↑

Learn more

Refine search

Degree $n$ Residue characteristic $p$ Discriminant exponent $c$ Galois group
Ramification index $e$ Residue field degree $f$ Galois tame degree $t$ Galois unram. degree $u$
Num. automorphisms Inertia subgroup Indices Assoc. Inertia Jump Set
Top Artin slope Galois Artin Visible Artin Wild inertia subgroup
Sort order Select

Results (1-50 of 186 matches)

Next   displayed columns for results
Label Polynomial $p$ $f$ $e$ $c$ Galois group $u$ $t$ Visible Artin slopes Visible Swan slopes Artin slope content Swan slope content Hidden Artin slopes Hidden Swan slopes Unram. Ext. Eisen. Poly. Ind. of Insep. Assoc. Inertia Resid. Poly Jump Set
2.4.1.0a1.1 $x^{4} + x + 1$ $2$ $4$ $1$ $0$ $C_4$ (as 4T1) $4$ $1$ $[\ ]$ $[\ ]$ $[\ ]^{4}$ $[\ ]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + t + 1$ $x - 2$ $[0]$ $[\ ]$ $[1]$
2.2.2.4a1.2 $( x^{2} + x + 1 )^{2} + 2 ( x^{2} + x + 1 ) + 4 x + 2$ $2$ $2$ $2$ $4$ $C_4$ (as 4T1) $2$ $1$ $[2]$ $[1]$ $[2]^{2}$ $[1]^{2}$ $[\ ]$ $[\ ]$ $t^{2} + t + 1$ $x^{2} + 2 x + 4 t + 2$ $[1, 0]$ $[1]$ $z + 1$ $[1, 4]$
2.2.2.6a1.2 $( x^{2} + x + 1 )^{2} + 8 x + 2$ $2$ $2$ $2$ $6$ $C_4$ (as 4T1) $2$ $1$ $[3]$ $[2]$ $[3]^{2}$ $[2]^{2}$ $[\ ]$ $[\ ]$ $t^{2} + t + 1$ $x^{2} + 8 t + 2$ $[2, 0]$ $[1]$ $z + 1$ $[1, 3]$
2.2.2.6a1.6 $( x^{2} + x + 1 )^{2} + 4 ( x^{2} + x + 1 ) + 8 x + 2$ $2$ $2$ $2$ $6$ $C_4$ (as 4T1) $2$ $1$ $[3]$ $[2]$ $[3]^{2}$ $[2]^{2}$ $[\ ]$ $[\ ]$ $t^{2} + t + 1$ $x^{2} + 4 x + 8 t + 2$ $[2, 0]$ $[1]$ $z + 1$ $[1, 3]$
2.1.4.11a1.9 $x^{4} + 4 x^{2} + 2$ $2$ $1$ $4$ $11$ $C_4$ (as 4T1) $1$ $1$ $[3, 4]$ $[2,3]$ $[3, 4]$ $[2,3]$ $[\ ]$ $[\ ]$ $t + 1$ $x^{4} + 4 x^{2} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.10 $x^{4} + 4 x^{2} + 18$ $2$ $1$ $4$ $11$ $C_4$ (as 4T1) $1$ $1$ $[3, 4]$ $[2,3]$ $[3, 4]$ $[2,3]$ $[\ ]$ $[\ ]$ $t + 1$ $x^{4} + 4 x^{2} + 18$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.11 $x^{4} + 8 x^{3} + 4 x^{2} + 2$ $2$ $1$ $4$ $11$ $C_4$ (as 4T1) $1$ $1$ $[3, 4]$ $[2,3]$ $[3, 4]$ $[2,3]$ $[\ ]$ $[\ ]$ $t + 1$ $x^{4} + 8 x^{3} + 4 x^{2} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.12 $x^{4} + 8 x^{3} + 4 x^{2} + 18$ $2$ $1$ $4$ $11$ $C_4$ (as 4T1) $1$ $1$ $[3, 4]$ $[2,3]$ $[3, 4]$ $[2,3]$ $[\ ]$ $[\ ]$ $t + 1$ $x^{4} + 8 x^{3} + 4 x^{2} + 18$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.15 $x^{4} + 4 x^{2} + 10$ $2$ $1$ $4$ $11$ $C_4$ (as 4T1) $1$ $1$ $[3, 4]$ $[2,3]$ $[3, 4]$ $[2,3]$ $[\ ]$ $[\ ]$ $t + 1$ $x^{4} + 4 x^{2} + 10$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.16 $x^{4} + 4 x^{2} + 26$ $2$ $1$ $4$ $11$ $C_4$ (as 4T1) $1$ $1$ $[3, 4]$ $[2,3]$ $[3, 4]$ $[2,3]$ $[\ ]$ $[\ ]$ $t + 1$ $x^{4} + 4 x^{2} + 26$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.17 $x^{4} + 8 x^{3} + 4 x^{2} + 10$ $2$ $1$ $4$ $11$ $C_4$ (as 4T1) $1$ $1$ $[3, 4]$ $[2,3]$ $[3, 4]$ $[2,3]$ $[\ ]$ $[\ ]$ $t + 1$ $x^{4} + 8 x^{3} + 4 x^{2} + 10$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.18 $x^{4} + 8 x^{3} + 4 x^{2} + 26$ $2$ $1$ $4$ $11$ $C_4$ (as 4T1) $1$ $1$ $[3, 4]$ $[2,3]$ $[3, 4]$ $[2,3]$ $[\ ]$ $[\ ]$ $t + 1$ $x^{4} + 8 x^{3} + 4 x^{2} + 26$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
