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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
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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
2.6.1.0a1.1 $x^{6} + x^{4} + x^{3} + x + 1$ $2$ $6$ $1$ $0$ $C_6$ (as 6T1) $6$ $1$ $[\ ]$ $[\ ]$ $[\ ]^{6}$ $[\ ]^{6}$ $[\ ]$ $[\ ]$ $t^{6} + t^{4} + t^{3} + t + 1$ $x - 2$ $[0]$ $[\ ]$ $[1]$
2.3.2.6a1.1 $( x^{3} + x + 1 )^{2} + 2 ( x^{3} + x + 1 ) + 2$ $2$ $3$ $2$ $6$ $C_6$ (as 6T1) $3$ $1$ $[2]$ $[1]$ $[2]^{3}$ $[1]^{3}$ $[\ ]$ $[\ ]$ $t^{3} + t + 1$ $x^{2} + 2 x + 2$ $[1, 0]$ $[1]$ $z + (t^2 + 1)$ $[1, 2]$
2.3.2.6a1.2 $( x^{3} + x + 1 )^{2} + 2 ( x^{3} + x + 1 ) + 6$ $2$ $3$ $2$ $6$ $C_6$ (as 6T1) $3$ $1$ $[2]$ $[1]$ $[2]^{3}$ $[1]^{3}$ $[\ ]$ $[\ ]$ $t^{3} + t + 1$ $x^{2} + 2 x + 6$ $[1, 0]$ $[1]$ $z + (t^2 + 1)$ $[1, 4]$
2.3.2.9a1.1 $( x^{3} + x + 1 )^{2} + 2$ $2$ $3$ $2$ $9$ $C_6$ (as 6T1) $3$ $1$ $[3]$ $[2]$ $[3]^{3}$ $[2]^{3}$ $[\ ]$ $[\ ]$ $t^{3} + t + 1$ $x^{2} + 2$ $[2, 0]$ $[1]$ $z + (t^2 + t + 1)$ $[1, 3]$
2.3.2.9a1.2 $( x^{3} + x + 1 )^{2} + 10$ $2$ $3$ $2$ $9$ $C_6$ (as 6T1) $3$ $1$ $[3]$ $[2]$ $[3]^{3}$ $[2]^{3}$ $[\ ]$ $[\ ]$ $t^{3} + t + 1$ $x^{2} + 10$ $[2, 0]$ $[1]$ $z + (t^2 + t + 1)$ $[1, 3]$
2.3.2.9a1.5 $( x^{3} + x + 1 )^{2} + 4 ( x^{3} + x + 1 ) + 2$ $2$ $3$ $2$ $9$ $C_6$ (as 6T1) $3$ $1$ $[3]$ $[2]$ $[3]^{3}$ $[2]^{3}$ $[\ ]$ $[\ ]$ $t^{3} + t + 1$ $x^{2} + 4 x + 2$ $[2, 0]$ $[1]$ $z + (t^2 + t + 1)$ $[1, 3]$
2.3.2.9a1.6 $( x^{3} + x + 1 )^{2} + 4 ( x^{3} + x + 1 ) + 10$ $2$ $3$ $2$ $9$ $C_6$ (as 6T1) $3$ $1$ $[3]$ $[2]$ $[3]^{3}$ $[2]^{3}$ $[\ ]$ $[\ ]$ $t^{3} + t + 1$ $x^{2} + 4 x + 10$ $[2, 0]$ $[1]$ $z + (t^2 + t + 1)$ $[1, 3]$
3.6.1.0a1.1 $x^{6} + 2 x^{4} + x^{2} + 2 x + 2$ $3$ $6$ $1$ $0$ $C_6$ (as 6T1) $6$ $1$ $[\ ]$ $[\ ]$ $[\ ]^{6}$ $[\ ]^{6}$ $[\ ]$ $[\ ]$ $t^{6} + 2 t^{4} + t^{2} + 2 t + 2$ $x - 3$ $[0]$ $[\ ]$ undefined
3.3.2.3a1.1 $( x^{3} + 2 x + 1 )^{2} + 3 x$ $3$ $3$ $2$ $3$ $C_6$ (as 6T1) $3$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{3}$ $[\ ]_{2}^{3}$ $[\ ]$ $[\ ]$ $t^{3} + 2 t + 1$ $x^{2} + 3 t$ $[0]$ $[1]$ $z + 2$ undefined
3.3.2.3a1.2 $( x^{3} + 2 x + 1 )^{2} + 3$ $3$ $3$ $2$ $3$ $C_6$ (as 6T1) $3$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{3}$ $[\ ]_{2}^{3}$ $[\ ]$ $[\ ]$ $t^{3} + 2 t + 1$ $x^{2} + 3$ $[0]$ $[1]$ $z + 2$ $[1]$
