Number Theory Tables, Department of Mathematics and Statistics, UNCG

Number of totally ramified extensions of Q2 of degree 20

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), (4,0), (20,0)}[ 1/3, 0 ](z+1, z16+1){1}415{ 960 }{ 20T173 }15·225·225·22
3{(1,3), (2,2), (4,0), (20,0)}[ 1, 0 ](z3+z+1, z16+1){1}12515·225·225·23
{(1,3), (4,0), (20,0)}[ 1, 0 ](z3+1, z16+1){2}45{ 320 }{ 20T80, 20T77 }25·225·22
5{(1,5), (2,2), (4,0), (20,0)}[ 3, 1, 0 ](z+1, z2+1, z16+1){22}4523 [3·2]5·235·235·24
{(1,5), (4,0), (20,0)}[ 5/3, 0 ](z+1, z16+1){1}415{ 240, ...}{ ..., 20T61 }25·235·23
7{(1,7), (2,2), (4,0), (20,0)}[ 5, 1, 0 ](z+1, z2+1, z16+1){22}4524 [?]5·245·245·25
{(1,7), (2,6), (4,0), (20,0)}[ 1, 3, 0 ](z+1, z2+1, z16+1){1}415225·235·23
{(1,7), (4,0), (20,0)}[ 7/3, 0 ](z+1, z16+1){1}415{ 960, ...}{ ..., 20T173 }225·245·24
9{(1,9), (2,2), (4,0), (20,0)}[ 7, 1, 0 ](z+1, z2+1, z16+1){2}45{ 163840, ...}{ 20T850, ... }25 [?]5·255·255·26
{(1,9), (2,6), (4,0), (20,0)}[ 3, 0 ](z3+z+1, z16+1){1}125225·245·24
{(1,9), (4,0), (20,0)}[ 3, 0 ](z3+1, z16+1){2}45235·245·24
11{(1,11), (2,2), (4,0), (20,0)}[ 9, 1, 0 ](z+1, z2+1, z16+1){22}4526 [?]5·265·265·27
{(1,11), (2,6), (4,0), (20,0)}[ 5, 3, 0 ](z+1, z2+1, z16+1){2}4525 [?]5·255·25
{(1,11), (2,10), (4,0), (20,0)}[ 1, 5, 0 ](z+1, z2+1, z16+1){1}415{ 960, ...}{ ..., 20T173 }235·245·24
{(1,11), (4,0), (20,0)}[ 11/3, 0 ](z+1, z16+1){1}415{ 960, ...}{ ..., 20T173 }235·255·25
13{(1,13), (2,2), (4,0), (20,0)}[ 11, 1, 0 ](z+1, z2+1, z16+1){2}4527 [?]5·275·275·28
{(1,13), (2,6), (4,0), (20,0)}[ 7, 3, 0 ](z+1, z2+1, z16+1){2}4526 [?]5·265·26
{(1,13), (2,10), (4,0), (20,0)}[ 3, 5, 0 ](z+1, z2+1, z16+1){1}415245·255·25
{(1,13), (4,0), (20,0)}[ 13/3, 0 ](z+1, z16+1){1}415{ 960, ...}{ ..., 20T173 }245·265·26
15{(1,15), (2,2), (4,0), (20,0)}[ 13, 1, 0 ](z+1, z2+1, z16+1){2}4528 [?]5·285·285·29
{(1,15), (2,6), (4,0), (20,0)}[ 9, 3, 0 ](z+1, z2+1, z16+1){2}4527 [?]5·275·27
{(1,15), (2,10), (4,0), (20,0)}[ 5, 0 ](z3+z+1, z16+1){1}125{ 240, ...}{ ..., 20T68 }245·265·26
{(1,15), (2,14), (4,0), (20,0)}[ 1, 7, 0 ](z+1, z2+1, z16+1){2}45245·255·25
{(1,15), (4,0), (20,0)}[ 5, 0 ](z3+1, z16+1){2}45{ 80, ...}{ ..., 20T19 }255·265·26
17{(1,17), (2,2), (4,0), (20,0)}[ 15, 1, 0 ](z+1, z2+1, z16+1){2}4529 [?]5·295·295·210
