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Properties

Label 7.2.1.0a1.1-1.7.7a
Base 7.2.1.0a1.1
Degree \(7\)
e \(7\)
f \(1\)
c \(7\)

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Defining polynomial

$x^{7} + 7a_{1} x + 7$

Invariants

Residue field characteristic: $7$
Degree: $7$
Base field: $\Q_{7}(\sqrt{3})$
Ramification index $e$: $7$
Residue field degree $f$: $1$
Discriminant exponent $c$: $7$
Absolute Artin slopes: $[\frac{7}{6}]$
Swan slopes: $[\frac{1}{6}]$
Means: $\langle\frac{1}{7}\rangle$
Rams: $(\frac{1}{6})$
Field count: $27$ (complete)
Ambiguity: $1$
Mass: $48$
Absolute Mass: $24$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 7 }$ within this relative family, not the relative extension.

Galois group: $F_7 \times C_2$ (show 6), $C_7^2:C_{12}$ (show 12), $D_7:F_7$ (show 9)
Hidden Artin slopes: $[\frac{7}{6}]_{6}$ (show 21), $[\ ]_{6}$ (show 6)
Indices of inseparability: $[1,0]$
Associated inertia: $[1]$
Jump Set: undefined

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 27

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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
7.2.7.14a1.1 $( x^{2} + 6 x + 3 )^{7} + 42 ( x^{2} + 6 x + 3 ) + 7$ $F_7 \times C_2$ (as 14T7) $84$ $2$ $[\frac{7}{6}]_{6}^{2}$ $[\frac{1}{6}]_{6}^{2}$ $[\ ]_{6}$ $[\ ]_{6}$ $[1, 0]$ $[1]$ $z + (5 t + 1)$ undefined
7.2.7.14a2.1 $( x^{2} + 6 x + 3 )^{7} + 35 ( x^{2} + 6 x + 3 ) + 7$ $F_7 \times C_2$ (as 14T7) $84$ $2$ $[\frac{7}{6}]_{6}^{2}$ $[\frac{1}{6}]_{6}^{2}$ $[\ ]_{6}$ $[\ ]_{6}$ $[1, 0]$ $[1]$ $z + (3 t + 2)$ undefined
7.2.7.14a3.1 $( x^{2} + 6 x + 3 )^{7} + 28 ( x^{2} + 6 x + 3 ) + 7$ $F_7 \times C_2$ (as 14T7) $84$ $2$ $[\frac{7}{6}]_{6}^{2}$ $[\frac{1}{6}]_{6}^{2}$ $[\ ]_{6}$ $[\ ]_{6}$ $[1, 0]$ $[1]$ $z + (t + 3)$ undefined
7.2.7.14a4.1 $( x^{2} + 6 x + 3 )^{7} + 21 ( x^{2} + 6 x + 3 ) + 7$ $F_7 \times C_2$ (as 14T7) $84$ $2$ $[\frac{7}{6}]_{6}^{2}$ $[\frac{1}{6}]_{6}^{2}$ $[\ ]_{6}$ $[\ ]_{6}$ $[1, 0]$ $[1]$ $z + (6 t + 4)$ undefined
7.2.7.14a5.1 $( x^{2} + 6 x + 3 )^{7} + 14 ( x^{2} + 6 x + 3 ) + 7$ $F_7 \times C_2$ (as 14T7) $84$ $2$ $[\frac{7}{6}]_{6}^{2}$ $[\frac{1}{6}]_{6}^{2}$ $[\ ]_{6}$ $[\ ]_{6}$ $[1, 0]$ $[1]$ $z + (4 t + 5)$ undefined
7.2.7.14a6.1 $( x^{2} + 6 x + 3 )^{7} + 7 ( x^{2} + 6 x + 3 ) + 7$ $F_7 \times C_2$ (as 14T7) $84$ $2$ $[\frac{7}{6}]_{6}^{2}$ $[\frac{1}{6}]_{6}^{2}$ $[\ ]_{6}$ $[\ ]_{6}$ $[1, 0]$ $[1]$ $z + (2 t + 6)$ undefined
7.2.7.14a7.1 $( x^{2} + 6 x + 3 )^{7} + 42 x ( x^{2} + 6 x + 3 ) + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[1, 0]$ $[1]$ $z + (6 t + 2)$ undefined
7.2.7.14a8.1 $( x^{2} + 6 x + 3 )^{7} + \left(42 x + 42\right) ( x^{2} + 6 x + 3 ) + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[1, 0]$ $[1]$ $z + (4 t + 3)$ undefined
7.2.7.14a9.1 $( x^{2} + 6 x + 3 )^{7} + \left(42 x + 21\right) ( x^{2} + 6 x + 3 ) + 7$ $D_7:F_7$ (as 14T24) $588$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[1, 0]$ $[1]$ $z + (5 t + 6)$ undefined
7.2.7.14a10.1 $( x^{2} + 6 x + 3 )^{7} + \left(42 x + 14\right) ( x^{2} + 6 x + 3 ) + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[1, 0]$ $[1]$ $z + 3 t$ undefined
