↑

Properties

Label 7.2.1.0a1.1-1.7.10a
Base 7.2.1.0a1.1
Degree \(7\)
e \(7\)
f \(1\)
c \(10\)

Related objects

Downloads

Learn more

Defining polynomial

$x^{7} + 7a_{4} x^{4} + 7$

Invariants

Residue field characteristic: $7$
Degree: $7$
Base field: $\Q_{7}(\sqrt{3})$
Ramification index $e$: $7$
Residue field degree $f$: $1$
Discriminant exponent $c$: $10$
Absolute Artin slopes: $[\frac{5}{3}]$
Swan slopes: $[\frac{2}{3}]$
Means: $\langle\frac{4}{7}\rangle$
Rams: $(\frac{2}{3})$
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: $F_7$ (show 3), $(C_7:C_3) \times C_2$ (show 3), $C_7:F_7$ (show 9), $C_7^2:C_{12}$ (show 12)
Hidden Artin slopes: $[\frac{5}{3}]^{2}_{3}$ (show 12), $[\frac{5}{3}]_{3}$ (show 9), $[\ ]_{3}$ (show 6)
Indices of inseparability: $[4,0]$
Associated inertia: $[1]$ (show 15), $[2]$ (show 12)
Jump Set: undefined

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
Sort order Select

Showing all 27

  displayed columns for results
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
7.2.7.20a1.1 $( x^{2} + 6 x + 3 )^{7} + 35 ( x^{2} + 6 x + 3 )^{4} + 7$ $F_7$ (as 14T4) $42$ $2$ $[\frac{5}{3}]_{3}^{2}$ $[\frac{2}{3}]_{3}^{2}$ $[\ ]_{3}$ $[\ ]_{3}$ $[4, 0]$ $[1]$ $z^2 + 2$ undefined
7.2.7.20a2.1 $( x^{2} + 6 x + 3 )^{7} + 21 ( x^{2} + 6 x + 3 )^{4} + 7$ $F_7$ (as 14T4) $42$ $2$ $[\frac{5}{3}]_{3}^{2}$ $[\frac{2}{3}]_{3}^{2}$ $[\ ]_{3}$ $[\ ]_{3}$ $[4, 0]$ $[1]$ $z^2 + 4$ undefined
7.2.7.20a3.1 $( x^{2} + 6 x + 3 )^{7} + 7 ( x^{2} + 6 x + 3 )^{4} + 7$ $(C_7:C_3) \times C_2$ (as 14T5) $42$ $2$ $[\frac{5}{3}]_{3}^{2}$ $[\frac{2}{3}]_{3}^{2}$ $[\ ]_{3}$ $[\ ]_{3}$ $[4, 0]$ $[1]$ $z^2 + 6$ undefined
7.2.7.20a4.1 $( x^{2} + 6 x + 3 )^{7} + 42 ( x^{2} + 6 x + 3 )^{4} + 7$ $F_7$ (as 14T4) $42$ $2$ $[\frac{5}{3}]_{3}^{2}$ $[\frac{2}{3}]_{3}^{2}$ $[\ ]_{3}$ $[\ ]_{3}$ $[4, 0]$ $[1]$ $z^2 + 1$ undefined
7.2.7.20a5.1 $( x^{2} + 6 x + 3 )^{7} + 28 ( x^{2} + 6 x + 3 )^{4} + 7$ $(C_7:C_3) \times C_2$ (as 14T5) $42$ $2$ $[\frac{5}{3}]_{3}^{2}$ $[\frac{2}{3}]_{3}^{2}$ $[\ ]_{3}$ $[\ ]_{3}$ $[4, 0]$ $[1]$ $z^2 + 3$ undefined
7.2.7.20a6.1 $( x^{2} + 6 x + 3 )^{7} + 14 ( x^{2} + 6 x + 3 )^{4} + 7$ $(C_7:C_3) \times C_2$ (as 14T5) $42$ $2$ $[\frac{5}{3}]_{3}^{2}$ $[\frac{2}{3}]_{3}^{2}$ $[\ ]_{3}$ $[\ ]_{3}$ $[4, 0]$ $[1]$ $z^2 + 5$ undefined
7.2.7.20a7.1 $( x^{2} + 6 x + 3 )^{7} + 35 x ( x^{2} + 6 x + 3 )^{4} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{4}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{4}$ $[\frac{5}{3}]^{2}_{3}$ $[\frac{2}{3}]^{2}_{3}$ $[4, 0]$ $[2]$ $z^2 + (5 t + 2)$ undefined
7.2.7.20a8.1 $( x^{2} + 6 x + 3 )^{7} + \left(35 x + 35\right) ( x^{2} + 6 x + 3 )^{4} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{4}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{4}$ $[\frac{5}{3}]^{2}_{3}$ $[\frac{2}{3}]^{2}_{3}$ $[4, 0]$ $[2]$ $z^2 + (5 t + 4)$ undefined
7.2.7.20a9.1 $( x^{2} + 6 x + 3 )^{7} + \left(35 x + 42\right) ( x^{2} + 6 x + 3 )^{4} + 7$ $C_7:F_7$ (as 14T14) $294$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{2}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{2}$ $[\frac{5}{3}]_{3}$ $[\frac{2}{3}]_{3}$ $[4, 0]$ $[1]$ $z^2 + (5 t + 3)$ undefined
7.2.7.20a10.1 $( x^{2} + 6 x + 3 )^{7} + \left(35 x + 28\right) ( x^{2} + 6 x + 3 )^{4} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{4}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{4}$ $[\frac{5}{3}]^{2}_{3}$ $[\frac{2}{3}]^{2}_{3}$ $[4, 0]$ $[2]$ $z^2 + (5 t + 5)$ undefined
