+2){1}42{ 136 }{ 17T4 }117
(z8+3){1}82{ 272 }{ 17T5 }117
(z8+4){1}22{ 68 }{ 17T3 }117
(z8+5){1}82{ 272 }{ 17T5 }117
(z8+6){1}82{ 272 }{ 17T5 }117
(z8+7){1}82{ 272 }{ 17T5 }117
(z8+8){1}42{ 136 }{ 17T4 }117
(z8+9){1}42{ 136 }{ 17T4 }117
(z8+10){1}82{ 272 }{ 17T5 }117
(z8+11){1}82{ 272 }{ 17T5 }117
(z8+12){1}82{ 272 }{ 17T5 }117
(z8+13){1}22{ 68 }{ 17T3 }117
(z8+14){1}82{ 272 }{ 17T5 }117
(z8+15){1}42{ 136 }{ 17T4 }117
(z8+16){1}12{ 34 }{ 17T2 }117
9{(1,9), (17,0)}[ 9/16 ](z+1){1}116{ 272 }{ 17T5 }11716·1716·17
(z+2){1}116{ 272 }{ 17T5 }117
(z+3){1}116{ 272 }{ 17T5 }117
(z+4){1}116{ 272 }{ 17T5 }117
(z+5){1}116{ 272 }{ 17T5 }117
(z+6){1}116{ 272 }{ 17T5 }117
(z+7){1}116{ 272 }{ 17T5 }117
(z+8){1}116{ 272 }{ 17T5 }117
(z+9){1}116{ 272 }{ 17T5 }117
(z+10){1}116{ 272 }{ 17T5 }117
(z+11){1}116{ 272 }{ 17T5 }117
(z+12){1}116{ 272 }{ 17T5 }117
(z+13){1}116{ 272 }{ 17T5 }117
(z+14){1}116{ 272 }{ 17T5 }117
(z+15){1}116{ 272 }{ 17T5 }117
(z+16){1}116{ 272 }{ 17T5 }117
10{(1,10), (17,0)}[ 5/8 ](z2+1){1}18{ 136 }{ 17T4 }11716·1716·17
(z2+2){1}18{ 136 }{ 17T4 }117
(z2+3){1}28{ 272 }{ 17T5 }117
(z2+4){1}18{ 136 }{ 17T4 }117
(z2+5){1}28{ 272 }{ 17T5 }117
(z2+6){1}28{ 272 }{ 17T5 }117
(z2+7){1}28{ 272 }{ 17T5 }117
(z2+8){1}18{ 136 }{ 17T4 }117
(z2+9){1}18{ 136 }{ 17T4 }117
(z2+10){1}28{ 272 }{ 17T5 }117
(z2+11){1}28{ 272 }{ 17T5 }117
(z2+12){1}28{ 272 }{ 17T5 }117
(z2+13){1}18{ 136 }{ 17T4 }117
(z2+14){1}28{ 272 }{ 17T5 }117
(z2+15){1}18{ 136 }{ 17T4 }117
(z2+16){1}18{ 136 }{ 17T4 }117
11{(1,11), (17,0)}[ 11/16 ](z+1){1}116{ 272 }{ 17T5 }11716·1716·17
(z+2){1}116{ 272 }{ 17T5 }117
(z+3){1}116{ 272 }{ 17T5 }117
(z+4){1}116{ 272 }{ 17T5 }117
(z+5){1}116{ 272 }{ 17T5 }117
(z+6){1}116{ 272 }{ 17T5 }117
(z+7){1}116{ 272 }{ 17T5 }117
(z+8){1}116{ 272 }{ 17T5 }117
(z+9){1}116{ 272 }{ 17T5 }117
(z+10){1}116{ 272 }{ 17T5 }117
(z+11){1}116{ 272 }{ 17T5 }117
(z+12){1}116{ 272 }{ 17T5 }117
(z+13){1}116{ 272 }{ 17T5 }117
(z+14){1}116{ 272 }{ 17T5 }117
(z+15){1}116{ 272 }{ 17T5 }117
(z+16){1}116{ 272 }{ 17T5 }117
12{(1,12), (17,0)}[ 3/4 ](z4+1){1}14{ 68 }{ 17T3 }11716·1716·17
(z4+2){1}24{ 136 }{ 17T4 }117
(z4+3){1}44{ 272 }{ 17T5 }117
(z4+4){1}14{ 68 }{ 17T3 }117
(z4+5){1}44{ 272 }{ 17T5 }117
(z4+6){1}44{ 272 }{ 17T5 }117
(z4+7){1}44{ 272 }{ 17T5 }117
(z4+8){1}24{ 136 }{ 17T4 }117
(z4+9){1}24{ 136 }{ 17T4 }117
(z4+10){1}44{ 272 }{ 17T5 }117
