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

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Degree $n$ Residue characteristic $p$ Discriminant exponent $c$ Galois group Num. automorphisms
Ramification index $e$ Residue field degree $f$ Galois tame degree $t$ Galois unram. degree $u$ Intermediate field
Inertia subgroup Indices Assoc. Inertia Jump Set
Top Artin slope Galois Artin Visible Artin Wild inertia subgroup
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Results (42 matches)

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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
2.2.5.8a1.1 $( x^{2} + x + 1 )^{5} + 2$ $2$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + t + 1$ $x^{5} + 2$ $[0]$ $[2]$ $z^4 + z^3 + 1$ $[5]$
3.2.5.8a1.1 $( x^{2} + 2 x + 2 )^{5} + 3$ $3$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 2 t + 2$ $x^{5} + 3$ $[0]$ $[2]$ $z^4 + 2 z^3 + z^2 + z + 2$ undefined
5.2.5.12a2.1 $( x^{2} + 4 x + 2 )^{5} + 20 ( x^{2} + 4 x + 2 )^{2} + 5$ $5$ $2$ $5$ $12$ $F_5$ (as 10T4) $2$ $2$ $[\frac{3}{2}]$ $[\frac{1}{2}]$ $[\frac{3}{2}]_{2}^{2}$ $[\frac{1}{2}]_{2}^{2}$ $[\ ]_{2}$ $[\ ]_{2}$ $t^{2} + 4 t + 2$ $x^{5} + 20 x^{2} + 5$ $[2, 0]$ $[1]$ $z^2 + 2$ undefined
5.2.5.12a3.1 $( x^{2} + 4 x + 2 )^{5} + 5 ( x^{2} + 4 x + 2 )^{2} + 5$ $5$ $2$ $5$ $12$ $F_5$ (as 10T4) $2$ $2$ $[\frac{3}{2}]$ $[\frac{1}{2}]$ $[\frac{3}{2}]_{2}^{2}$ $[\frac{1}{2}]_{2}^{2}$ $[\ ]_{2}$ $[\ ]_{2}$ $t^{2} + 4 t + 2$ $x^{5} + 5 x^{2} + 5$ $[2, 0]$ $[1]$ $z^2 + 3$ undefined
5.2.5.16a2.1 $( x^{2} + 4 x + 2 )^{5} + 10 ( x^{2} + 4 x + 2 )^{4} + 5$ $5$ $2$ $5$ $16$ $F_5$ (as 10T4) $4$ $1$ $[2]$ $[1]$ $[2]^{4}$ $[1]^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 4 t + 2$ $x^{5} + 10 x^{4} + 5$ $[4, 0]$ $[2]$ $z^4 + 2$ undefined
5.2.5.16a3.1 $( x^{2} + 4 x + 2 )^{5} + 15 ( x^{2} + 4 x + 2 )^{4} + 5$ $5$ $2$ $5$ $16$ $F_5$ (as 10T4) $4$ $1$ $[2]$ $[1]$ $[2]^{4}$ $[1]^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 4 t + 2$ $x^{5} + 15 x^{4} + 5$ $[4, 0]$ $[2]$ $z^4 + 3$ undefined
5.1.10.11a2.1 $x^{10} + 5 x^{2} + 5$ $5$ $1$ $10$ $11$ $F_5$ (as 10T4) $1$ $4$ $[\frac{5}{4}]$ $[\frac{1}{4}]$ $[\frac{5}{4}]_{4}$ $[\frac{1}{4}]_{4}$ $[\ ]_{2}$ $[\ ]_{2}$ $t + 3$ $x^{10} + 5 x^{2} + 5$ $[2, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.1.10.11a2.2 $x^{10} + 20 x^{2} + 5$ $5$ $1$ $10$ $11$ $F_5$ (as 10T4) $1$ $4$ $[\frac{5}{4}]$ $[\frac{1}{4}]$ $[\frac{5}{4}]_{4}$ $[\frac{1}{4}]_{4}$ $[\ ]_{2}$ $[\ ]_{2}$ $t + 3$ $x^{10} + 20 x^{2} + 5$ $[2, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.1.10.11a2.3 $x^{10} + 10 x^{2} + 10$ $5$ $1$ $10$ $11$ $F_5$ (as 10T4) $1$ $4$ $[\frac{5}{4}]$ $[\frac{1}{4}]$ $[\frac{5}{4}]_{4}$ $[\frac{1}{4}]_{4}$ $[\ ]_{2}$ $[\ ]_{2}$ $t + 3$ $x^{10} + 10 x^{2} + 10$ $[2, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.1.10.11a2.4 $x^{10} + 15 x^{2} + 10$ $5$ $1$ $10$ $11$ $F_5$ (as 10T4) $1$ $4$ $[\frac{5}{4}]$ $[\frac{1}{4}]$ $[\frac{5}{4}]_{4}$ $[\frac{1}{4}]_{4}$ $[\ ]_{2}$ $[\ ]_{2}$ $t + 3$ $x^{10} + 15 x^{2} + 10$ $[2, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.1.10.15a2.1 $x^{10} + 10 x^{6} + 5$ $5$ $1$ $10$ $15$ $F_5$ (as 10T4) $1$ $4$ $[\frac{7}{4}]$ $[\frac{3}{4}]$ $[\frac{7}{4}]_{4}$ $[\frac{3}{4}]_{4}$ $[\ ]_{2}$ $[\ ]_{2}$ $t + 3$ $x^{10} + 10 x^{6} + 5$ $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.1.10.15a2.4 $x^{10} + 15 x^{6} + 5$ $5$ $1$ $10$ $15$ $F_5$ (as 10T4) $1$ $4$ $[\frac{7}{4}]$ $[\frac{3}{4}]$ $[\frac{7}{4}]_{4}$ $[\frac{3}{4}]_{4}$ $[\ ]_{2}$ $[\ ]_{2}$ $t + 3$ $x^{10} + 15 x^{6} + 5$ $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.1.10.15a2.7 $x^{10} + 5 x^{6} + 10$ $5$ $1$ $10$ $15$ $F_5$ (as 10T4) $1$ $4$ $[\frac{7}{4}]$ $[\frac{3}{4}]$ $[\frac{7}{4}]_{4}$ $[\frac{3}{4}]_{4}$ $[\ ]_{2}$ $[\ ]_{2}$ $t + 3$ $x^{10} + 5 x^{6} + 10$ $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.1.10.15a2.10 $x^{10} + 20 x^{6} + 10$ $5$ $1$ $10$ $15$ $F_5$ (as 10T4) $1$ $4$ $[\frac{7}{4}]$ $[\frac{3}{4}]$ $[\frac{7}{4}]_{4}$ $[\frac{3}{4}]_{4}$ $[\ ]_{2}$ $[\ ]_{2}$ $t + 3$ $x^{10} + 20 x^{6} + 10$ $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.1.10.19a2.1 $x^{10} + 5$ $5$ $1$ $10$ $19$ $F_5$ (as 10T4) $1$ $4$ $[\frac{9}{4}]$ $[\frac{5}{4}]$ $[\frac{9}{4}]_{4}$ $[\frac{5}{4}]_{4}$ $[\ ]_{2}$ $[\ ]_{2}$ $t + 3$ $x^{10} + 5$ $[10, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.1.10.19a2.2 $x^{10} + 25 x^{2} + 5$ $5$ $1$ $10$ $19$ $F_5$ (as 10T4) $1$ $4$ $[\frac{9}{4}]$ $[\frac{5}{4}]$ $[\frac{9}{4}]_{4}$ $[\frac{5}{4}]_{4}$ $[\ ]_{2}$ $[\ ]_{2}$ $t + 3$ $x^{10} + 25 x^{2} + 5$ $[10, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.1.10.19a2.3 $x^{10} + 50 x^{2} + 5$ $5$ $1$ $10$ $19$ $F_5$ (as 10T4) $1$ $4$ $[\frac{9}{4}]$ $[\frac{5}{4}]$ $[\frac{9}{4}]_{4}$ $[\frac{5}{4}]_{4}$ $[\ ]_{2}$ $[\ ]_{2}$ $t + 3$ $x^{10} + 50 x^{2} + 5$ $[10, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.1.10.19a2.4 $x^{10} + 75 x^{2} + 5$ $5$ $1$ $10$ $19$ $F_5$ (as 10T4) $1$ $4$ $[\frac{9}{4}]$ $[\frac{5}{4}]$ $[\frac{9}{4}]_{4}$ $[\frac{5}{4}]_{4}$ $[\ ]_{2}$ $[\ ]_{2}$ $t + 3$ $x^{10} + 75 x^{2} + 5$ $[10, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.1.10.19a2.5 $x^{10} + 100 x^{2} + 5$ $5$ $1$ $10$ $19$ $F_5$ (as 10T4) $1$ $4$ $[\frac{9}{4}]$ $[\frac{5}{4}]$ $[\frac{9}{4}]_{4}$ $[\frac{5}{4}]_{4}$ $[\ ]_{2}$ $[\ ]_{2}$ $t + 3$ $x^{10} + 100 x^{2} + 5$ $[10, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
7.2.5.8a1.1 $( x^{2} + 6 x + 3 )^{5} + 7$ $7$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 6 t + 3$ $x^{5} + 7$ $[0]$ $[2]$ $z^4 + 5 z^3 + 3 z^2 + 3 z + 5$ undefined
13.2.5.8a1.1 $( x^{2} + 12 x + 2 )^{5} + 13$ $13$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 12 t + 2$ $x^{5} + 13$ $[0]$ $[2]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
17.2.5.8a1.1 $( x^{2} + 16 x + 3 )^{5} + 17$ $17$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 16 t + 3$ $x^{5} + 17$ $[0]$ $[2]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
23.2.5.8a1.1 $( x^{2} + 21 x + 5 )^{5} + 23$ $23$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 21 t + 5$ $x^{5} + 23$ $[0]$ $[2]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
37.2.5.8a1.1 $( x^{2} + 33 x + 2 )^{5} + 37$ $37$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 33 t + 2$ $x^{5} + 37$ $[0]$ $[2]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
43.2.5.8a1.1 $( x^{2} + 42 x + 3 )^{5} + 43$ $43$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 42 t + 3$ $x^{5} + 43$ $[0]$ $[2]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