3.2.3.8a2.1 $( x^{2} + 2 x + 2 )^{3} + 6 ( x^{2} + 2 x + 2 )^{2} + 3$ $3$ $2$ $3$ $8$ $C_6$ (as 6T1) $2$ $1$ $[2]$ $[1]$ $[2]^{2}$ $[1]^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 2 t + 2$ $x^{3} + 6 x^{2} + 3$ $[2, 0]$ $[1]$ $z^2 + 1$ undefined
3.2.3.8a2.2 $( x^{2} + 2 x + 2 )^{3} + 6 ( x^{2} + 2 x + 2 )^{2} + 12$ $3$ $2$ $3$ $8$ $C_6$ (as 6T1) $2$ $1$ $[2]$ $[1]$ $[2]^{2}$ $[1]^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 2 t + 2$ $x^{3} + 6 x^{2} + 12$ $[2, 0]$ $[1]$ $z^2 + 1$ undefined
3.2.3.8a2.3 $( x^{2} + 2 x + 2 )^{3} + 6 ( x^{2} + 2 x + 2 )^{2} + 21$ $3$ $2$ $3$ $8$ $C_6$ (as 6T1) $2$ $1$ $[2]$ $[1]$ $[2]^{2}$ $[1]^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 2 t + 2$ $x^{3} + 6 x^{2} + 21$ $[2, 0]$ $[1]$ $z^2 + 1$ undefined
3.1.6.9a1.1 $x^{6} + 6 x^{4} + 3$ $3$ $1$ $6$ $9$ $C_6$ (as 6T1) $1$ $2$ $[2]$ $[1]$ $[2]_{2}$ $[1]_{2}$ $[\ ]$ $[\ ]$ $t + 1$ $x^{6} + 6 x^{4} + 3$ $[4, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ $[1, 3]$
3.1.6.9a1.2 $x^{6} + 6 x^{4} + 12$ $3$ $1$ $6$ $9$ $C_6$ (as 6T1) $1$ $2$ $[2]$ $[1]$ $[2]_{2}$ $[1]_{2}$ $[\ ]$ $[\ ]$ $t + 1$ $x^{6} + 6 x^{4} + 12$ $[4, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ $[1, 9]$
3.1.6.9a1.3 $x^{6} + 6 x^{4} + 21$ $3$ $1$ $6$ $9$ $C_6$ (as 6T1) $1$ $2$ $[2]$ $[1]$ $[2]_{2}$ $[1]_{2}$ $[\ ]$ $[\ ]$ $t + 1$ $x^{6} + 6 x^{4} + 21$ $[4, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ $[1, 9]$
3.1.6.9a1.7 $x^{6} + 3 x^{4} + 6$ $3$ $1$ $6$ $9$ $C_6$ (as 6T1) $1$ $2$ $[2]$ $[1]$ $[2]_{2}$ $[1]_{2}$ $[\ ]$ $[\ ]$ $t + 1$ $x^{6} + 3 x^{4} + 6$ $[4, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ undefined
3.1.6.9a1.8 $x^{6} + 3 x^{4} + 15$ $3$ $1$ $6$ $9$ $C_6$ (as 6T1) $1$ $2$ $[2]$ $[1]$ $[2]_{2}$ $[1]_{2}$ $[\ ]$ $[\ ]$ $t + 1$ $x^{6} + 3 x^{4} + 15$ $[4, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ undefined
3.1.6.9a1.9 $x^{6} + 3 x^{4} + 24$ $3$ $1$ $6$ $9$ $C_6$ (as 6T1) $1$ $2$ $[2]$ $[1]$ $[2]_{2}$ $[1]_{2}$ $[\ ]$ $[\ ]$ $t + 1$ $x^{6} + 3 x^{4} + 24$ $[4, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ undefined
