Properties

Label 149.1.5.4a1.1-1.4.3a
Base 149.1.5.4a1.1
Degree \(4\)
e \(4\)
f \(1\)
c \(3\)

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Defining polynomial

$x^{4} + d_{0} \pi$

Invariants

Residue field characteristic: $149$
Degree: $4$
Base field: 149.1.5.4a1.1
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $3$
Absolute Artin slopes: $[\ ]$
Swan slopes: $[\ ]$
Means: $\langle\ \rangle$
Rams: $(\ )$
Field count: $4$ (complete)
Ambiguity: $4$
Mass: $1$
Absolute Mass: $1$

Varying

These invariants are all associated to absolute extensions of $\Q_{ 149 }$ within this relative family, not the relative extension.

Galois group: $C_4\times D_5$
Hidden Artin slopes: $[\ ]^{2}$
Indices of inseparability: $[0]$
Associated inertia: $[2]$
Jump Set: undefined

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Label Polynomial $/ \Q_p$ Galois group $/ \Q_p$ Galois degree $/ \Q_p$ $\#\Aut(K/\Q_p)$ Artin slope content $/ \Q_p$ Swan slope content $/ \Q_p$ Hidden Artin slopes $/ \Q_p$ Hidden Swan slopes $/ \Q_p$ Ind. of Insep. $/ \Q_p$ Assoc. Inertia $/ \Q_p$ Resid. Poly Jump Set
149.1.20.19a1.1 $x^{20} + 149$ $C_4\times D_5$ (as 20T6) $40$ $4$ $[\ ]_{20}^{2}$ $[\ ]_{20}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[2]$ $z^{19} + 20 z^{18} + 41 z^{17} + 97 z^{16} + 77 z^{15} + 8 z^{14} + 20 z^{13} + 40 z^{12} + 65 z^{11} + 37 z^{10} + 145 z^9 + 37 z^8 + 65 z^7 + 40 z^6 + 20 z^5 + 8 z^4 + 77 z^3 + 97 z^2 + 41 z + 20$ undefined
149.1.20.19a1.2 $x^{20} + 298$ $C_4\times D_5$ (as 20T6) $40$ $4$ $[\ ]_{20}^{2}$ $[\ ]_{20}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[2]$ $z^{19} + 20 z^{18} + 41 z^{17} + 97 z^{16} + 77 z^{15} + 8 z^{14} + 20 z^{13} + 40 z^{12} + 65 z^{11} + 37 z^{10} + 145 z^9 + 37 z^8 + 65 z^7 + 40 z^6 + 20 z^5 + 8 z^4 + 77 z^3 + 97 z^2 + 41 z + 20$ undefined
149.1.20.19a1.3 $x^{20} + 596$ $C_4\times D_5$ (as 20T6) $40$ $4$ $[\ ]_{20}^{2}$ $[\ ]_{20}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[2]$ $z^{19} + 20 z^{18} + 41 z^{17} + 97 z^{16} + 77 z^{15} + 8 z^{14} + 20 z^{13} + 40 z^{12} + 65 z^{11} + 37 z^{10} + 145 z^9 + 37 z^8 + 65 z^7 + 40 z^6 + 20 z^5 + 8 z^4 + 77 z^3 + 97 z^2 + 41 z + 20$ undefined
149.1.20.19a1.4 $x^{20} + 1192$ $C_4\times D_5$ (as 20T6) $40$ $4$ $[\ ]_{20}^{2}$ $[\ ]_{20}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[2]$ $z^{19} + 20 z^{18} + 41 z^{17} + 97 z^{16} + 77 z^{15} + 8 z^{14} + 20 z^{13} + 40 z^{12} + 65 z^{11} + 37 z^{10} + 145 z^9 + 37 z^8 + 65 z^7 + 40 z^6 + 20 z^5 + 8 z^4 + 77 z^3 + 97 z^2 + 41 z + 20$ undefined
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