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Properties

Label 7.1.14.15a
Base 7.1.1.0a1.1
Degree \(14\)
e \(14\)
f \(1\)
c \(15\)

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Defining polynomial

$x^{14} + 7 a_{2} x^{2} + 7 d_{0}$

Invariants

Residue field characteristic: $7$
Degree: $14$
Base field: $\Q_{7}$
Ramification index $e$: $14$
Residue field degree $f$: $1$
Discriminant exponent $c$: $15$
Artin slopes: $[\frac{7}{6}]$
Swan slopes: $[\frac{1}{6}]$
Means: $\langle\frac{1}{7}\rangle$
Rams: $(\frac{1}{3})$
Field count: $12$ (complete)
Ambiguity: $2$
Mass: $6$
Absolute Mass: $6$

Diagrams

Varying

Indices of inseparability: $[2,0]$
Associated inertia: $[1,1]$ (show 6), $[1,2]$ (show 6)
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 12

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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
7.1.14.15a1.1 6 $x^{14} + 21 x^{2} + 7$ $F_7 \times C_2$ (as 14T7) $84$ $2$ $[\frac{7}{6}]_{6}^{2}$ $[\frac{1}{6}]_{6}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[2, 0]$ $[1, 2]$ $z^7 + 2,2 z^2 + 1$ undefined
7.1.14.15a1.2 6 $x^{14} + 35 x^{2} + 7$ $F_7 \times C_2$ (as 14T7) $84$ $2$ $[\frac{7}{6}]_{6}^{2}$ $[\frac{1}{6}]_{6}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[2, 0]$ $[1, 2]$ $z^7 + 2,2 z^2 + 4$ undefined
7.1.14.15a1.3 6 $x^{14} + 42 x^{2} + 7$ $F_7 \times C_2$ (as 14T7) $84$ $2$ $[\frac{7}{6}]_{6}^{2}$ $[\frac{1}{6}]_{6}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[2, 0]$ $[1, 2]$ $z^7 + 2,2 z^2 + 2$ undefined
7.1.14.15a1.4 6 $x^{14} + 7 x^{2} + 21$ $F_7 \times C_2$ (as 14T7) $84$ $2$ $[\frac{7}{6}]_{6}^{2}$ $[\frac{1}{6}]_{6}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[2, 0]$ $[1, 2]$ $z^7 + 2,2 z^2 + 4$ undefined
7.1.14.15a1.5 6 $x^{14} + 14 x^{2} + 21$ $F_7 \times C_2$ (as 14T7) $84$ $2$ $[\frac{7}{6}]_{6}^{2}$ $[\frac{1}{6}]_{6}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[2, 0]$ $[1, 2]$ $z^7 + 2,2 z^2 + 1$ undefined
7.1.14.15a1.6 6 $x^{14} + 28 x^{2} + 21$ $F_7 \times C_2$ (as 14T7) $84$ $2$ $[\frac{7}{6}]_{6}^{2}$ $[\frac{1}{6}]_{6}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[2, 0]$ $[1, 2]$ $z^7 + 2,2 z^2 + 2$ undefined
7.1.14.15a2.1 6 $x^{14} + 7 x^{2} + 7$ $F_7$ (as 14T4) $42$ $2$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[\ ]_{3}$ $[\ ]_{3}$ $[2, 0]$ $[1, 1]$ $z^7 + 2,2 z^2 + 5$ undefined
7.1.14.15a2.2 6 $x^{14} + 14 x^{2} + 7$ $F_7$ (as 14T4) $42$ $2$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[\ ]_{3}$ $[\ ]_{3}$ $[2, 0]$ $[1, 1]$ $z^7 + 2,2 z^2 + 3$ undefined
7.1.14.15a2.3 6 $x^{14} + 28 x^{2} + 7$ $F_7$ (as 14T4) $42$ $2$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[\ ]_{3}$ $[\ ]_{3}$ $[2, 0]$ $[1, 1]$ $z^7 + 2,2 z^2 + 6$ undefined
7.1.14.15a2.4 6 $x^{14} + 21 x^{2} + 21$ $F_7$ (as 14T4) $42$ $2$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[\ ]_{3}$ $[\ ]_{3}$ $[2, 0]$ $[1, 1]$ $z^7 + 2,2 z^2 + 5$ undefined
7.1.14.15a2.5 6 $x^{14} + 35 x^{2} + 21$ $F_7$ (as 14T4) $42$ $2$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[\ ]_{3}$ $[\ ]_{3}$ $[2, 0]$ $[1, 1]$ $z^7 + 2,2 z^2 + 6$ undefined
7.1.14.15a2.6 6 $x^{14} + 42 x^{2} + 21$ $F_7$ (as 14T4) $42$ $2$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[\ ]_{3}$ $[\ ]_{3}$ $[2, 0]$ $[1, 1]$ $z^7 + 2,2 z^2 + 3$ undefined
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