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Properties

Label 73.1.8.7a
Base 73.1.1.0a1.1
Degree \(8\)
e \(8\)
f \(1\)
c \(7\)

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Defining polynomial

$x^{8} + 73d_{0}$

Invariants

Residue field characteristic: $73$
Degree: $8$
Base field: $\Q_{73}$
Ramification index $e$: $8$
Residue field degree $f$: $1$
Discriminant exponent $c$: $7$
Artin slopes: $[\ ]$
Swan slopes: $[\ ]$
Means: $\langle\ \rangle$
Rams: $(\ )$
Field count: $8$ (complete)
Ambiguity: $8$
Mass: $1$
Absolute Mass: $1$

Varying

Indices of inseparability: $[0]$
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 8

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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
73.1.8.7a1.1 8 $x^{8} + 73$ $C_8$ (as 8T1) $8$ $8$ $[\ ]_{8}$ $[\ ]_{8}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^7 + 8 z^6 + 28 z^5 + 56 z^4 + 70 z^3 + 56 z^2 + 28 z + 8$ undefined
73.1.8.7a1.2 8 $x^{8} + 219$ $C_8$ (as 8T1) $8$ $8$ $[\ ]_{8}$ $[\ ]_{8}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^7 + 8 z^6 + 28 z^5 + 56 z^4 + 70 z^3 + 56 z^2 + 28 z + 8$ undefined
73.1.8.7a1.3 8 $x^{8} + 365$ $C_8$ (as 8T1) $8$ $8$ $[\ ]_{8}$ $[\ ]_{8}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^7 + 8 z^6 + 28 z^5 + 56 z^4 + 70 z^3 + 56 z^2 + 28 z + 8$ undefined
73.1.8.7a1.4 8 $x^{8} + 1095$ $C_8$ (as 8T1) $8$ $8$ $[\ ]_{8}$ $[\ ]_{8}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^7 + 8 z^6 + 28 z^5 + 56 z^4 + 70 z^3 + 56 z^2 + 28 z + 8$ undefined
73.1.8.7a1.5 8 $x^{8} + 1825$ $C_8$ (as 8T1) $8$ $8$ $[\ ]_{8}$ $[\ ]_{8}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^7 + 8 z^6 + 28 z^5 + 56 z^4 + 70 z^3 + 56 z^2 + 28 z + 8$ undefined
73.1.8.7a1.6 8 $x^{8} + 2993$ $C_8$ (as 8T1) $8$ $8$ $[\ ]_{8}$ $[\ ]_{8}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^7 + 8 z^6 + 28 z^5 + 56 z^4 + 70 z^3 + 56 z^2 + 28 z + 8$ undefined
73.1.8.7a1.7 8 $x^{8} + 3796$ $C_8$ (as 8T1) $8$ $8$ $[\ ]_{8}$ $[\ ]_{8}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^7 + 8 z^6 + 28 z^5 + 56 z^4 + 70 z^3 + 56 z^2 + 28 z + 8$ undefined
73.1.8.7a1.8 8 $x^{8} + 4307$ $C_8$ (as 8T1) $8$ $8$ $[\ ]_{8}$ $[\ ]_{8}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^7 + 8 z^6 + 28 z^5 + 56 z^4 + 70 z^3 + 56 z^2 + 28 z + 8$ undefined
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