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Properties

Label 7.2.7.22a
Base 7.1.1.0a1.1
Degree \(14\)
e \(7\)
f \(2\)
c \(22\)

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Defining polynomial over unramified subextension

$x^{7} + 7a_{5} x^{5} + 7$

Invariants

Residue field characteristic: $7$
Degree: $14$
Base field: $\Q_{7}$
Ramification index $e$: $7$
Residue field degree $f$: $2$
Discriminant exponent $c$: $22$
Artin slopes: $[\frac{11}{6}]$
Swan slopes: $[\frac{5}{6}]$
Means: $\langle\frac{5}{7}\rangle$
Rams: $(\frac{5}{6})$
Field count: $27$ (complete)
Ambiguity: $2$
Mass: $48$
Absolute Mass: $24$

Diagrams

Varying

Indices of inseparability: $[5,0]$
Associated inertia: $[1]$
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 27

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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
7.2.7.22a1.1 1 $( x^{2} + 6 x + 3 )^{7} + 28 ( x^{2} + 6 x + 3 )^{5} + 7$ $F_7 \times C_2$ (as 14T7) $84$ $2$ $[\frac{11}{6}]_{6}^{2}$ $[\frac{5}{6}]_{6}^{2}$ $[\ ]_{6}$ $[\ ]_{6}$ $[5, 0]$ $[1]$ $z + (3 t + 2)$ undefined
7.2.7.22a2.1 1 $( x^{2} + 6 x + 3 )^{7} + 7 ( x^{2} + 6 x + 3 )^{5} + 7$ $F_7 \times C_2$ (as 14T7) $84$ $2$ $[\frac{11}{6}]_{6}^{2}$ $[\frac{5}{6}]_{6}^{2}$ $[\ ]_{6}$ $[\ ]_{6}$ $[5, 0]$ $[1]$ $z + (6 t + 4)$ undefined
7.2.7.22a3.1 1 $( x^{2} + 6 x + 3 )^{7} + 35 ( x^{2} + 6 x + 3 )^{5} + 7$ $F_7 \times C_2$ (as 14T7) $84$ $2$ $[\frac{11}{6}]_{6}^{2}$ $[\frac{5}{6}]_{6}^{2}$ $[\ ]_{6}$ $[\ ]_{6}$ $[5, 0]$ $[1]$ $z + (2 t + 6)$ undefined
7.2.7.22a4.1 1 $( x^{2} + 6 x + 3 )^{7} + 14 ( x^{2} + 6 x + 3 )^{5} + 7$ $F_7 \times C_2$ (as 14T7) $84$ $2$ $[\frac{11}{6}]_{6}^{2}$ $[\frac{5}{6}]_{6}^{2}$ $[\ ]_{6}$ $[\ ]_{6}$ $[5, 0]$ $[1]$ $z + (5 t + 1)$ undefined
7.2.7.22a5.1 1 $( x^{2} + 6 x + 3 )^{7} + 42 ( x^{2} + 6 x + 3 )^{5} + 7$ $F_7 \times C_2$ (as 14T7) $84$ $2$ $[\frac{11}{6}]_{6}^{2}$ $[\frac{5}{6}]_{6}^{2}$ $[\ ]_{6}$ $[\ ]_{6}$ $[5, 0]$ $[1]$ $z + (t + 3)$ undefined
7.2.7.22a6.1 1 $( x^{2} + 6 x + 3 )^{7} + 21 ( x^{2} + 6 x + 3 )^{5} + 7$ $F_7 \times C_2$ (as 14T7) $84$ $2$ $[\frac{11}{6}]_{6}^{2}$ $[\frac{5}{6}]_{6}^{2}$ $[\ ]_{6}$ $[\ ]_{6}$ $[5, 0]$ $[1]$ $z + (4 t + 5)$ undefined
7.2.7.22a7.1 1 $( x^{2} + 6 x + 3 )^{7} + 28 x ( x^{2} + 6 x + 3 )^{5} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{11}{6}, \frac{11}{6}]_{6}^{2}$ $[\frac{5}{6},\frac{5}{6}]_{6}^{2}$ $[\frac{11}{6}]_{6}$ $[\frac{5}{6}]_{6}$ $[5, 0]$ $[1]$ $z + (5 t + 4)$ undefined
7.2.7.22a8.1 1 $( x^{2} + 6 x + 3 )^{7} + \left(28 x + 28\right) ( x^{2} + 6 x + 3 )^{5} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{11}{6}, \frac{11}{6}]_{6}^{2}$ $[\frac{5}{6},\frac{5}{6}]_{6}^{2}$ $[\frac{11}{6}]_{6}$ $[\frac{5}{6}]_{6}$ $[5, 0]$ $[1]$ $z + (t + 6)$ undefined