5.6.1.0a1.1 $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ $5$ $6$ $1$ $0$ $C_6$ (as 6T1) $6$ $1$ $[\ ]$ $[\ ]$ $[\ ]^{6}$ $[\ ]^{6}$ $[\ ]$ $[\ ]$ $t^{6} + t^{4} + 4 t^{3} + t^{2} + 2$ $x - 5$ $[0]$ $[\ ]$ undefined
5.3.2.3a1.1 $( x^{3} + 3 x + 3 )^{2} + 5 x$ $5$ $3$ $2$ $3$ $C_6$ (as 6T1) $3$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{3}$ $[\ ]_{2}^{3}$ $[\ ]$ $[\ ]$ $t^{3} + 3 t + 3$ $x^{2} + 5 t$ $[0]$ $[1]$ $z + 2$ undefined
5.3.2.3a1.2 $( x^{3} + 3 x + 3 )^{2} + 5$ $5$ $3$ $2$ $3$ $C_6$ (as 6T1) $3$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{3}$ $[\ ]_{2}^{3}$ $[\ ]$ $[\ ]$ $t^{3} + 3 t + 3$ $x^{2} + 5$ $[0]$ $[1]$ $z + 2$ undefined
7.6.1.0a1.1 $x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$ $7$ $6$ $1$ $0$ $C_6$ (as 6T1) $6$ $1$ $[\ ]$ $[\ ]$ $[\ ]^{6}$ $[\ ]^{6}$ $[\ ]$ $[\ ]$ $t^{6} + t^{4} + 5 t^{3} + 4 t^{2} + 6 t + 3$ $x - 7$ $[0]$ $[\ ]$ undefined
7.3.2.3a1.1 $( x^{3} + 6 x^{2} + 4 )^{2} + 7 x$ $7$ $3$ $2$ $3$ $C_6$ (as 6T1) $3$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{3}$ $[\ ]_{2}^{3}$ $[\ ]$ $[\ ]$ $t^{3} + 6 t^{2} + 4$ $x^{2} + 7 t$ $[0]$ $[1]$ $z + 2$ undefined
7.3.2.3a1.2 $( x^{3} + 6 x^{2} + 4 )^{2} + 7$ $7$ $3$ $2$ $3$ $C_6$ (as 6T1) $3$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{3}$ $[\ ]_{2}^{3}$ $[\ ]$ $[\ ]$ $t^{3} + 6 t^{2} + 4$ $x^{2} + 7$ $[0]$ $[1]$ $z + 2$ undefined
7.2.3.4a1.1 $( x^{2} + 6 x + 3 )^{3} + 7 x$ $7$ $2$ $3$ $4$ $C_6$ (as 6T1) $2$ $3$ $[\ ]$ $[\ ]$ $[\ ]_{3}^{2}$ $[\ ]_{3}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 6 t + 3$ $x^{3} + 7 t$ $[0]$ $[1]$ $z^2 + 3 z + 3$ undefined
7.2.3.4a1.2 $( x^{2} + 6 x + 3 )^{3} + 7$ $7$ $2$ $3$ $4$ $C_6$ (as 6T1) $2$ $3$ $[\ ]$ $[\ ]$ $[\ ]_{3}^{2}$ $[\ ]_{3}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 6 t + 3$ $x^{3} + 7$ $[0]$ $[1]$ $z^2 + 3 z + 3$ undefined
7.2.3.4a1.3 $( x^{2} + 6 x + 3 )^{3} + 7 x + 28$ $7$ $2$ $3$ $4$ $C_6$ (as 6T1) $2$ $3$ $[\ ]$ $[\ ]$ $[\ ]_{3}^{2}$ $[\ ]_{3}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 6 t + 3$ $x^{3} + 7 t + 28$ $[0]$ $[1]$ $z^2 + 3 z + 3$ undefined
7.1.6.5a1.1 $x^{6} + 7$ $7$ $1$ $6$ $5$ $C_6$ (as 6T1) $1$ $6$ $[\ ]$ $[\ ]$ $[\ ]_{6}$ $[\ ]_{6}$ $[\ ]$ $[\ ]$ $t + 4$ $x^{6} + 7$ $[0]$ $[1]$ $z^5 + 6 z^4 + z^3 + 6 z^2 + z + 6$ $[1]$
