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Properties

Label 17.1.17.28a
Base 17.1.1.0a1.1
Degree \(17\)
e \(17\)
f \(1\)
c \(28\)

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Defining polynomial

$x^{17} + 17a_{12} x^{12} + 17$

Invariants

Residue field characteristic: $17$
Degree: $17$
Base field: $\Q_{17}$
Ramification index $e$: $17$
Residue field degree $f$: $1$
Discriminant exponent $c$: $28$
Artin slopes: $[\frac{7}{4}]$
Swan slopes: $[\frac{3}{4}]$
Means: $\langle\frac{12}{17}\rangle$
Rams: $(\frac{3}{4})$
Field count: $16$ (complete)
Ambiguity: $1$
Mass: $16$
Absolute Mass: $16$

Diagrams

Varying

Indices of inseparability: $[12,0]$
Associated inertia: $[1]$ (show 4), $[2]$ (show 4), $[4]$ (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 16

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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
17.1.17.28a1.1 4 $x^{17} + 102 x^{12} + 17$ $C_{17}:C_{4}$ (as 17T3) $68$ $1$ $[\frac{7}{4}]_{4}$ $[\frac{3}{4}]_{4}$ $[\ ]_{4}$ $[\ ]_{4}$ $[12, 0]$ $[1]$ $z^4 + 13$ undefined
17.1.17.28a1.2 4 $x^{17} + 119 x^{12} + 17$ $C_{17}:C_{4}$ (as 17T3) $68$ $1$ $[\frac{7}{4}]_{4}$ $[\frac{3}{4}]_{4}$ $[\ ]_{4}$ $[\ ]_{4}$ $[12, 0]$ $[1]$ $z^4 + 1$ undefined
17.1.17.28a1.3 4 $x^{17} + 170 x^{12} + 17$ $C_{17}:C_{4}$ (as 17T3) $68$ $1$ $[\frac{7}{4}]_{4}$ $[\frac{3}{4}]_{4}$ $[\ ]_{4}$ $[\ ]_{4}$ $[12, 0]$ $[1]$ $z^4 + 16$ undefined
17.1.17.28a1.4 4 $x^{17} + 187 x^{12} + 17$ $C_{17}:C_{4}$ (as 17T3) $68$ $1$ $[\frac{7}{4}]_{4}$ $[\frac{3}{4}]_{4}$ $[\ ]_{4}$ $[\ ]_{4}$ $[12, 0]$ $[1]$ $z^4 + 4$ undefined
17.1.17.28a2.1 4 $x^{17} + 17 x^{12} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{7}{4}]_{4}^{4}$ $[\frac{3}{4}]_{4}^{4}$ $[\ ]^{4}_{4}$ $[\ ]^{4}_{4}$ $[12, 0]$ $[4]$ $z^4 + 5$ undefined
17.1.17.28a2.2 4 $x^{17} + 68 x^{12} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{7}{4}]_{4}^{4}$ $[\frac{3}{4}]_{4}^{4}$ $[\ ]^{4}_{4}$ $[\ ]^{4}_{4}$ $[12, 0]$ $[4]$ $z^4 + 3$ undefined
17.1.17.28a2.3 4 $x^{17} + 221 x^{12} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{7}{4}]_{4}^{4}$ $[\frac{3}{4}]_{4}^{4}$ $[\ ]^{4}_{4}$ $[\ ]^{4}_{4}$ $[12, 0]$ $[4]$ $z^4 + 14$ undefined
17.1.17.28a2.4 4 $x^{17} + 272 x^{12} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{7}{4}]_{4}^{4}$ $[\frac{3}{4}]_{4}^{4}$ $[\ ]^{4}_{4}$ $[\ ]^{4}_{4}$ $[12, 0]$ $[4]$ $z^4 + 12$ undefined
17.1.17.28a3.1 4 $x^{17} + 51 x^{12} + 17$ $C_{17}:C_{8}$ (as 17T4) $136$ $1$ $[\frac{7}{4}]_{4}^{2}$ $[\frac{3}{4}]_{4}^{2}$ $[\ ]^{2}_{4}$ $[\ ]^{2}_{4}$ $[12, 0]$ $[2]$ $z^4 + 15$ undefined
17.1.17.28a3.2 4 $x^{17} + 85 x^{12} + 17$ $C_{17}:C_{8}$ (as 17T4) $136$ $1$ $[\frac{7}{4}]_{4}^{2}$ $[\frac{3}{4}]_{4}^{2}$ $[\ ]^{2}_{4}$ $[\ ]^{2}_{4}$ $[12, 0]$ $[2]$ $z^4 + 8$ undefined
17.1.17.28a3.3 4 $x^{17} + 204 x^{12} + 17$ $C_{17}:C_{8}$ (as 17T4) $136$ $1$ $[\frac{7}{4}]_{4}^{2}$ $[\frac{3}{4}]_{4}^{2}$ $[\ ]^{2}_{4}$ $[\ ]^{2}_{4}$ $[12, 0]$ $[2]$ $z^4 + 9$ undefined
17.1.17.28a3.4 4 $x^{17} + 238 x^{12} + 17$ $C_{17}:C_{8}$ (as 17T4) $136$ $1$ $[\frac{7}{4}]_{4}^{2}$ $[\frac{3}{4}]_{4}^{2}$ $[\ ]^{2}_{4}$ $[\ ]^{2}_{4}$ $[12, 0]$ $[2]$ $z^4 + 2$ undefined
17.1.17.28a4.1 4 $x^{17} + 34 x^{12} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{7}{4}]_{4}^{4}$ $[\frac{3}{4}]_{4}^{4}$ $[\ ]^{4}_{4}$ $[\ ]^{4}_{4}$ $[12, 0]$ $[4]$ $z^4 + 10$ undefined
17.1.17.28a4.2 4 $x^{17} + 136 x^{12} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{7}{4}]_{4}^{4}$ $[\frac{3}{4}]_{4}^{4}$ $[\ ]^{4}_{4}$ $[\ ]^{4}_{4}$ $[12, 0]$ $[4]$ $z^4 + 6$ undefined
17.1.17.28a4.3 4 $x^{17} + 153 x^{12} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{7}{4}]_{4}^{4}$ $[\frac{3}{4}]_{4}^{4}$ $[\ ]^{4}_{4}$ $[\ ]^{4}_{4}$ $[12, 0]$ $[4]$ $z^4 + 11$ undefined
17.1.17.28a4.4 4 $x^{17} + 255 x^{12} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{7}{4}]_{4}^{4}$ $[\frac{3}{4}]_{4}^{4}$ $[\ ]^{4}_{4}$ $[\ ]^{4}_{4}$ $[12, 0]$ $[4]$ $z^4 + 7$ undefined
  displayed columns for results