7.2.7.22a9.1 1 $( x^{2} + 6 x + 3 )^{7} + \left(28 x + 14\right) ( x^{2} + 6 x + 3 )^{5} + 7$ $D_7:F_7$ (as 14T24) $588$ $1$ $[\frac{11}{6}, \frac{11}{6}]_{6}^{2}$ $[\frac{5}{6},\frac{5}{6}]_{6}^{2}$ $[\frac{11}{6}]_{6}$ $[\frac{5}{6}]_{6}$ $[5, 0]$ $[1]$ $z + (3 t + 5)$ undefined
7.2.7.22a10.1 1 $( x^{2} + 6 x + 3 )^{7} + \left(28 x + 42\right) ( x^{2} + 6 x + 3 )^{5} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{11}{6}, \frac{11}{6}]_{6}^{2}$ $[\frac{5}{6},\frac{5}{6}]_{6}^{2}$ $[\frac{11}{6}]_{6}$ $[\frac{5}{6}]_{6}$ $[5, 0]$ $[1]$ $z + 6 t$ undefined
7.2.7.22a11.1 1 $( x^{2} + 6 x + 3 )^{7} + 7 x ( x^{2} + 6 x + 3 )^{5} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{11}{6}, \frac{11}{6}]_{6}^{2}$ $[\frac{5}{6},\frac{5}{6}]_{6}^{2}$ $[\frac{11}{6}]_{6}$ $[\frac{5}{6}]_{6}$ $[5, 0]$ $[1]$ $z + (3 t + 1)$ undefined
7.2.7.22a12.1 1 $( x^{2} + 6 x + 3 )^{7} + \left(7 x + 28\right) ( x^{2} + 6 x + 3 )^{5} + 7$ $D_7:F_7$ (as 14T24) $588$ $1$ $[\frac{11}{6}, \frac{11}{6}]_{6}^{2}$ $[\frac{5}{6},\frac{5}{6}]_{6}^{2}$ $[\frac{11}{6}]_{6}$ $[\frac{5}{6}]_{6}$ $[5, 0]$ $[1]$ $z + (6 t + 3)$ undefined
7.2.7.22a13.1 1 $( x^{2} + 6 x + 3 )^{7} + \left(7 x + 35\right) ( x^{2} + 6 x + 3 )^{5} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{11}{6}, \frac{11}{6}]_{6}^{2}$ $[\frac{5}{6},\frac{5}{6}]_{6}^{2}$ $[\frac{11}{6}]_{6}$ $[\frac{5}{6}]_{6}$ $[5, 0]$ $[1]$ $z + 5 t$ undefined
7.2.7.22a14.1 1 $( x^{2} + 6 x + 3 )^{7} + \left(7 x + 14\right) ( x^{2} + 6 x + 3 )^{5} + 7$ $D_7:F_7$ (as 14T24) $588$ $1$ $[\frac{11}{6}, \frac{11}{6}]_{6}^{2}$ $[\frac{5}{6},\frac{5}{6}]_{6}^{2}$ $[\frac{11}{6}]_{6}$ $[\frac{5}{6}]_{6}$ $[5, 0]$ $[1]$ $z + (t + 2)$ undefined
7.2.7.22a15.1 1 $( x^{2} + 6 x + 3 )^{7} + \left(7 x + 21\right) ( x^{2} + 6 x + 3 )^{5} + 7$ $D_7:F_7$ (as 14T24) $588$ $1$ $[\frac{11}{6}, \frac{11}{6}]_{6}^{2}$ $[\frac{5}{6},\frac{5}{6}]_{6}^{2}$ $[\frac{11}{6}]_{6}$ $[\frac{5}{6}]_{6}$ $[5, 0]$ $[1]$ $z + 6$ undefined
7.2.7.22a16.1 1 $( x^{2} + 6 x + 3 )^{7} + 35 x ( x^{2} + 6 x + 3 )^{5} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{11}{6}, \frac{11}{6}]_{6}^{2}$ $[\frac{5}{6},\frac{5}{6}]_{6}^{2}$ $[\frac{11}{6}]_{6}$ $[\frac{5}{6}]_{6}$ $[5, 0]$ $[1]$ $z + (t + 5)$ undefined
7.2.7.22a17.1 1 $( x^{2} + 6 x + 3 )^{7} + \left(35 x + 28\right) ( x^{2} + 6 x + 3 )^{5} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{11}{6}, \frac{11}{6}]_{6}^{2}$ $[\frac{5}{6},\frac{5}{6}]_{6}^{2}$ $[\frac{11}{6}]_{6}$ $[\frac{5}{6}]_{6}$ $[5, 0]$ $[1]$ $z + 4 t$ undefined
