The results below are complete, since the LMFDB contains all p-adic fields of degree at most 23 and residue characteristic at most 199

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Results (30 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
151.20.1.0a1.1 $x^{20} - x + 134$ $151$ $20$ $1$ $0$ $C_{20}$ (as 20T1) $20$ $1$ $[\ ]$ $[\ ]$ $[\ ]^{20}$ $[\ ]^{20}$ $[\ ]$ $[\ ]$ $t^{20} - t + 134$ $x - 151$ $[0]$ $[\ ]$ undefined
151.10.2.10a1.1 $( x^{10} + x^{6} + 21 x^{5} + 104 x^{4} + 49 x^{3} + 20 x^{2} + 142 x + 6 )^{2} + 151 x$ $151$ $10$ $2$ $10$ $C_{20}$ (as 20T1) $10$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{10}$ $[\ ]_{2}^{10}$ $[\ ]$ $[\ ]$ $t^{10} + t^{6} + 21 t^{5} + 104 t^{4} + 49 t^{3} + 20 t^{2} + 142 t + 6$ $x^{2} + 151 t$ $[0]$ $[1]$ $z + 2$ undefined
151.10.2.10a1.2 $( x^{10} + x^{6} + 21 x^{5} + 104 x^{4} + 49 x^{3} + 20 x^{2} + 142 x + 6 )^{2} + 151$ $151$ $10$ $2$ $10$ $C_2\times C_{10}$ (as 20T3) $10$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{10}$ $[\ ]_{2}^{10}$ $[\ ]$ $[\ ]$ $t^{10} + t^{6} + 21 t^{5} + 104 t^{4} + 49 t^{3} + 20 t^{2} + 142 t + 6$ $x^{2} + 151$ $[0]$ $[1]$ $z + 2$ undefined
151.5.4.15a1.1 $( x^{5} + 11 x + 145 )^{4} + 151 x$ $151$ $5$ $4$ $15$ $C_5\times D_4$ (as 20T12) $10$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{10}$ $[\ ]_{4}^{10}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{5} + 11 t + 145$ $x^{4} + 151 t$ $[0]$ $[2]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
151.5.4.15a1.2 $( x^{5} + 11 x + 145 )^{4} + 151$ $151$ $5$ $4$ $15$ $C_5\times D_4$ (as 20T12) $10$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{10}$ $[\ ]_{4}^{10}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{5} + 11 t + 145$ $x^{4} + 151$ $[0]$ $[2]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
151.4.5.16a1.1 $( x^{4} + 13 x^{2} + 89 x + 6 )^{5} + 151 x^{3}$ $151$ $4$ $5$ $16$ $C_{20}$ (as 20T1) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 13 t^{2} + 89 t + 6$ $x^{5} + 151 t^{3}$ $[0]$ $[1]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
151.4.5.16a1.2 $( x^{4} + 13 x^{2} + 89 x + 6 )^{5} + 151 x^{2}$ $151$ $4$ $5$ $16$ $C_{20}$ (as 20T1) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 13 t^{2} + 89 t + 6$ $x^{5} + 151 t^{2}$ $[0]$ $[1]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
151.4.5.16a1.3 $( x^{4} + 13 x^{2} + 89 x + 6 )^{5} + 151 x$ $151$ $4$ $5$ $16$ $C_{20}$ (as 20T1) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 13 t^{2} + 89 t + 6$ $x^{5} + 151 t$ $[0]$ $[1]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
151.4.5.16a1.4 $( x^{4} + 13 x^{2} + 89 x + 6 )^{5} + 151$ $151$ $4$ $5$ $16$ $C_{20}$ (as 20T1) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 13 t^{2} + 89 t + 6$ $x^{5} + 151$ $[0]$ $[1]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
