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Properties

Label 17.1.17.31a
Base 17.1.1.0a1.1
Degree \(17\)
e \(17\)
f \(1\)
c \(31\)

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Defining polynomial

$x^{17} + 17a_{15} x^{15} + 17$

Invariants

Residue field characteristic: $17$
Degree: $17$
Base field: $\Q_{17}$
Ramification index $e$: $17$
Residue field degree $f$: $1$
Discriminant exponent $c$: $31$
Artin slopes: $[\frac{31}{16}]$
Swan slopes: $[\frac{15}{16}]$
Means: $\langle\frac{15}{17}\rangle$
Rams: $(\frac{15}{16})$
Field count: $16$ (complete)
Ambiguity: $1$
Mass: $16$
Absolute Mass: $16$

Diagrams

Varying

Indices of inseparability: $[15,0]$
Associated inertia: $[1]$
Jump Set: undefined

Galois groups and Hidden Artin slopes

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
17.1.17.31a1.1 16 $x^{17} + 17 x^{15} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{31}{16}]_{16}$ $[\frac{15}{16}]_{16}$ $[\ ]_{16}$ $[\ ]_{16}$ $[15, 0]$ $[1]$ $z + 2$ undefined
17.1.17.31a1.2 16 $x^{17} + 34 x^{15} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{31}{16}]_{16}$ $[\frac{15}{16}]_{16}$ $[\ ]_{16}$ $[\ ]_{16}$ $[15, 0]$ $[1]$ $z + 4$ undefined
17.1.17.31a1.3 16 $x^{17} + 51 x^{15} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{31}{16}]_{16}$ $[\frac{15}{16}]_{16}$ $[\ ]_{16}$ $[\ ]_{16}$ $[15, 0]$ $[1]$ $z + 6$ undefined
17.1.17.31a1.4 16 $x^{17} + 68 x^{15} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{31}{16}]_{16}$ $[\frac{15}{16}]_{16}$ $[\ ]_{16}$ $[\ ]_{16}$ $[15, 0]$ $[1]$ $z + 8$ undefined
17.1.17.31a1.5 16 $x^{17} + 85 x^{15} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{31}{16}]_{16}$ $[\frac{15}{16}]_{16}$ $[\ ]_{16}$ $[\ ]_{16}$ $[15, 0]$ $[1]$ $z + 10$ undefined
17.1.17.31a1.6 16 $x^{17} + 102 x^{15} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{31}{16}]_{16}$ $[\frac{15}{16}]_{16}$ $[\ ]_{16}$ $[\ ]_{16}$ $[15, 0]$ $[1]$ $z + 12$ undefined
17.1.17.31a1.7 16 $x^{17} + 119 x^{15} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{31}{16}]_{16}$ $[\frac{15}{16}]_{16}$ $[\ ]_{16}$ $[\ ]_{16}$ $[15, 0]$ $[1]$ $z + 14$ undefined
17.1.17.31a1.8 16 $x^{17} + 136 x^{15} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{31}{16}]_{16}$ $[\frac{15}{16}]_{16}$ $[\ ]_{16}$ $[\ ]_{16}$ $[15, 0]$ $[1]$ $z + 16$ undefined
17.1.17.31a1.9 16 $x^{17} + 153 x^{15} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{31}{16}]_{16}$ $[\frac{15}{16}]_{16}$ $[\ ]_{16}$ $[\ ]_{16}$ $[15, 0]$ $[1]$ $z + 1$ undefined
17.1.17.31a1.10 16 $x^{17} + 170 x^{15} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{31}{16}]_{16}$ $[\frac{15}{16}]_{16}$ $[\ ]_{16}$ $[\ ]_{16}$ $[15, 0]$ $[1]$ $z + 3$ undefined
17.1.17.31a1.11 16 $x^{17} + 187 x^{15} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{31}{16}]_{16}$ $[\frac{15}{16}]_{16}$ $[\ ]_{16}$ $[\ ]_{16}$ $[15, 0]$ $[1]$ $z + 5$ undefined
17.1.17.31a1.12 16 $x^{17} + 204 x^{15} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{31}{16}]_{16}$ $[\frac{15}{16}]_{16}$ $[\ ]_{16}$ $[\ ]_{16}$ $[15, 0]$ $[1]$ $z + 7$ undefined
17.1.17.31a1.13 16 $x^{17} + 221 x^{15} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{31}{16}]_{16}$ $[\frac{15}{16}]_{16}$ $[\ ]_{16}$ $[\ ]_{16}$ $[15, 0]$ $[1]$ $z + 9$ undefined
17.1.17.31a1.14 16 $x^{17} + 238 x^{15} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{31}{16}]_{16}$ $[\frac{15}{16}]_{16}$ $[\ ]_{16}$ $[\ ]_{16}$ $[15, 0]$ $[1]$ $z + 11$ undefined
17.1.17.31a1.15 16 $x^{17} + 255 x^{15} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{31}{16}]_{16}$ $[\frac{15}{16}]_{16}$ $[\ ]_{16}$ $[\ ]_{16}$ $[15, 0]$ $[1]$ $z + 13$ undefined
17.1.17.31a1.16 16 $x^{17} + 272 x^{15} + 17$ $F_{17}$ (as 17T5) $272$ $1$ $[\frac{31}{16}]_{16}$ $[\frac{15}{16}]_{16}$ $[\ ]_{16}$ $[\ ]_{16}$ $[15, 0]$ $[1]$ $z + 15$ undefined
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