{(1,17), (2,6), (4,0), (20,0)}[ 11, 3, 0 ](z+1, z2+1, z16+1){2}4528 [?]5·285·28
{(1,17), (2,10), (4,0), (20,0)}[ 7, 5, 0 ](z+1, z2+1, z16+1){2}4527 [?]5·275·27
{(1,17), (2,14), (4,0), (20,0)}[ 3, 7, 0 ](z+1, z2+1, z16+1){1}415255·265·26
{(1,17), (4,0), (20,0)}[ 17/3, 0 ](z+1, z16+1){1}415{ 960, ...}{ ..., 20T173 }255·275·27
19{(1,19), (2,2), (4,0), (20,0)}[ 17, 1, 0 ](z+1, z2+1, z16+1){2}45210 [?]5·2105·2105·211
{(1,19), (2,6), (4,0), (20,0)}[ 13, 3, 0 ](z+1, z2+1, z16+1){2}4529 [?]5·295·29
{(1,19), (2,10), (4,0), (20,0)}[ 9, 5, 0 ](z+1, z2+1, z16+1){2}4528 [?]5·285·28
{(1,19), (2,14), (4,0), (20,0)}[ 5, 7, 0 ](z+1, z2+1, z16+1){1}41526 [.]5·275·27
{(1,19), (2,18), (4,0), (20,0)}[ 1, 9, 0 ](z+1, z2+1, z16+1){1}41525 [.]5·265·26
{(1,19), (4,0), (20,0)}[ 19/3, 0 ](z+1, z16+1){1}415{ 960, ...}{ ..., 20T173 }265·285·28
21{(1,21), (2,2), (4,0), (20,0)}[ 19, 1, 0 ](z+1, z2+1, z16+1){22,2}45{ 640, ...}{ 20T137, ..., 20T129 }211 [?]5·2115·2115·212
{(1,21), (2,6), (4,0), (20,0)}[ 15, 3, 0 ](z+1, z2+1, z16+1){2}45210 [?]5·2105·210
{(1,21), (2,10), (4,0), (20,0)}[ 11, 5, 0 ](z+1, z2+1, z16+1){2}4529 [?]5·295·29
{(1,21), (2,14), (4,0), (20,0)}[ 7, 0 ](z3+z+1, z16+1){1}125265·285·28
{(1,21), (2,18), (4,0), (20,0)}[ 3, 9, 0 ](z+1, z2+1, z16+1){2}45265·275·27
22{(1,22), (2,2), (4,0), (20,0)}[ 20, 1, 0 ](z+1, z2+1, z16+1){2}45212 [?]5·2125·2125·212
23{(1,23), (2,6), (4,0), (20,0)}[ 17, 3, 0 ](z+1, z2+1, z16+1){22}45211 [?]5·2115·2115·212
{(1,23), (2,10), (4,0), (20,0)}[ 13, 5, 0 ](z+1, z2+1, z16+1){2}45210 [?]5·2105·210
{(1,23), (2,14), (4,0), (20,0)}[ 9, 7, 0 ](z+1, z2+1, z16+1){2}4529 [?]5·295·29
{(1,23), (2,18), (4,0), (20,0)}[ 5, 9, 0 ](z+1, z2+1, z16+1){1}41527 [.]5·285·28
{(1,23), (2,20), (4,0), (20,0)}[ 3, 10, 0 ](z+1, z2+1, z16+1){1}415{ 960, ...}{ ..., 20T173 }275·285·28
25{(1,25), (2,6), (4,0), (20,0)}[ 19, 3, 0 ](z+1, z2+1, z16+1){2}45212 [?]5·2125·2125·213
{(1,25), (2,10), (4,0), (20,0)}[ 15, 5, 0 ](z+1, z2+1, z16+1){22}45{ 80, ...}{ 20T16, ... }211 [?]5·2115·211
{(1,25), (2,14), (4,0), (20,0)}[ 11, 7, 0 ](z+1, z2+1, z16+1){2}45210 [?]5·2105·210
{(1,25), (2,18), (4,0), (20,0)}[ 7, 9, 0 ](z+1, z2+1, z16+1){1}41528 [.]5·295·29
{(1,25), (2,20), (4,0), (20,0)}[ 5, 10, 0 ](z+1, z2+1, z16+1){1}415{ 240, ...}{ ..., 20T61 }28 [.]5·295·29
