Number Theory Tables, Department of Mathematics and Statistics, UNCG

Number of totally ramified extensions of Q2 of degree 12

Introduction

Polynomial Invariants #Aut Splitting Field Number of
j Ramification Polygon Slopes Residual Polynomials fT eT #Gal Gal Polynomials Extensions
1{(1,1), (4,0), (12,0)}[ 1/3, 0 ](z+1, z8+1){1}69{ 3456 }{ 12T254 }13·223·223·22
3{(1,3), (2,2), (4,0), (12,0)}[ 1, 0 ](z3+z+1, z8+1){1}6313·223·223·23
{(1,3), (4,0), (12,0)}[ 1, 0 ](z3+1, z8+1){2}23{ 24, 48 }{ 12T8, 12T27 }23·223·22
5{(1,5), (2,2), (4,0), (12,0)}[ 3, 1, 0 ](z+1, z2+1, z8+1){2,4}23{ 96, 192, ...}{ 12T64, ..., 12T98, 12T62, 12T65 }23 [3·2]3·233·233·24
{(1,5), (4,0), (12,0)}[ 5/3, 0 ](z+1, z8+1){1}69{ 3456 }{ 12T254 }23·233·23
7{(1,7), (2,2), (4,0), (12,0)}[ 5, 1, 0 ](z+1, z2+1, z8+1){2,4}23{ 48, 192, ...}{ ..., 12T24, 12T97, 12T27, 12T98, 12T23, 12T96, 12T22 }24 [3·22]3·243·243·25
{(1,7), (2,6), (4,0), (12,0)}[ 1, 3, 0 ](z+1, z2+1, z8+1){1}69{ 3456 }{ 12T254 }223·233·23
{(1,7), (4,0), (12,0)}[ 7/3, 0 ](z+1, z8+1){1}69{ 3456 }{ 12T254 }223·243·24
9{(1,9), (2,2), (4,0), (12,0)}[ 7, 1, 0 ](z+1, z2+1, z8+1){2}2325 [24]3·253·253·26
{(1,9), (2,6), (4,0), (12,0)}[ 3, 0 ](z3+z+1, z8+1){1}63{ 72, 1152 }{ 12T43, 12T206 }223·243·24
{(1,9), (4,0), (12,0)}[ 3, 0 ](z3+1, z8+1){2}23{ 24, 96 }{ 12T13, 12T52, 12T49 }233·243·24
11{(1,11), (2,2), (4,0), (12,0)}[ 9, 1, 0 ](z+1, z2+1, z8+1){2,4}23{ 96, 192, ...}{ 12T100, 12T96, 12T67, ..., 12T65, 12T103, 12T98, 12T68, 12T97, 12T66, 12T64, 12T62, 12T102 }26 [5·23]3·263·263·27
{(1,11), (2,6), (4,0), (12,0)}[ 5, 3, 0 ](z+1, z2+1, z8+1){2}23{ 192, ...}{ 12T113, ..., 12T109 }25 [24]3·253·25
{(1,11), (2,10), (4,0), (12,0)}[ 1, 5, 0 ](z+1, z2+1, z8+1){1}69{ 3456 }{ 12T254 }233·243·24
{(1,11), (4,0), (12,0)}[ 11/3, 0 ](z+1, z8+1){1}69{ 3456 }{ 12T254 }233·253·25
13{(1,13), (2,2), (4,0), (12,0)}[ 11, 1, 0 ](z+1, z2+1, z8+1){2,4}23{ 48, 192, ...}{ 12T100, 12T96, ..., 12T27, 12T23, 12T103, 12T98, 12T97, 12T24, 12T22, 12T102 }27 [5·24]3·273·273·28
{(1,13), (2,6), (4,0), (12,0)}[ 7, 3, 0 ](z+1, z2+1, z8+1){2,4}23{ 48, 192, 384, ...}{ 12T113, ..., 12T24, 12T23, 12T109, 12T148, 12T101 }26 [5·23]3·263·26
{(1,13), (2,10), (4,0), (12,0)}[ 3, 5, 0 ](z+1, z2+1, z8+1){1}69{ 3456 }{ 12T254 }24 [.]3·253·25
14{(1,14), (2,2), (4,0), (12,0)}[ 12, 1, 0 ](z+1, z2+1, z8+1){2}2328 [27]3·283·283·28
15{(1,15), (2,6), (4,0), (12,0)}[ 9, 3, 0 ](z+1, z2+1, z8+1){2,4}23{ 24, 96, 384, ...}{ 12T13, ..., 12T49, 12T48, 12T148, 12T10 }27 [5·24]3·273·273·28
{(1,15), (2,10), (4,0), (12,0)}[ 5, 0 ](z3+z+1, z8+1){1}63{ 288, 1152 }{ 12T128, 12T206 }243·263·26
{(1,15), (2,12), (4,0), (12,0)}[ 3, 6, 0 ](z+1, z2+1, z8+1){2}23{ 24, 48, 96, 192, 384 }{ 12T8, 12T100, 12T49, 12T66, 12T148, 12T22 }25 [.]3·263·26
17{(1,17), (2,6), (4,0), (12,0)}[ 11, 3, 0 ](z+1, z2+1, z8+1){2}23{ 192, ...}{ ..., 12T109 }28 [27]3·283·283·29
{(1,17), (2,10), (4,0), (12,0)}[ 7, 5, 0 ](z+1, z2+1, z8+1){2,4}23{ 48, 96, 192, 768, ...}{ ..., 12T24, 12T103, 12T184, 12T23, 12T48 }27 [3·25]3·273·27
{(1,17), (2,12), (4,0), (12,0)}[ 5, 6, 0 ](z+1, z2+1, z8+1){1}6926 [.]3·273·27
18{(1,18), (2,6), (4,0), (12,0)}[ 12, 3, 0 ](z+1, z2+1, z8+1){2,3,4,6,12}2329 [?]3·293·293·29
19{(1,19), (2,10), (4,0), (12,0)}[ 9, 5, 0 ](z+1, z2+1, z8+1){2}2328 [27]3·283·283·29
{(1,19), (2,12), (4,0), (12,0)}[ 7, 6, 0 ](z+1, z2+1, z8+1){2}23{ 192, ...}{ 12T113, ..., 12T109 }28 [27]3·283·28
21{(1,21), (2,10), (4,0), (12,0)}[ 11, 5, 0 ](z+1, z2+1, z8+1){2,3,4,6,12}2329 [?]3·293·293·210
{(1,21), (2,12), (4,0), (12,0)}[ 9, 6, 0 ](z+1, z2+1, z8+1){2,3,4,6,12}2329 [?]3·293·29
22{(1,22), (2,10), (4,0), (12,0)}[ 12, 5, 0 ](z+1, z2+1, z8+1){2,3,4,6,12}23210 [?]3·2103·2103·210
23{(1,23), (2,12), (4,0), (12,0)}[ 11, 6, 0 ](z+1, z2+1, z8+1){2,3,4,6,12}23210 [?]3·2103·2103·210
24{(1,24), (2,12), (4,0), (12,0)}[ 12, 6, 0 ](z+1, z2+1, z8+1){2,3,4,6,12}23211 [?]3·2113·2113·211