7.1.6.5a1.2 $x^{6} + 14$ $7$ $1$ $6$ $5$ $C_6$ (as 6T1) $1$ $6$ $[\ ]$ $[\ ]$ $[\ ]_{6}$ $[\ ]_{6}$ $[\ ]$ $[\ ]$ $t + 4$ $x^{6} + 14$ $[0]$ $[1]$ $z^5 + 6 z^4 + z^3 + 6 z^2 + z + 6$ undefined
7.1.6.5a1.3 $x^{6} + 21$ $7$ $1$ $6$ $5$ $C_6$ (as 6T1) $1$ $6$ $[\ ]$ $[\ ]$ $[\ ]_{6}$ $[\ ]_{6}$ $[\ ]$ $[\ ]$ $t + 4$ $x^{6} + 21$ $[0]$ $[1]$ $z^5 + 6 z^4 + z^3 + 6 z^2 + z + 6$ undefined
7.1.6.5a1.4 $x^{6} + 28$ $7$ $1$ $6$ $5$ $C_6$ (as 6T1) $1$ $6$ $[\ ]$ $[\ ]$ $[\ ]_{6}$ $[\ ]_{6}$ $[\ ]$ $[\ ]$ $t + 4$ $x^{6} + 28$ $[0]$ $[1]$ $z^5 + 6 z^4 + z^3 + 6 z^2 + z + 6$ undefined
7.1.6.5a1.5 $x^{6} + 35$ $7$ $1$ $6$ $5$ $C_6$ (as 6T1) $1$ $6$ $[\ ]$ $[\ ]$ $[\ ]_{6}$ $[\ ]_{6}$ $[\ ]$ $[\ ]$ $t + 4$ $x^{6} + 35$ $[0]$ $[1]$ $z^5 + 6 z^4 + z^3 + 6 z^2 + z + 6$ undefined
7.1.6.5a1.6 $x^{6} + 42$ $7$ $1$ $6$ $5$ $C_6$ (as 6T1) $1$ $6$ $[\ ]$ $[\ ]$ $[\ ]_{6}$ $[\ ]_{6}$ $[\ ]$ $[\ ]$ $t + 4$ $x^{6} + 42$ $[0]$ $[1]$ $z^5 + 6 z^4 + z^3 + 6 z^2 + z + 6$ undefined
11.6.1.0a1.1 $x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$ $11$ $6$ $1$ $0$ $C_6$ (as 6T1) $6$ $1$ $[\ ]$ $[\ ]$ $[\ ]^{6}$ $[\ ]^{6}$ $[\ ]$ $[\ ]$ $t^{6} + 3 t^{4} + 4 t^{3} + 6 t^{2} + 7 t + 2$ $x - 11$ $[0]$ $[\ ]$ undefined
11.3.2.3a1.1 $( x^{3} + 2 x + 9 )^{2} + 11 x$ $11$ $3$ $2$ $3$ $C_6$ (as 6T1) $3$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{3}$ $[\ ]_{2}^{3}$ $[\ ]$ $[\ ]$ $t^{3} + 2 t + 9$ $x^{2} + 11 t$ $[0]$ $[1]$ $z + 2$ undefined
11.3.2.3a1.2 $( x^{3} + 2 x + 9 )^{2} + 11$ $11$ $3$ $2$ $3$ $C_6$ (as 6T1) $3$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{3}$ $[\ ]_{2}^{3}$ $[\ ]$ $[\ ]$ $t^{3} + 2 t + 9$ $x^{2} + 11$ $[0]$ $[1]$ $z + 2$ undefined
13.6.1.0a1.1 $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ $13$ $6$ $1$ $0$ $C_6$ (as 6T1) $6$ $1$ $[\ ]$ $[\ ]$ $[\ ]^{6}$ $[\ ]^{6}$ $[\ ]$ $[\ ]$ $t^{6} + 10 t^{3} + 11 t^{2} + 11 t + 2$ $x - 13$ $[0]$ $[\ ]$ undefined
13.3.2.3a1.1 $( x^{3} + 2 x + 11 )^{2} + 13 x$ $13$ $3$ $2$ $3$ $C_6$ (as 6T1) $3$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{3}$ $[\ ]_{2}^{3}$ $[\ ]$ $[\ ]$ $t^{3} + 2 t + 11$ $x^{2} + 13 t$ $[0]$ $[1]$ $z + 2$ undefined
13.3.2.3a1.2 $( x^{3} + 2 x + 11 )^{2} + 13$ $13$ $3$ $2$ $3$ $C_6$ (as 6T1) $3$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{3}$ $[\ ]_{2}^{3}$ $[\ ]$ $[\ ]$ $t^{3} + 2 t + 11$ $x^{2} + 13$ $[0]$ $[1]$ $z + 2$ undefined
