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Properties

Label 17.1.17.32a
Base 17.1.1.0a1.1
Degree \(17\)
e \(17\)
f \(1\)
c \(32\)

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Defining polynomial

$x^{17} + 17 a_{16} x^{16} + 289 c_{17} + 17$

Invariants

Residue field characteristic: $17$
Degree: $17$
Base field: $\Q_{17}$
Ramification index $e$: $17$
Residue field degree $f$: $1$
Discriminant exponent $c$: $32$
Artin slopes: $[2]$
Swan slopes: $[1]$
Means: $\langle\frac{16}{17}\rangle$
Rams: $(1)$
Field count: $32$ (complete)
Ambiguity: $17$
Mass: $16$
Absolute Mass: $16$

Diagrams

Varying

Indices of inseparability: $[16,0]$
Associated inertia: $[1]$ (show 17), $[2]$ (show 1), $[4]$ (show 2), $[8]$ (show 4), $[16]$ (show 8)
Jump Set: undefined

Galois groups and Hidden Artin slopes

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Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
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Showing all 32

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Label Packet size Polynomial Galois group Galois degree $\#\Aut(K/\Q_p)$ Artin slope content Swan slope content Hidden Artin slopes Hidden Swan slopes Ind. of Insep. Assoc. Inertia Resid. Poly Jump Set
17.1.17.32a1.1 1 $x^{17} + 17 x^{16} + 17$ $D_{17}$ (as 17T2) $34$ $1$ $[2]^{2}$ $[1]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[16, 0]$ $[2]$ $z^{16} + 1$ undefined
17.1.17.32a2.1 1 $x^{17} + 34 x^{16} + 17$ $C_{17}:C_{8}$ (as 17T4) $136$ $1$ $[2]^{8}$ $[1]^{8}$ $[\ ]^{8}$ $[\ ]^{8}$ $[16, 0]$ $[8]$ $z^{16} + 2$ undefined
17.1.17.32a3.1 1 $x^{17} + 51 x^{16} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[2]^{16}$ $[1]^{16}$ $[\ ]^{16}$ $[\ ]^{16}$ $[16, 0]$ $[16]$ $z^{16} + 3$ undefined
17.1.17.32a4.1 1 $x^{17} + 68 x^{16} + 17$ $C_{17}:C_{4}$ (as 17T3) $68$ $1$ $[2]^{4}$ $[1]^{4}$ $[\ ]^{4}$ $[\ ]^{4}$ $[16, 0]$ $[4]$ $z^{16} + 4$ undefined
17.1.17.32a5.1 1 $x^{17} + 85 x^{16} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[2]^{16}$ $[1]^{16}$ $[\ ]^{16}$ $[\ ]^{16}$ $[16, 0]$ $[16]$ $z^{16} + 5$ undefined
17.1.17.32a6.1 1 $x^{17} + 102 x^{16} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[2]^{16}$ $[1]^{16}$ $[\ ]^{16}$ $[\ ]^{16}$ $[16, 0]$ $[16]$ $z^{16} + 6$ undefined
17.1.17.32a7.1 1 $x^{17} + 119 x^{16} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[2]^{16}$ $[1]^{16}$ $[\ ]^{16}$ $[\ ]^{16}$ $[16, 0]$ $[16]$ $z^{16} + 7$ undefined
17.1.17.32a8.1 1 $x^{17} + 136 x^{16} + 17$ $C_{17}:C_{8}$ (as 17T4) $136$ $1$ $[2]^{8}$ $[1]^{8}$ $[\ ]^{8}$ $[\ ]^{8}$ $[16, 0]$ $[8]$ $z^{16} + 8$ undefined
17.1.17.32a9.1 1 $x^{17} + 153 x^{16} + 17$ $C_{17}:C_{8}$ (as 17T4) $136$ $1$ $[2]^{8}$ $[1]^{8}$ $[\ ]^{8}$ $[\ ]^{8}$ $[16, 0]$ $[8]$ $z^{16} + 9$ undefined
17.1.17.32a10.1 1 $x^{17} + 170 x^{16} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[2]^{16}$ $[1]^{16}$ $[\ ]^{16}$ $[\ ]^{16}$ $[16, 0]$ $[16]$ $z^{16} + 10$ undefined
17.1.17.32a11.1 1 $x^{17} + 187 x^{16} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[2]^{16}$ $[1]^{16}$ $[\ ]^{16}$ $[\ ]^{16}$ $[16, 0]$ $[16]$ $z^{16} + 11$ undefined
17.1.17.32a12.1 1 $x^{17} + 204 x^{16} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[2]^{16}$ $[1]^{16}$ $[\ ]^{16}$ $[\ ]^{16}$ $[16, 0]$ $[16]$ $z^{16} + 12$ undefined