7.2.7.22a18.1 1 $( x^{2} + 6 x + 3 )^{7} + \left(35 x + 35\right) ( x^{2} + 6 x + 3 )^{5} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{11}{6}, \frac{11}{6}]_{6}^{2}$ $[\frac{5}{6},\frac{5}{6}]_{6}^{2}$ $[\frac{11}{6}]_{6}$ $[\frac{5}{6}]_{6}$ $[5, 0]$ $[1]$ $z + (3 t + 4)$ undefined
7.2.7.22a19.1 1 $( x^{2} + 6 x + 3 )^{7} + \left(35 x + 42\right) ( x^{2} + 6 x + 3 )^{5} + 7$ $D_7:F_7$ (as 14T24) $588$ $1$ $[\frac{11}{6}, \frac{11}{6}]_{6}^{2}$ $[\frac{5}{6},\frac{5}{6}]_{6}^{2}$ $[\frac{11}{6}]_{6}$ $[\frac{5}{6}]_{6}$ $[5, 0]$ $[1]$ $z + (2 t + 1)$ undefined
7.2.7.22a20.1 1 $( x^{2} + 6 x + 3 )^{7} + 14 x ( x^{2} + 6 x + 3 )^{5} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{11}{6}, \frac{11}{6}]_{6}^{2}$ $[\frac{5}{6},\frac{5}{6}]_{6}^{2}$ $[\frac{11}{6}]_{6}$ $[\frac{5}{6}]_{6}$ $[5, 0]$ $[1]$ $z + (6 t + 2)$ undefined
7.2.7.22a21.1 1 $( x^{2} + 6 x + 3 )^{7} + \left(14 x + 7\right) ( x^{2} + 6 x + 3 )^{5} + 7$ $D_7:F_7$ (as 14T24) $588$ $1$ $[\frac{11}{6}, \frac{11}{6}]_{6}^{2}$ $[\frac{5}{6},\frac{5}{6}]_{6}^{2}$ $[\frac{11}{6}]_{6}$ $[\frac{5}{6}]_{6}$ $[5, 0]$ $[1]$ $z + (5 t + 6)$ undefined
7.2.7.22a22.1 1 $( x^{2} + 6 x + 3 )^{7} + \left(14 x + 42\right) ( x^{2} + 6 x + 3 )^{5} + 7$ $D_7:F_7$ (as 14T24) $588$ $1$ $[\frac{11}{6}, \frac{11}{6}]_{6}^{2}$ $[\frac{5}{6},\frac{5}{6}]_{6}^{2}$ $[\frac{11}{6}]_{6}$ $[\frac{5}{6}]_{6}$ $[5, 0]$ $[1]$ $z + 5$ undefined
7.2.7.22a23.1 1 $( x^{2} + 6 x + 3 )^{7} + 42 x ( x^{2} + 6 x + 3 )^{5} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{11}{6}, \frac{11}{6}]_{6}^{2}$ $[\frac{5}{6},\frac{5}{6}]_{6}^{2}$ $[\frac{11}{6}]_{6}$ $[\frac{5}{6}]_{6}$ $[5, 0]$ $[1]$ $z + (4 t + 6)$ undefined
7.2.7.22a24.1 1 $( x^{2} + 6 x + 3 )^{7} + \left(42 x + 14\right) ( x^{2} + 6 x + 3 )^{5} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{11}{6}, \frac{11}{6}]_{6}^{2}$ $[\frac{5}{6},\frac{5}{6}]_{6}^{2}$ $[\frac{11}{6}]_{6}$ $[\frac{5}{6}]_{6}$ $[5, 0]$ $[1]$ $z + 2 t$ undefined
7.2.7.22a25.1 1 $( x^{2} + 6 x + 3 )^{7} + 21 x ( x^{2} + 6 x + 3 )^{5} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{11}{6}, \frac{11}{6}]_{6}^{2}$ $[\frac{5}{6},\frac{5}{6}]_{6}^{2}$ $[\frac{11}{6}]_{6}$ $[\frac{5}{6}]_{6}$ $[5, 0]$ $[1]$ $z + (2 t + 3)$ undefined
7.2.7.22a26.1 1 $( x^{2} + 6 x + 3 )^{7} + \left(21 x + 35\right) ( x^{2} + 6 x + 3 )^{5} + 7$ $D_7:F_7$ (as 14T24) $588$ $1$ $[\frac{11}{6}, \frac{11}{6}]_{6}^{2}$ $[\frac{5}{6},\frac{5}{6}]_{6}^{2}$ $[\frac{11}{6}]_{6}$ $[\frac{5}{6}]_{6}$ $[5, 0]$ $[1]$ $z + (4 t + 2)$ undefined
7.2.7.22a27.1 1 $( x^{2} + 6 x + 3 )^{7} + \left(21 x + 14\right) ( x^{2} + 6 x + 3 )^{5} + 7$ $D_7:F_7$ (as 14T24) $588$ $1$ $[\frac{11}{6}, \frac{11}{6}]_{6}^{2}$ $[\frac{5}{6},\frac{5}{6}]_{6}^{2}$ $[\frac{11}{6}]_{6}$ $[\frac{5}{6}]_{6}$ $[5, 0]$ $[1]$ $z + 4$ undefined
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