13.2.3.4a1.1 $( x^{2} + 12 x + 2 )^{3} + 13 x$ $13$ $2$ $3$ $4$ $C_6$ (as 6T1) $2$ $3$ $[\ ]$ $[\ ]$ $[\ ]_{3}^{2}$ $[\ ]_{3}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 12 t + 2$ $x^{3} + 13 t$ $[0]$ $[1]$ $z^2 + 3 z + 3$ undefined
13.2.3.4a1.2 $( x^{2} + 12 x + 2 )^{3} + 13$ $13$ $2$ $3$ $4$ $C_6$ (as 6T1) $2$ $3$ $[\ ]$ $[\ ]$ $[\ ]_{3}^{2}$ $[\ ]_{3}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 12 t + 2$ $x^{3} + 13$ $[0]$ $[1]$ $z^2 + 3 z + 3$ undefined
13.2.3.4a1.3 $( x^{2} + 12 x + 2 )^{3} + 13 x + 143$ $13$ $2$ $3$ $4$ $C_6$ (as 6T1) $2$ $3$ $[\ ]$ $[\ ]$ $[\ ]_{3}^{2}$ $[\ ]_{3}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 12 t + 2$ $x^{3} + 13 t + 143$ $[0]$ $[1]$ $z^2 + 3 z + 3$ undefined
13.1.6.5a1.1 $x^{6} + 13$ $13$ $1$ $6$ $5$ $C_6$ (as 6T1) $1$ $6$ $[\ ]$ $[\ ]$ $[\ ]_{6}$ $[\ ]_{6}$ $[\ ]$ $[\ ]$ $t + 11$ $x^{6} + 13$ $[0]$ $[1]$ $z^5 + 6 z^4 + 2 z^3 + 7 z^2 + 2 z + 6$ undefined
13.1.6.5a1.2 $x^{6} + 26$ $13$ $1$ $6$ $5$ $C_6$ (as 6T1) $1$ $6$ $[\ ]$ $[\ ]$ $[\ ]_{6}$ $[\ ]_{6}$ $[\ ]$ $[\ ]$ $t + 11$ $x^{6} + 26$ $[0]$ $[1]$ $z^5 + 6 z^4 + 2 z^3 + 7 z^2 + 2 z + 6$ undefined
13.1.6.5a1.3 $x^{6} + 39$ $13$ $1$ $6$ $5$ $C_6$ (as 6T1) $1$ $6$ $[\ ]$ $[\ ]$ $[\ ]_{6}$ $[\ ]_{6}$ $[\ ]$ $[\ ]$ $t + 11$ $x^{6} + 39$ $[0]$ $[1]$ $z^5 + 6 z^4 + 2 z^3 + 7 z^2 + 2 z + 6$ undefined
13.1.6.5a1.4 $x^{6} + 52$ $13$ $1$ $6$ $5$ $C_6$ (as 6T1) $1$ $6$ $[\ ]$ $[\ ]$ $[\ ]_{6}$ $[\ ]_{6}$ $[\ ]$ $[\ ]$ $t + 11$ $x^{6} + 52$ $[0]$ $[1]$ $z^5 + 6 z^4 + 2 z^3 + 7 z^2 + 2 z + 6$ undefined
13.1.6.5a1.5 $x^{6} + 78$ $13$ $1$ $6$ $5$ $C_6$ (as 6T1) $1$ $6$ $[\ ]$ $[\ ]$ $[\ ]_{6}$ $[\ ]_{6}$ $[\ ]$ $[\ ]$ $t + 11$ $x^{6} + 78$ $[0]$ $[1]$ $z^5 + 6 z^4 + 2 z^3 + 7 z^2 + 2 z + 6$ undefined
13.1.6.5a1.6 $x^{6} + 104$ $13$ $1$ $6$ $5$ $C_6$ (as 6T1) $1$ $6$ $[\ ]$ $[\ ]$ $[\ ]_{6}$ $[\ ]_{6}$ $[\ ]$ $[\ ]$ $t + 11$ $x^{6} + 104$ $[0]$ $[1]$ $z^5 + 6 z^4 + 2 z^3 + 7 z^2 + 2 z + 6$ undefined
17.6.1.0a1.1 $x^{6} + 2 x^{4} + 10 x^{2} + 3 x + 3$ $17$ $6$ $1$ $0$ $C_6$ (as 6T1) $6$ $1$ $[\ ]$ $[\ ]$ $[\ ]^{6}$ $[\ ]^{6}$ $[\ ]$ $[\ ]$ $t^{6} + 2 t^{4} + 10 t^{2} + 3 t + 3$ $x - 17$ $[0]$ $[\ ]$ undefined
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