17.1.17.32a13.1 1 $x^{17} + 221 x^{16} + 17$ $C_{17}:C_{4}$ (as 17T3) $68$ $1$ $[2]^{4}$ $[1]^{4}$ $[\ ]^{4}$ $[\ ]^{4}$ $[16, 0]$ $[4]$ $z^{16} + 13$ undefined
17.1.17.32a14.1 1 $x^{17} + 238 x^{16} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[2]^{16}$ $[1]^{16}$ $[\ ]^{16}$ $[\ ]^{16}$ $[16, 0]$ $[16]$ $z^{16} + 14$ undefined
17.1.17.32a15.1 1 $x^{17} + 255 x^{16} + 17$ $C_{17}:C_{8}$ (as 17T4) $136$ $1$ $[2]^{8}$ $[1]^{8}$ $[\ ]^{8}$ $[\ ]^{8}$ $[16, 0]$ $[8]$ $z^{16} + 15$ undefined
17.1.17.32a16.1 17 $x^{17} + 272 x^{16} + 17$ $C_{17}$ (as 17T1) $17$ $17$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[16, 0]$ $[1]$ $z^{16} + 16$ undefined
17.1.17.32a16.2 17 $x^{17} + 272 x^{16} + 306$ $C_{17}$ (as 17T1) $17$ $17$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[16, 0]$ $[1]$ $z^{16} + 16$ undefined
17.1.17.32a16.3 17 $x^{17} + 272 x^{16} + 595$ $C_{17}$ (as 17T1) $17$ $17$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[16, 0]$ $[1]$ $z^{16} + 16$ undefined
17.1.17.32a16.4 17 $x^{17} + 272 x^{16} + 884$ $C_{17}$ (as 17T1) $17$ $17$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[16, 0]$ $[1]$ $z^{16} + 16$ undefined
17.1.17.32a16.5 17 $x^{17} + 272 x^{16} + 1173$ $C_{17}$ (as 17T1) $17$ $17$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[16, 0]$ $[1]$ $z^{16} + 16$ undefined
17.1.17.32a16.6 17 $x^{17} + 272 x^{16} + 1462$ $C_{17}$ (as 17T1) $17$ $17$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[16, 0]$ $[1]$ $z^{16} + 16$ undefined
17.1.17.32a16.7 17 $x^{17} + 272 x^{16} + 1751$ $C_{17}$ (as 17T1) $17$ $17$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[16, 0]$ $[1]$ $z^{16} + 16$ undefined
17.1.17.32a16.8 17 $x^{17} + 272 x^{16} + 2040$ $C_{17}$ (as 17T1) $17$ $17$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[16, 0]$ $[1]$ $z^{16} + 16$ undefined
17.1.17.32a16.9 17 $x^{17} + 272 x^{16} + 2329$ $C_{17}$ (as 17T1) $17$ $17$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[16, 0]$ $[1]$ $z^{16} + 16$ undefined
17.1.17.32a16.10 17 $x^{17} + 272 x^{16} + 2618$ $C_{17}$ (as 17T1) $17$ $17$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[16, 0]$ $[1]$ $z^{16} + 16$ undefined
17.1.17.32a16.11 17 $x^{17} + 272 x^{16} + 2907$ $C_{17}$ (as 17T1) $17$ $17$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[16, 0]$ $[1]$ $z^{16} + 16$ undefined
17.1.17.32a16.12 17 $x^{17} + 272 x^{16} + 3196$ $C_{17}$ (as 17T1) $17$ $17$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[16, 0]$ $[1]$ $z^{16} + 16$ undefined
17.1.17.32a16.13 17 $x^{17} + 272 x^{16} + 3485$ $C_{17}$ (as 17T1) $17$ $17$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[16, 0]$ $[1]$ $z^{16} + 16$ undefined
17.1.17.32a16.14 17 $x^{17} + 272 x^{16} + 3774$ $C_{17}$ (as 17T1) $17$ $17$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[16, 0]$ $[1]$ $z^{16} + 16$ undefined
17.1.17.32a16.15 17 $x^{17} + 272 x^{16} + 4063$ $C_{17}$ (as 17T1) $17$ $17$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[16, 0]$ $[1]$ $z^{16} + 16$ undefined
17.1.17.32a16.16 17 $x^{17} + 272 x^{16} + 4352$ $C_{17}$ (as 17T1) $17$ $17$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[16, 0]$ $[1]$ $z^{16} + 16$ undefined
17.1.17.32a16.17 17 $x^{17} + 272 x^{16} + 4641$ $C_{17}$ (as 17T1) $17$ $17$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[16, 0]$ $[1]$ $z^{16} + 16$ undefined
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