(z4+11){1}44{ 272 }{ 17T5 }117
(z4+12){1}44{ 272 }{ 17T5 }117
(z4+13){1}14{ 68 }{ 17T3 }117
(z4+14){1}44{ 272 }{ 17T5 }117
(z4+15){1}24{ 136 }{ 17T4 }117
(z4+16){1}14{ 68 }{ 17T3 }117
13{(1,13), (17,0)}[ 13/16 ](z+1){1}116{ 272 }{ 17T5 }11716·1716·17
(z+2){1}116{ 272 }{ 17T5 }117
(z+3){1}116{ 272 }{ 17T5 }117
(z+4){1}116{ 272 }{ 17T5 }117
(z+5){1}116{ 272 }{ 17T5 }117
(z+6){1}116{ 272 }{ 17T5 }117
(z+7){1}116{ 272 }{ 17T5 }117
(z+8){1}116{ 272 }{ 17T5 }117
(z+9){1}116{ 272 }{ 17T5 }117
(z+10){1}116{ 272 }{ 17T5 }117
(z+11){1}116{ 272 }{ 17T5 }117
(z+12){1}116{ 272 }{ 17T5 }117
(z+13){1}116{ 272 }{ 17T5 }117
(z+14){1}116{ 272 }{ 17T5 }117
(z+15){1}116{ 272 }{ 17T5 }117
(z+16){1}116{ 272 }{ 17T5 }117
14{(1,14), (17,0)}[ 7/8 ](z2+1){1}18{ 136 }{ 17T4 }11716·1716·17
(z2+2){1}18{ 136 }{ 17T4 }117
(z2+3){1}28{ 272 }{ 17T5 }117
(z2+4){1}18{ 136 }{ 17T4 }117
(z2+5){1}28{ 272 }{ 17T5 }117
(z2+6){1}28{ 272 }{ 17T5 }117
(z2+7){1}28{ 272 }{ 17T5 }117
(z2+8){1}18{ 136 }{ 17T4 }117
(z2+9){1}18{ 136 }{ 17T4 }117
(z2+10){1}28{ 272 }{ 17T5 }117
(z2+11){1}28{ 272 }{ 17T5 }117
(z2+12){1}28{ 272 }{ 17T5 }117
(z2+13){1}18{ 136 }{ 17T4 }117
(z2+14){1}28{ 272 }{ 17T5 }117
(z2+15){1}18{ 136 }{ 17T4 }117
(z2+16){1}18{ 136 }{ 17T4 }117
15{(1,15), (17,0)}[ 15/16 ](z+1){1}116{ 272 }{ 17T5 }11716·1716·17
(z+2){1}116{ 272 }{ 17T5 }117
(z+3){1}116{ 272 }{ 17T5 }117
(z+4){1}116{ 272 }{ 17T5 }117
(z+5){1}116{ 272 }{ 17T5 }117
(z+6){1}116{ 272 }{ 17T5 }117
(z+7){1}116{ 272 }{ 17T5 }117
(z+8){1}116{ 272 }{ 17T5 }117
(z+9){1}116{ 272 }{ 17T5 }117
(z+10){1}116{ 272 }{ 17T5 }117
(z+11){1}116{ 272 }{ 17T5 }117
(z+12){1}116{ 272 }{ 17T5 }117
(z+13){1}116{ 272 }{ 17T5 }117
(z+14){1}116{ 272 }{ 17T5 }117
(z+15){1}116{ 272 }{ 17T5 }117
(z+16){1}116{ 272 }{ 17T5 }117
16{(1,16), (17,0)}[ 1 ](z16+1){1}21{ 34 }{ 17T2 }11716·1716·17
(z16+2){1}81{ 136 }{ 17T4 }117
(z16+3){1}161{ 272 }{ 17T5 }117
(z16+4){1}41{ 68 }{ 17T3 }117
(z16+5){1}161{ 272 }{ 17T5 }117
(z16+6){1}161{ 272 }{ 17T5 }117
(z16+7){1}161{ 272 }{ 17T5 }117
(z16+8){1}81{ 136 }{ 17T4 }117
(z16+9){1}81{ 136 }{ 17T4 }117
(z16+10){1}161{ 272 }{ 17T5 }117
(z16+11){1}161{ 272 }{ 17T5 }117
(z16+12){1}161{ 272 }{ 17T5 }117
(z16+13){1}41{ 68 }{ 17T3 }117
(z16+14){1}161{ 272 }{ 17T5 }117
(z16+15){1}81{ 136 }{ 17T4 }117
(z16+16){17}11{ 17 }{ 17T1 }1717
17{(1,17), (17,0)}[ 17/16 ](z+16){1}116{ 272 }{ 17T5 }17172172172