151.4.5.16a1.5 $( x^{4} + 13 x^{2} + 89 x + 6 )^{5} + 20838 x^{2} + 9362 x + 21895$ $151$ $4$ $5$ $16$ $C_{20}$ (as 20T1) $4$ $5$ $[\ ]$ $[\ ]$ $[\ ]_{5}^{4}$ $[\ ]_{5}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 13 t^{2} + 89 t + 6$ $x^{5} + 20838 t^{2} + 9362 t + 21895$ $[0]$ $[1]$ $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ undefined
151.2.10.18a1.1 $( x^{2} + 149 x + 6 )^{10} + 151 x$ $151$ $2$ $10$ $18$ $C_{20}$ (as 20T1) $2$ $10$ $[\ ]$ $[\ ]$ $[\ ]_{10}^{2}$ $[\ ]_{10}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 149 t + 6$ $x^{10} + 151 t$ $[0]$ $[1]$ $z^9 + 10 z^8 + 45 z^7 + 120 z^6 + 59 z^5 + 101 z^4 + 59 z^3 + 120 z^2 + 45 z + 10$ undefined
151.2.10.18a1.2 $( x^{2} + 149 x + 6 )^{10} + 151$ $151$ $2$ $10$ $18$ $C_2\times C_{10}$ (as 20T3) $2$ $10$ $[\ ]$ $[\ ]$ $[\ ]_{10}^{2}$ $[\ ]_{10}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 149 t + 6$ $x^{10} + 151$ $[0]$ $[1]$ $z^9 + 10 z^8 + 45 z^7 + 120 z^6 + 59 z^5 + 101 z^4 + 59 z^3 + 120 z^2 + 45 z + 10$ undefined
151.2.10.18a1.3 $( x^{2} + 149 x + 6 )^{10} + 20385 x + 1812$ $151$ $2$ $10$ $18$ $C_2\times C_{10}$ (as 20T3) $2$ $10$ $[\ ]$ $[\ ]$ $[\ ]_{10}^{2}$ $[\ ]_{10}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 149 t + 6$ $x^{10} + 20385 t + 1812$ $[0]$ $[1]$ $z^9 + 10 z^8 + 45 z^7 + 120 z^6 + 59 z^5 + 101 z^4 + 59 z^3 + 120 z^2 + 45 z + 10$ undefined
151.2.10.18a1.4 $( x^{2} + 149 x + 6 )^{10} + 19781 x + 14496$ $151$ $2$ $10$ $18$ $C_{20}$ (as 20T1) $2$ $10$ $[\ ]$ $[\ ]$ $[\ ]_{10}^{2}$ $[\ ]_{10}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 149 t + 6$ $x^{10} + 19781 t + 14496$ $[0]$ $[1]$ $z^9 + 10 z^8 + 45 z^7 + 120 z^6 + 59 z^5 + 101 z^4 + 59 z^3 + 120 z^2 + 45 z + 10$ undefined
151.2.10.18a1.5 $( x^{2} + 149 x + 6 )^{10} + 12231 x + 17667$ $151$ $2$ $10$ $18$ $C_{20}$ (as 20T1) $2$ $10$ $[\ ]$ $[\ ]$ $[\ ]_{10}^{2}$ $[\ ]_{10}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 149 t + 6$ $x^{10} + 12231 t + 17667$ $[0]$ $[1]$ $z^9 + 10 z^8 + 45 z^7 + 120 z^6 + 59 z^5 + 101 z^4 + 59 z^3 + 120 z^2 + 45 z + 10$ undefined
151.2.10.18a1.6 $( x^{2} + 149 x + 6 )^{10} + 19328 x + 17818$ $151$ $2$ $10$ $18$ $C_2\times C_{10}$ (as 20T3) $2$ $10$ $[\ ]$ $[\ ]$ $[\ ]_{10}^{2}$ $[\ ]_{10}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 149 t + 6$ $x^{10} + 19328 t + 17818$ $[0]$ $[1]$ $z^9 + 10 z^8 + 45 z^7 + 120 z^6 + 59 z^5 + 101 z^4 + 59 z^3 + 120 z^2 + 45 z + 10$ undefined
151.2.10.18a1.7 $( x^{2} + 149 x + 6 )^{10} + 8456 x + 18120$ $151$ $2$ $10$ $18$ $C_2\times C_{10}$ (as 20T3) $2$ $10$ $[\ ]$ $[\ ]$ $[\ ]_{10}^{2}$ $[\ ]_{10}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 149 t + 6$ $x^{10} + 8456 t + 18120$ $[0]$ $[1]$ $z^9 + 10 z^8 + 45 z^7 + 120 z^6 + 59 z^5 + 101 z^4 + 59 z^3 + 120 z^2 + 45 z + 10$ undefined
