Properties

Label 29.1.20.19a
Base 29.1.1.0a1.1
Degree \(20\)
e \(20\)
f \(1\)
c \(19\)

Related objects

Downloads

Learn more

Defining polynomial

$x^{20} + 29d_{0}$

Invariants

Residue field characteristic: $29$
Degree: $20$
Base field: $\Q_{29}$
Ramification index $e$: $20$
Residue field degree $f$: $1$
Discriminant exponent $c$: $19$
Artin slopes: $[\ ]$
Swan slopes: $[\ ]$
Means: $\langle\ \rangle$
Rams: $(\ )$
Field count: $4$ (complete)
Ambiguity: $4$
Mass: $1$
Absolute Mass: $1$

Varying

Indices of inseparability: $[0]$
Associated inertia: $[2]$
Jump Set: undefined

Galois groups and Hidden Artin slopes

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
Sort order Select

Showing all 4

  displayed columns for results
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
29.1.20.19a1.1 4 $x^{20} + 29$ $C_4\times D_5$ (as 20T6) $40$ $4$ $[\ ]_{20}^{2}$ $[\ ]_{20}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[2]$ $z^{19} + 20 z^{18} + 16 z^{17} + 9 z^{16} + 2 z^{15} + 18 z^{14} + 16 z^{13} + 3 z^{12} + 23 z^{11} + 21 z^{10} + 26 z^9 + 21 z^8 + 23 z^7 + 3 z^6 + 16 z^5 + 18 z^4 + 2 z^3 + 9 z^2 + 16 z + 20$ undefined
29.1.20.19a1.2 4 $x^{20} + 58$ $C_4\times D_5$ (as 20T6) $40$ $4$ $[\ ]_{20}^{2}$ $[\ ]_{20}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[2]$ $z^{19} + 20 z^{18} + 16 z^{17} + 9 z^{16} + 2 z^{15} + 18 z^{14} + 16 z^{13} + 3 z^{12} + 23 z^{11} + 21 z^{10} + 26 z^9 + 21 z^8 + 23 z^7 + 3 z^6 + 16 z^5 + 18 z^4 + 2 z^3 + 9 z^2 + 16 z + 20$ undefined
29.1.20.19a1.3 4 $x^{20} + 116$ $C_4\times D_5$ (as 20T6) $40$ $4$ $[\ ]_{20}^{2}$ $[\ ]_{20}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[2]$ $z^{19} + 20 z^{18} + 16 z^{17} + 9 z^{16} + 2 z^{15} + 18 z^{14} + 16 z^{13} + 3 z^{12} + 23 z^{11} + 21 z^{10} + 26 z^9 + 21 z^8 + 23 z^7 + 3 z^6 + 16 z^5 + 18 z^4 + 2 z^3 + 9 z^2 + 16 z + 20$ undefined
29.1.20.19a1.4 4 $x^{20} + 232$ $C_4\times D_5$ (as 20T6) $40$ $4$ $[\ ]_{20}^{2}$ $[\ ]_{20}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[2]$ $z^{19} + 20 z^{18} + 16 z^{17} + 9 z^{16} + 2 z^{15} + 18 z^{14} + 16 z^{13} + 3 z^{12} + 23 z^{11} + 21 z^{10} + 26 z^9 + 21 z^8 + 23 z^7 + 3 z^6 + 16 z^5 + 18 z^4 + 2 z^3 + 9 z^2 + 16 z + 20$ undefined
  displayed columns for results