↑

Learn more

Note: Search results may be incomplete due to uncomputed quantities: galois_label (239087 objects)

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 (25 matches)

  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
2.4.4.44a1.1147 $( x^{4} + x + 1 )^{4} + 4 ( x^{4} + x + 1 )^{2} + 2$ $2$ $4$ $4$ $44$ $C_4^2$ (as 16T4) $4$ $1$ $[3, 4]$ $[2,3]$ $[3, 4]^{4}$ $[2,3]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + t + 1$ $x^{4} + 4 x^{2} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1155 $( x^{4} + x + 1 )^{4} + 8 ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 2$ $2$ $4$ $4$ $44$ $C_4^2$ (as 16T4) $4$ $1$ $[3, 4]$ $[2,3]$ $[3, 4]^{4}$ $[2,3]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + t + 1$ $x^{4} + 8 x^{3} + 4 x^{2} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.8.48c1.402 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + \left(8 x + 12\right) ( x^{2} + x + 1 ) + 12 x + 2$ $2$ $2$ $8$ $48$ $C_4^2$ (as 16T4) $2$ $1$ $[2, 3, 4]$ $[1,2,3]$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]$ $[\ ]$ $t^{2} + t + 1$ $x^{8} + 8 t x^{7} + 4 t x^{6} + 8 x^{5} + 2 x^{4} + \left(8 t + 8\right) x^{3} + 4 x^{2} + 8 x + 12 t + 2$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.2.8.48c1.403 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + \left(8 x + 12\right) ( x^{2} + x + 1 ) + 28 x + 2$ $2$ $2$ $8$ $48$ $C_4^2$ (as 16T4) $2$ $1$ $[2, 3, 4]$ $[1,2,3]$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]$ $[\ ]$ $t^{2} + t + 1$ $x^{8} + 8 t x^{7} + 4 t x^{6} + 8 x^{5} + 2 x^{4} + \left(8 t + 8\right) x^{3} + 4 x^{2} + 8 x + 28 t + 2$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
5.4.4.12a1.4 $( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ $5$ $4$ $4$ $12$ $C_4^2$ (as 16T4) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 4 t^{2} + 4 t + 2$ $x^{4} + 5$ $[0]$ $[1]$ $z^3 + 4 z^2 + z + 4$ $[1]$
13.4.4.12a1.4 $( x^{4} + 3 x^{2} + 12 x + 2 )^{4} + 13$ $13$ $4$ $4$ $12$ $C_4^2$ (as 16T4) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 3 t^{2} + 12 t + 2$ $x^{4} + 13$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
17.4.4.12a1.4 $( x^{4} + 7 x^{2} + 10 x + 3 )^{4} + 17$ $17$ $4$ $4$ $12$ $C_4^2$ (as 16T4) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 7 t^{2} + 10 t + 3$ $x^{4} + 17$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
29.4.4.12a1.4 $( x^{4} + 2 x^{2} + 15 x + 2 )^{4} + 29$ $29$ $4$ $4$ $12$ $C_4^2$ (as 16T4) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 2 t^{2} + 15 t + 2$ $x^{4} + 29$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
37.4.4.12a1.4 $( x^{4} + 6 x^{2} + 24 x + 2 )^{4} + 37$ $37$ $4$ $4$ $12$ $C_4^2$ (as 16T4) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 6 t^{2} + 24 t + 2$ $x^{4} + 37$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
41.4.4.12a1.4 $( x^{4} + 23 x + 6 )^{4} + 41$ $41$ $4$ $4$ $12$ $C_4^2$ (as 16T4) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 23 t + 6$ $x^{4} + 41$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
53.4.4.12a1.4 $( x^{4} + 9 x^{2} + 38 x + 2 )^{4} + 53$ $53$ $4$ $4$ $12$ $C_4^2$ (as 16T4) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 9 t^{2} + 38 t + 2$ $x^{4} + 53$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
61.4.4.12a1.4 $( x^{4} + 3 x^{2} + 40 x + 2 )^{4} + 61$ $61$ $4$ $4$ $12$ $C_4^2$ (as 16T4) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 3 t^{2} + 40 t + 2$ $x^{4} + 61$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
73.4.4.12a1.4 $( x^{4} + 16 x^{2} + 56 x + 5 )^{4} + 73$ $73$ $4$ $4$ $12$ $C_4^2$ (as 16T4) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 16 t^{2} + 56 t + 5$ $x^{4} + 73$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
89.4.4.12a1.4 $( x^{4} + 4 x^{2} + 72 x + 3 )^{4} + 89$ $89$ $4$ $4$ $12$ $C_4^2$ (as 16T4) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 4 t^{2} + 72 t + 3$ $x^{4} + 89$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
97.4.4.12a1.4 $( x^{4} + 6 x^{2} + 80 x + 5 )^{4} + 97$ $97$ $4$ $4$ $12$ $C_4^2$ (as 16T4) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 6 t^{2} + 80 t + 5$ $x^{4} + 97$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
101.4.4.12a1.4 $( x^{4} + x^{2} + 78 x + 2 )^{4} + 101$ $101$ $4$ $4$ $12$ $C_4^2$ (as 16T4) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + t^{2} + 78 t + 2$ $x^{4} + 101$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
109.4.4.12a1.4 $( x^{4} + 11 x^{2} + 98 x + 6 )^{4} + 109$ $109$ $4$ $4$ $12$ $C_4^2$ (as 16T4) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 11 t^{2} + 98 t + 6$ $x^{4} + 109$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
113.4.4.12a1.4 $( x^{4} + 62 x + 3 )^{4} + 113$ $113$ $4$ $4$ $12$ $C_4^2$ (as 16T4) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 62 t + 3$ $x^{4} + 113$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
137.4.4.12a1.4 $( x^{4} + x^{2} + 95 x + 3 )^{4} + 137$ $137$ $4$ $4$ $12$ $C_4^2$ (as 16T4) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + t^{2} + 95 t + 3$ $x^{4} + 137$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
149.4.4.12a1.4 $( x^{4} + 7 x^{2} + 107 x + 2 )^{4} + 149$ $149$ $4$ $4$ $12$ $C_4^2$ (as 16T4) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 7 t^{2} + 107 t + 2$ $x^{4} + 149$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
157.4.4.12a1.4 $( x^{4} + 11 x^{2} + 136 x + 5 )^{4} + 157$ $157$ $4$ $4$ $12$ $C_4^2$ (as 16T4) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 11 t^{2} + 136 t + 5$ $x^{4} + 157$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
173.4.4.12a1.4 $( x^{4} + x^{2} + 102 x + 2 )^{4} + 173$ $173$ $4$ $4$ $12$ $C_4^2$ (as 16T4) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + t^{2} + 102 t + 2$ $x^{4} + 173$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
181.4.4.12a1.4 $( x^{4} + 6 x^{2} + 105 x + 2 )^{4} + 181$ $181$ $4$ $4$ $12$ $C_4^2$ (as 16T4) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 6 t^{2} + 105 t + 2$ $x^{4} + 181$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
193.4.4.12a1.4 $( x^{4} + 6 x^{2} + 148 x + 5 )^{4} + 193$ $193$ $4$ $4$ $12$ $C_4^2$ (as 16T4) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 6 t^{2} + 148 t + 5$ $x^{4} + 193$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
197.4.4.12a1.4 $( x^{4} + 16 x^{2} + 124 x + 2 )^{4} + 197$ $197$ $4$ $4$ $12$ $C_4^2$ (as 16T4) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 16 t^{2} + 124 t + 2$ $x^{4} + 197$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
  displayed columns for results