151.2.10.18a1.8 $( x^{2} + 149 x + 6 )^{10} + 10872 x + 20838$ $151$ $2$ $10$ $18$ $C_{20}$ (as 20T1) $2$ $10$ $[\ ]$ $[\ ]$ $[\ ]_{10}^{2}$ $[\ ]_{10}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 149 t + 6$ $x^{10} + 10872 t + 20838$ $[0]$ $[1]$ $z^9 + 10 z^8 + 45 z^7 + 120 z^6 + 59 z^5 + 101 z^4 + 59 z^3 + 120 z^2 + 45 z + 10$ undefined
151.2.10.18a1.9 $( x^{2} + 149 x + 6 )^{10} + 22499 x + 20989$ $151$ $2$ $10$ $18$ $C_{20}$ (as 20T1) $2$ $10$ $[\ ]$ $[\ ]$ $[\ ]_{10}^{2}$ $[\ ]_{10}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 149 t + 6$ $x^{10} + 22499 t + 20989$ $[0]$ $[1]$ $z^9 + 10 z^8 + 45 z^7 + 120 z^6 + 59 z^5 + 101 z^4 + 59 z^3 + 120 z^2 + 45 z + 10$ undefined
151.2.10.18a1.10 $( x^{2} + 149 x + 6 )^{10} + 302 x + 21895$ $151$ $2$ $10$ $18$ $C_2\times C_{10}$ (as 20T3) $2$ $10$ $[\ ]$ $[\ ]$ $[\ ]_{10}^{2}$ $[\ ]_{10}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 149 t + 6$ $x^{10} + 302 t + 21895$ $[0]$ $[1]$ $z^9 + 10 z^8 + 45 z^7 + 120 z^6 + 59 z^5 + 101 z^4 + 59 z^3 + 120 z^2 + 45 z + 10$ undefined
151.1.20.19a1.1 $x^{20} + 151$ $151$ $1$ $20$ $19$ $C_5\times D_4$ (as 20T12) $2$ $20$ $[\ ]$ $[\ ]$ $[\ ]_{20}^{2}$ $[\ ]_{20}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $t + 145$ $x^{20} + 151$ $[0]$ $[2]$ $z^{19} + 20 z^{18} + 39 z^{17} + 83 z^{16} + 13 z^{15} + 102 z^{14} + 104 z^{13} + 57 z^{12} + 36 z^{11} + 48 z^{10} + 83 z^9 + 48 z^8 + 36 z^7 + 57 z^6 + 104 z^5 + 102 z^4 + 13 z^3 + 83 z^2 + 39 z + 20$ undefined
151.1.20.19a1.2 $x^{20} + 906$ $151$ $1$ $20$ $19$ $C_5\times D_4$ (as 20T12) $2$ $20$ $[\ ]$ $[\ ]$ $[\ ]_{20}^{2}$ $[\ ]_{20}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $t + 145$ $x^{20} + 906$ $[0]$ $[2]$ $z^{19} + 20 z^{18} + 39 z^{17} + 83 z^{16} + 13 z^{15} + 102 z^{14} + 104 z^{13} + 57 z^{12} + 36 z^{11} + 48 z^{10} + 83 z^9 + 48 z^8 + 36 z^7 + 57 z^6 + 104 z^5 + 102 z^4 + 13 z^3 + 83 z^2 + 39 z + 20$ undefined
151.1.20.19a1.3 $x^{20} + 5436$ $151$ $1$ $20$ $19$ $C_5\times D_4$ (as 20T12) $2$ $20$ $[\ ]$ $[\ ]$ $[\ ]_{20}^{2}$ $[\ ]_{20}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $t + 145$ $x^{20} + 5436$ $[0]$ $[2]$ $z^{19} + 20 z^{18} + 39 z^{17} + 83 z^{16} + 13 z^{15} + 102 z^{14} + 104 z^{13} + 57 z^{12} + 36 z^{11} + 48 z^{10} + 83 z^9 + 48 z^8 + 36 z^7 + 57 z^6 + 104 z^5 + 102 z^4 + 13 z^3 + 83 z^2 + 39 z + 20$ undefined
151.1.20.19a1.4 $x^{20} + 6493$ $151$ $1$ $20$ $19$ $C_5\times D_4$ (as 20T12) $2$ $20$ $[\ ]$ $[\ ]$ $[\ ]_{20}^{2}$ $[\ ]_{20}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $t + 145$ $x^{20} + 6493$ $[0]$ $[2]$ $z^{19} + 20 z^{18} + 39 z^{17} + 83 z^{16} + 13 z^{15} + 102 z^{14} + 104 z^{13} + 57 z^{12} + 36 z^{11} + 48 z^{10} + 83 z^9 + 48 z^8 + 36 z^7 + 57 z^6 + 104 z^5 + 102 z^4 + 13 z^3 + 83 z^2 + 39 z + 20$ undefined