47.2.5.8a1.1 $( x^{2} + 45 x + 5 )^{5} + 47$ $47$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 45 t + 5$ $x^{5} + 47$ $[0]$ $[2]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
53.2.5.8a1.1 $( x^{2} + 49 x + 2 )^{5} + 53$ $53$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 49 t + 2$ $x^{5} + 53$ $[0]$ $[2]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
67.2.5.8a1.1 $( x^{2} + 63 x + 2 )^{5} + 67$ $67$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 63 t + 2$ $x^{5} + 67$ $[0]$ $[2]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
73.2.5.8a1.1 $( x^{2} + 70 x + 5 )^{5} + 73$ $73$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 70 t + 5$ $x^{5} + 73$ $[0]$ $[2]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
83.2.5.8a1.1 $( x^{2} + 82 x + 2 )^{5} + 83$ $83$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 82 t + 2$ $x^{5} + 83$ $[0]$ $[2]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
97.2.5.8a1.1 $( x^{2} + 96 x + 5 )^{5} + 97$ $97$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 96 t + 5$ $x^{5} + 97$ $[0]$ $[2]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
103.2.5.8a1.1 $( x^{2} + 102 x + 5 )^{5} + 103$ $103$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 102 t + 5$ $x^{5} + 103$ $[0]$ $[2]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
107.2.5.8a1.1 $( x^{2} + 103 x + 2 )^{5} + 107$ $107$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 103 t + 2$ $x^{5} + 107$ $[0]$ $[2]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
113.2.5.8a1.1 $( x^{2} + 101 x + 3 )^{5} + 113$ $113$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 101 t + 3$ $x^{5} + 113$ $[0]$ $[2]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
127.2.5.8a1.1 $( x^{2} + 126 x + 3 )^{5} + 127$ $127$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 126 t + 3$ $x^{5} + 127$ $[0]$ $[2]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
137.2.5.8a1.1 $( x^{2} + 131 x + 3 )^{5} + 137$ $137$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 131 t + 3$ $x^{5} + 137$ $[0]$ $[2]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
157.2.5.8a1.1 $( x^{2} + 152 x + 5 )^{5} + 157$ $157$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 152 t + 5$ $x^{5} + 157$ $[0]$ $[2]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
163.2.5.8a1.1 $( x^{2} + 159 x + 2 )^{5} + 163$ $163$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 159 t + 2$ $x^{5} + 163$ $[0]$ $[2]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
167.2.5.8a1.1 $( x^{2} + 166 x + 5 )^{5} + 167$ $167$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 166 t + 5$ $x^{5} + 167$ $[0]$ $[2]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
173.2.5.8a1.1 $( x^{2} + 169 x + 2 )^{5} + 173$ $173$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 169 t + 2$ $x^{5} + 173$ $[0]$ $[2]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
193.2.5.8a1.1 $( x^{2} + 192 x + 5 )^{5} + 193$ $193$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 192 t + 5$ $x^{5} + 193$ $[0]$ $[2]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
197.2.5.8a1.1 $( x^{2} + 192 x + 2 )^{5} + 197$ $197$ $2$ $5$ $8$ $F_5$ (as 10T4) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 192 t + 2$ $x^{5} + 197$ $[0]$ $[2]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
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