7.2.7.14a11.1 $( x^{2} + 6 x + 3 )^{7} + 35 x ( x^{2} + 6 x + 3 ) + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[1, 0]$ $[1]$ $z + (5 t + 4)$ undefined
7.2.7.14a12.1 $( x^{2} + 6 x + 3 )^{7} + \left(35 x + 42\right) ( x^{2} + 6 x + 3 ) + 7$ $D_7:F_7$ (as 14T24) $588$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[1, 0]$ $[1]$ $z + (3 t + 5)$ undefined
7.2.7.14a13.1 $( x^{2} + 6 x + 3 )^{7} + \left(35 x + 28\right) ( x^{2} + 6 x + 3 ) + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[1, 0]$ $[1]$ $z + 6 t$ undefined
7.2.7.14a14.1 $( x^{2} + 6 x + 3 )^{7} + \left(35 x + 21\right) ( x^{2} + 6 x + 3 ) + 7$ $D_7:F_7$ (as 14T24) $588$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[1, 0]$ $[1]$ $z + (4 t + 1)$ undefined
7.2.7.14a15.1 $( x^{2} + 6 x + 3 )^{7} + \left(35 x + 7\right) ( x^{2} + 6 x + 3 ) + 7$ $D_7:F_7$ (as 14T24) $588$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[1, 0]$ $[1]$ $z + 3$ undefined
7.2.7.14a16.1 $( x^{2} + 6 x + 3 )^{7} + 28 x ( x^{2} + 6 x + 3 ) + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[1, 0]$ $[1]$ $z + (4 t + 6)$ undefined
7.2.7.14a17.1 $( x^{2} + 6 x + 3 )^{7} + \left(28 x + 42\right) ( x^{2} + 6 x + 3 ) + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[1, 0]$ $[1]$ $z + 2 t$ undefined
7.2.7.14a18.1 $( x^{2} + 6 x + 3 )^{7} + \left(28 x + 28\right) ( x^{2} + 6 x + 3 ) + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[1, 0]$ $[1]$ $z + (5 t + 2)$ undefined
7.2.7.14a19.1 $( x^{2} + 6 x + 3 )^{7} + \left(28 x + 14\right) ( x^{2} + 6 x + 3 ) + 7$ $D_7:F_7$ (as 14T24) $588$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[1, 0]$ $[1]$ $z + (t + 4)$ undefined
7.2.7.14a20.1 $( x^{2} + 6 x + 3 )^{7} + 21 x ( x^{2} + 6 x + 3 ) + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[1, 0]$ $[1]$ $z + (3 t + 1)$ undefined
7.2.7.14a21.1 $( x^{2} + 6 x + 3 )^{7} + \left(21 x + 35\right) ( x^{2} + 6 x + 3 ) + 7$ $D_7:F_7$ (as 14T24) $588$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[1, 0]$ $[1]$ $z + (6 t + 3)$ undefined
7.2.7.14a22.1 $( x^{2} + 6 x + 3 )^{7} + \left(21 x + 14\right) ( x^{2} + 6 x + 3 ) + 7$ $D_7:F_7$ (as 14T24) $588$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[1, 0]$ $[1]$ $z + 6$ undefined
7.2.7.14a23.1 $( x^{2} + 6 x + 3 )^{7} + 14 x ( x^{2} + 6 x + 3 ) + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[1, 0]$ $[1]$ $z + (2 t + 3)$ undefined
7.2.7.14a24.1 $( x^{2} + 6 x + 3 )^{7} + \left(14 x + 21\right) ( x^{2} + 6 x + 3 ) + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[1, 0]$ $[1]$ $z + t$ undefined
7.2.7.14a25.1 $( x^{2} + 6 x + 3 )^{7} + 7 x ( x^{2} + 6 x + 3 ) + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[1, 0]$ $[1]$ $z + (t + 5)$ undefined
7.2.7.14a26.1 $( x^{2} + 6 x + 3 )^{7} + \left(7 x + 28\right) ( x^{2} + 6 x + 3 ) + 7$ $D_7:F_7$ (as 14T24) $588$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[1, 0]$ $[1]$ $z + (2 t + 1)$ undefined
7.2.7.14a27.1 $( x^{2} + 6 x + 3 )^{7} + \left(7 x + 21\right) ( x^{2} + 6 x + 3 ) + 7$ $D_7:F_7$ (as 14T24) $588$ $1$ $[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ $[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ $[\frac{7}{6}]_{6}$ $[\frac{1}{6}]_{6}$ $[1, 0]$ $[1]$ $z + 2$ undefined
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