3.4.1.0a1.1 $x^{4} + 2 x^{3} + 2$ $3$ $4$ $1$ $0$ $C_4$ (as 4T1) $4$ $1$ $[\ ]$ $[\ ]$ $[\ ]^{4}$ $[\ ]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 2 t^{3} + 2$ $x - 3$ $[0]$ $[\ ]$ undefined
3.2.2.2a1.1 $( x^{2} + 2 x + 2 )^{2} + 3 x$ $3$ $2$ $2$ $2$ $C_4$ (as 4T1) $2$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{2}$ $[\ ]_{2}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 2 t + 2$ $x^{2} + 3 t$ $[0]$ $[1]$ $z + 2$ undefined
5.4.1.0a1.1 $x^{4} + 4 x^{2} + 4 x + 2$ $5$ $4$ $1$ $0$ $C_4$ (as 4T1) $4$ $1$ $[\ ]$ $[\ ]$ $[\ ]^{4}$ $[\ ]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 4 t^{2} + 4 t + 2$ $x - 5$ $[0]$ $[\ ]$ undefined
5.2.2.2a1.1 $( x^{2} + 4 x + 2 )^{2} + 5 x$ $5$ $2$ $2$ $2$ $C_4$ (as 4T1) $2$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{2}$ $[\ ]_{2}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 4 t + 2$ $x^{2} + 5 t$ $[0]$ $[1]$ $z + 2$ undefined
5.1.4.3a1.1 $x^{4} + 5$ $5$ $1$ $4$ $3$ $C_4$ (as 4T1) $1$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}$ $[\ ]_{4}$ $[\ ]$ $[\ ]$ $t + 3$ $x^{4} + 5$ $[0]$ $[1]$ $z^3 + 4 z^2 + z + 4$ $[1]$
5.1.4.3a1.2 $x^{4} + 10$ $5$ $1$ $4$ $3$ $C_4$ (as 4T1) $1$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}$ $[\ ]_{4}$ $[\ ]$ $[\ ]$ $t + 3$ $x^{4} + 10$ $[0]$ $[1]$ $z^3 + 4 z^2 + z + 4$ undefined
5.1.4.3a1.3 $x^{4} + 15$ $5$ $1$ $4$ $3$ $C_4$ (as 4T1) $1$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}$ $[\ ]_{4}$ $[\ ]$ $[\ ]$ $t + 3$ $x^{4} + 15$ $[0]$ $[1]$ $z^3 + 4 z^2 + z + 4$ undefined
5.1.4.3a1.4 $x^{4} + 20$ $5$ $1$ $4$ $3$ $C_4$ (as 4T1) $1$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}$ $[\ ]_{4}$ $[\ ]$ $[\ ]$ $t + 3$ $x^{4} + 20$ $[0]$ $[1]$ $z^3 + 4 z^2 + z + 4$ undefined
7.4.1.0a1.1 $x^{4} + 5 x^{2} + 4 x + 3$ $7$ $4$ $1$ $0$ $C_4$ (as 4T1) $4$ $1$ $[\ ]$ $[\ ]$ $[\ ]^{4}$ $[\ ]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 5 t^{2} + 4 t + 3$ $x - 7$ $[0]$ $[\ ]$ undefined
7.2.2.2a1.1 $( x^{2} + 6 x + 3 )^{2} + 7 x$ $7$ $2$ $2$ $2$ $C_4$ (as 4T1) $2$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{2}$ $[\ ]_{2}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 6 t + 3$ $x^{2} + 7 t$ $[0]$ $[1]$ $z + 2$ undefined
11.4.1.0a1.1 $x^{4} + 8 x^{2} + 10 x + 2$ $11$ $4$ $1$ $0$ $C_4$ (as 4T1) $4$ $1$ $[\ ]$ $[\ ]$ $[\ ]^{4}$ $[\ ]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 8 t^{2} + 10 t + 2$ $x - 11$ $[0]$ $[\ ]$ undefined
11.2.2.2a1.1 $( x^{2} + 7 x + 2 )^{2} + 11 x$ $11$ $2$ $2$ $2$ $C_4$ (as 4T1) $2$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{2}$ $[\ ]_{2}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 7 t + 2$ $x^{2} + 11 t$ $[0]$ $[1]$ $z + 2$ undefined