26{(1,26), (2,6), (4,0), (20,0)}[ 20, 3, 0 ](z+1, z2+1, z16+1){2}45213 [?]5·2135·2135·213
27{(1,27), (2,10), (4,0), (20,0)}[ 17, 5, 0 ](z+1, z2+1, z16+1){2}45212 [?]5·2125·2125·213
{(1,27), (2,14), (4,0), (20,0)}[ 13, 7, 0 ](z+1, z2+1, z16+1){22}45211 [?]5·2115·211
{(1,27), (2,18), (4,0), (20,0)}[ 9, 0 ](z3+z+1, z16+1){1}125285·2105·210
{(1,27), (2,20), (4,0), (20,0)}[ 7, 10, 0 ](z+1, z2+1, z16+1){2}4529 [.]5·2105·210
29{(1,29), (2,10), (4,0), (20,0)}[ 19, 5, 0 ](z+1, z2+1, z16+1){2}45213 [?]5·2135·2135·214
{(1,29), (2,14), (4,0), (20,0)}[ 15, 7, 0 ](z+1, z2+1, z16+1){2}45212 [?]5·2125·212
{(1,29), (2,18), (4,0), (20,0)}[ 11, 9, 0 ](z+1, z2+1, z16+1){22}45{ 640, ...}{ 20T137, ... }211 [?]5·2115·211
{(1,29), (2,20), (4,0), (20,0)}[ 9, 10, 0 ](z+1, z2+1, z16+1){1}415210 [.]5·2115·211
30{(1,30), (2,10), (4,0), (20,0)}[ 20, 5, 0 ](z+1, z2+1, z16+1){2}45{ 160, ...}{ 20T42, ... }214 [?]5·2145·2145·214
31{(1,31), (2,14), (4,0), (20,0)}[ 17, 7, 0 ](z+1, z2+1, z16+1){2}45213 [?]5·2135·2135·214
{(1,31), (2,18), (4,0), (20,0)}[ 13, 9, 0 ](z+1, z2+1, z16+1){2}45212 [?]5·2125·212
{(1,31), (2,20), (4,0), (20,0)}[ 11, 10, 0 ](z+1, z2+1, z16+1){2}45212 [?]5·2125·212
33{(1,33), (2,14), (4,0), (20,0)}[ 19, 7, 0 ](z+1, z2+1, z16+1){2}45214 [?]5·2145·2145·215
{(1,33), (2,18), (4,0), (20,0)}[ 15, 9, 0 ](z+1, z2+1, z16+1){2}45213 [?]5·2135·213
{(1,33), (2,20), (4,0), (20,0)}[ 13, 10, 0 ](z+1, z2+1, z16+1){2}45213 [?]5·2135·213
34{(1,34), (2,14), (4,0), (20,0)}[ 20, 7, 0 ](z+1, z2+1, z16+1){2}45215 [?]5·2155·2155·215
35{(1,35), (2,18), (4,0), (20,0)}[ 17, 9, 0 ](z+1, z2+1, z16+1){2}45214 [?]5·2145·2145·215
{(1,35), (2,20), (4,0), (20,0)}[ 15, 10, 0 ](z+1, z2+1, z16+1){2}45{ 160, ...}{ 20T42, ... }214 [?]5·2145·214
37{(1,37), (2,18), (4,0), (20,0)}[ 19, 9, 0 ](z+1, z2+1, z16+1){2}45215 [?]5·2155·2155·216
{(1,37), (2,20), (4,0), (20,0)}[ 17, 10, 0 ](z+1, z2+1, z16+1){2}45215 [?]5·2155·215
38{(1,38), (2,18), (4,0), (20,0)}[ 20, 9, 0 ](z+1, z2+1, z16+1){2}45216 [?]5·2165·2165·216
39{(1,39), (2,20), (4,0), (20,0)}[ 19, 10, 0 ](z+1, z2+1, z16+1){2}45216 [?]5·2165·2165·216
40{(1,40), (2,20), (4,0), (20,0)}[ 20, 10, 0 ](z+1, z2+1, z16+1){2}45{ 160, ...}{ 20T42, ... }217 [?]5·2175·2175·217