151.1.20.19a1.5 $x^{20} + 9815$ $151$ $1$ $20$ $19$ $C_5\times D_4$ (as 20T12) $2$ $20$ $[\ ]$ $[\ ]$ $[\ ]_{20}^{2}$ $[\ ]_{20}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $t + 145$ $x^{20} + 9815$ $[0]$ $[2]$ $z^{19} + 20 z^{18} + 39 z^{17} + 83 z^{16} + 13 z^{15} + 102 z^{14} + 104 z^{13} + 57 z^{12} + 36 z^{11} + 48 z^{10} + 83 z^9 + 48 z^8 + 36 z^7 + 57 z^6 + 104 z^5 + 102 z^4 + 13 z^3 + 83 z^2 + 39 z + 20$ undefined
151.1.20.19a1.6 $x^{20} + 11325$ $151$ $1$ $20$ $19$ $C_5\times D_4$ (as 20T12) $2$ $20$ $[\ ]$ $[\ ]$ $[\ ]_{20}^{2}$ $[\ ]_{20}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $t + 145$ $x^{20} + 11325$ $[0]$ $[2]$ $z^{19} + 20 z^{18} + 39 z^{17} + 83 z^{16} + 13 z^{15} + 102 z^{14} + 104 z^{13} + 57 z^{12} + 36 z^{11} + 48 z^{10} + 83 z^9 + 48 z^8 + 36 z^7 + 57 z^6 + 104 z^5 + 102 z^4 + 13 z^3 + 83 z^2 + 39 z + 20$ undefined
151.1.20.19a1.7 $x^{20} + 13288$ $151$ $1$ $20$ $19$ $C_5\times D_4$ (as 20T12) $2$ $20$ $[\ ]$ $[\ ]$ $[\ ]_{20}^{2}$ $[\ ]_{20}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $t + 145$ $x^{20} + 13288$ $[0]$ $[2]$ $z^{19} + 20 z^{18} + 39 z^{17} + 83 z^{16} + 13 z^{15} + 102 z^{14} + 104 z^{13} + 57 z^{12} + 36 z^{11} + 48 z^{10} + 83 z^9 + 48 z^8 + 36 z^7 + 57 z^6 + 104 z^5 + 102 z^4 + 13 z^3 + 83 z^2 + 39 z + 20$ undefined
151.1.20.19a1.8 $x^{20} + 16157$ $151$ $1$ $20$ $19$ $C_5\times D_4$ (as 20T12) $2$ $20$ $[\ ]$ $[\ ]$ $[\ ]_{20}^{2}$ $[\ ]_{20}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $t + 145$ $x^{20} + 16157$ $[0]$ $[2]$ $z^{19} + 20 z^{18} + 39 z^{17} + 83 z^{16} + 13 z^{15} + 102 z^{14} + 104 z^{13} + 57 z^{12} + 36 z^{11} + 48 z^{10} + 83 z^9 + 48 z^8 + 36 z^7 + 57 z^6 + 104 z^5 + 102 z^4 + 13 z^3 + 83 z^2 + 39 z + 20$ undefined
151.1.20.19a1.9 $x^{20} + 20083$ $151$ $1$ $20$ $19$ $C_5\times D_4$ (as 20T12) $2$ $20$ $[\ ]$ $[\ ]$ $[\ ]_{20}^{2}$ $[\ ]_{20}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $t + 145$ $x^{20} + 20083$ $[0]$ $[2]$ $z^{19} + 20 z^{18} + 39 z^{17} + 83 z^{16} + 13 z^{15} + 102 z^{14} + 104 z^{13} + 57 z^{12} + 36 z^{11} + 48 z^{10} + 83 z^9 + 48 z^8 + 36 z^7 + 57 z^6 + 104 z^5 + 102 z^4 + 13 z^3 + 83 z^2 + 39 z + 20$ undefined
151.1.20.19a1.10 $x^{20} + 22348$ $151$ $1$ $20$ $19$ $C_5\times D_4$ (as 20T12) $2$ $20$ $[\ ]$ $[\ ]$ $[\ ]_{20}^{2}$ $[\ ]_{20}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $t + 145$ $x^{20} + 22348$ $[0]$ $[2]$ $z^{19} + 20 z^{18} + 39 z^{17} + 83 z^{16} + 13 z^{15} + 102 z^{14} + 104 z^{13} + 57 z^{12} + 36 z^{11} + 48 z^{10} + 83 z^9 + 48 z^8 + 36 z^7 + 57 z^6 + 104 z^5 + 102 z^4 + 13 z^3 + 83 z^2 + 39 z + 20$ undefined
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