13.4.1.0a1.1 $x^{4} + 3 x^{2} + 12 x + 2$ $13$ $4$ $1$ $0$ $C_4$ (as 4T1) $4$ $1$ $[\ ]$ $[\ ]$ $[\ ]^{4}$ $[\ ]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 3 t^{2} + 12 t + 2$ $x - 13$ $[0]$ $[\ ]$ undefined
13.2.2.2a1.1 $( x^{2} + 12 x + 2 )^{2} + 13 x$ $13$ $2$ $2$ $2$ $C_4$ (as 4T1) $2$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{2}$ $[\ ]_{2}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 12 t + 2$ $x^{2} + 13 t$ $[0]$ $[1]$ $z + 2$ undefined
13.1.4.3a1.1 $x^{4} + 13$ $13$ $1$ $4$ $3$ $C_4$ (as 4T1) $1$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}$ $[\ ]_{4}$ $[\ ]$ $[\ ]$ $t + 11$ $x^{4} + 13$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
13.1.4.3a1.2 $x^{4} + 26$ $13$ $1$ $4$ $3$ $C_4$ (as 4T1) $1$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}$ $[\ ]_{4}$ $[\ ]$ $[\ ]$ $t + 11$ $x^{4} + 26$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
13.1.4.3a1.3 $x^{4} + 52$ $13$ $1$ $4$ $3$ $C_4$ (as 4T1) $1$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}$ $[\ ]_{4}$ $[\ ]$ $[\ ]$ $t + 11$ $x^{4} + 52$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
13.1.4.3a1.4 $x^{4} + 104$ $13$ $1$ $4$ $3$ $C_4$ (as 4T1) $1$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}$ $[\ ]_{4}$ $[\ ]$ $[\ ]$ $t + 11$ $x^{4} + 104$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
17.4.1.0a1.1 $x^{4} + 7 x^{2} + 10 x + 3$ $17$ $4$ $1$ $0$ $C_4$ (as 4T1) $4$ $1$ $[\ ]$ $[\ ]$ $[\ ]^{4}$ $[\ ]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 7 t^{2} + 10 t + 3$ $x - 17$ $[0]$ $[\ ]$ undefined
17.2.2.2a1.1 $( x^{2} + 16 x + 3 )^{2} + 17 x$ $17$ $2$ $2$ $2$ $C_4$ (as 4T1) $2$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{2}$ $[\ ]_{2}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 16 t + 3$ $x^{2} + 17 t$ $[0]$ $[1]$ $z + 2$ undefined
17.1.4.3a1.1 $x^{4} + 17$ $17$ $1$ $4$ $3$ $C_4$ (as 4T1) $1$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}$ $[\ ]_{4}$ $[\ ]$ $[\ ]$ $t + 14$ $x^{4} + 17$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
17.1.4.3a1.2 $x^{4} + 51$ $17$ $1$ $4$ $3$ $C_4$ (as 4T1) $1$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}$ $[\ ]_{4}$ $[\ ]$ $[\ ]$ $t + 14$ $x^{4} + 51$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
17.1.4.3a1.3 $x^{4} + 153$ $17$ $1$ $4$ $3$ $C_4$ (as 4T1) $1$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}$ $[\ ]_{4}$ $[\ ]$ $[\ ]$ $t + 14$ $x^{4} + 153$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
17.1.4.3a1.4 $x^{4} + 170$ $17$ $1$ $4$ $3$ $C_4$ (as 4T1) $1$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}$ $[\ ]_{4}$ $[\ ]$ $[\ ]$ $t + 14$ $x^{4} + 170$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
19.4.1.0a1.1 $x^{4} + 2 x^{2} + 11 x + 2$ $19$ $4$ $1$ $0$ $C_4$ (as 4T1) $4$ $1$ $[\ ]$ $[\ ]$ $[\ ]^{4}$ $[\ ]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 2 t^{2} + 11 t + 2$ $x - 19$ $[0]$ $[\ ]$ undefined
19.2.2.2a1.1 $( x^{2} + 18 x + 2 )^{2} + 19 x$ $19$ $2$ $2$ $2$ $C_4$ (as 4T1) $2$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{2}$ $[\ ]_{2}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 18 t + 2$ $x^{2} + 19 t$ $[0]$ $[1]$ $z + 2$ undefined
23.4.1.0a1.1 $x^{4} + 3 x^{2} + 19 x + 5$ $23$ $4$ $1$ $0$ $C_4$ (as 4T1) $4$ $1$ $[\ ]$ $[\ ]$ $[\ ]^{4}$ $[\ ]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 3 t^{2} + 19 t + 5$ $x - 23$ $[0]$ $[\ ]$ undefined
23.2.2.2a1.1 $( x^{2} + 21 x + 5 )^{2} + 23 x$ $23$ $2$ $2$ $2$ $C_4$ (as 4T1) $2$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{2}$ $[\ ]_{2}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 21 t + 5$ $x^{2} + 23 t$ $[0]$ $[1]$ $z + 2$ undefined
29.4.1.0a1.1 $x^{4} + 2 x^{2} + 15 x + 2$ $29$ $4$ $1$ $0$ $C_4$ (as 4T1) $4$ $1$ $[\ ]$ $[\ ]$ $[\ ]^{4}$ $[\ ]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 2 t^{2} + 15 t + 2$ $x - 29$ $[0]$ $[\ ]$ undefined
29.2.2.2a1.1 $( x^{2} + 24 x + 2 )^{2} + 29 x$ $29$ $2$ $2$ $2$ $C_4$ (as 4T1) $2$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{2}$ $[\ ]_{2}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 24 t + 2$ $x^{2} + 29 t$ $[0]$ $[1]$ $z + 2$ undefined
29.1.4.3a1.1 $x^{4} + 29$ $29$ $1$ $4$ $3$ $C_4$ (as 4T1) $1$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}$ $[\ ]_{4}$ $[\ ]$ $[\ ]$ $t + 27$ $x^{4} + 29$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
29.1.4.3a1.2 $x^{4} + 58$ $29$ $1$ $4$ $3$ $C_4$ (as 4T1) $1$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}$ $[\ ]_{4}$ $[\ ]$ $[\ ]$ $t + 27$ $x^{4} + 58$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
29.1.4.3a1.3 $x^{4} + 116$ $29$ $1$ $4$ $3$ $C_4$ (as 4T1) $1$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}$ $[\ ]_{4}$ $[\ ]$ $[\ ]$ $t + 27$ $x^{4} + 116$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
29.1.4.3a1.4 $x^{4} + 232$ $29$ $1$ $4$ $3$ $C_4$ (as 4T1) $1$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}$ $[\ ]_{4}$ $[\ ]$ $[\ ]$ $t + 27$ $x^{4} + 232$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
31.4.1.0a1.1 $x^{4} + 3 x^{2} + 16 x + 3$ $31$ $4$ $1$ $0$ $C_4$ (as 4T1) $4$ $1$ $[\ ]$ $[\ ]$ $[\ ]^{4}$ $[\ ]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 3 t^{2} + 16 t + 3$ $x - 31$ $[0]$ $[\ ]$ undefined
31.2.2.2a1.1 $( x^{2} + 29 x + 3 )^{2} + 31 x$ $31$ $2$ $2$ $2$ $C_4$ (as 4T1) $2$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{2}$ $[\ ]_{2}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 29 t + 3$ $x^{2} + 31 t$ $[0]$ $[1]$ $z + 2$ undefined
37.4.1.0a1.1 $x^{4} + 6 x^{2} + 24 x + 2$ $37$ $4$ $1$ $0$ $C_4$ (as 4T1) $4$ $1$ $[\ ]$ $[\ ]$ $[\ ]^{4}$ $[\ ]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 6 t^{2} + 24 t + 2$ $x - 37$ $[0]$ $[\ ]$ undefined
37.2.2.2a1.1 $( x^{2} + 33 x + 2 )^{2} + 37 x$ $37$ $2$ $2$ $2$ $C_4$ (as 4T1) $2$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{2}$ $[\ ]_{2}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 33 t + 2$ $x^{2} + 37 t$ $[0]$ $[1]$ $z + 2$ undefined
Next   displayed columns for results