7.2.7.20a11.1 $( x^{2} + 6 x + 3 )^{7} + 21 x ( x^{2} + 6 x + 3 )^{4} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{4}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{4}$ $[\frac{5}{3}]^{2}_{3}$ $[\frac{2}{3}]^{2}_{3}$ $[4, 0]$ $[2]$ $z^2 + (3 t + 4)$ undefined
7.2.7.20a12.1 $( x^{2} + 6 x + 3 )^{7} + \left(21 x + 35\right) ( x^{2} + 6 x + 3 )^{4} + 7$ $C_7:F_7$ (as 14T14) $294$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{2}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{2}$ $[\frac{5}{3}]_{3}$ $[\frac{2}{3}]_{3}$ $[4, 0]$ $[1]$ $z^2 + (3 t + 6)$ undefined
7.2.7.20a13.1 $( x^{2} + 6 x + 3 )^{7} + \left(21 x + 7\right) ( x^{2} + 6 x + 3 )^{4} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{4}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{4}$ $[\frac{5}{3}]^{2}_{3}$ $[\frac{2}{3}]^{2}_{3}$ $[4, 0]$ $[2]$ $z^2 + (3 t + 3)$ undefined
7.2.7.20a14.1 $( x^{2} + 6 x + 3 )^{7} + \left(21 x + 42\right) ( x^{2} + 6 x + 3 )^{4} + 7$ $C_7:F_7$ (as 14T14) $294$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{2}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{2}$ $[\frac{5}{3}]_{3}$ $[\frac{2}{3}]_{3}$ $[4, 0]$ $[1]$ $z^2 + (3 t + 5)$ undefined
7.2.7.20a15.1 $( x^{2} + 6 x + 3 )^{7} + \left(21 x + 14\right) ( x^{2} + 6 x + 3 )^{4} + 7$ $C_7:F_7$ (as 14T14) $294$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{2}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{2}$ $[\frac{5}{3}]_{3}$ $[\frac{2}{3}]_{3}$ $[4, 0]$ $[1]$ $z^2 + (3 t + 2)$ undefined
7.2.7.20a16.1 $( x^{2} + 6 x + 3 )^{7} + 7 x ( x^{2} + 6 x + 3 )^{4} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{4}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{4}$ $[\frac{5}{3}]^{2}_{3}$ $[\frac{2}{3}]^{2}_{3}$ $[4, 0]$ $[2]$ $z^2 + (t + 6)$ undefined
7.2.7.20a17.1 $( x^{2} + 6 x + 3 )^{7} + \left(7 x + 35\right) ( x^{2} + 6 x + 3 )^{4} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{4}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{4}$ $[\frac{5}{3}]^{2}_{3}$ $[\frac{2}{3}]^{2}_{3}$ $[4, 0]$ $[2]$ $z^2 + (t + 1)$ undefined
7.2.7.20a18.1 $( x^{2} + 6 x + 3 )^{7} + \left(7 x + 7\right) ( x^{2} + 6 x + 3 )^{4} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{4}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{4}$ $[\frac{5}{3}]^{2}_{3}$ $[\frac{2}{3}]^{2}_{3}$ $[4, 0]$ $[2]$ $z^2 + (t + 5)$ undefined
7.2.7.20a19.1 $( x^{2} + 6 x + 3 )^{7} + \left(7 x + 28\right) ( x^{2} + 6 x + 3 )^{4} + 7$ $C_7:F_7$ (as 14T14) $294$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{2}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{2}$ $[\frac{5}{3}]_{3}$ $[\frac{2}{3}]_{3}$ $[4, 0]$ $[1]$ $z^2 + (t + 2)$ undefined
7.2.7.20a20.1 $( x^{2} + 6 x + 3 )^{7} + 42 x ( x^{2} + 6 x + 3 )^{4} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{4}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{4}$ $[\frac{5}{3}]^{2}_{3}$ $[\frac{2}{3}]^{2}_{3}$ $[4, 0]$ $[2]$ $z^2 + (6 t + 1)$ undefined
7.2.7.20a21.1 $( x^{2} + 6 x + 3 )^{7} + \left(42 x + 21\right) ( x^{2} + 6 x + 3 )^{4} + 7$ $C_7:F_7$ (as 14T14) $294$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{2}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{2}$ $[\frac{5}{3}]_{3}$ $[\frac{2}{3}]_{3}$ $[4, 0]$ $[1]$ $z^2 + (6 t + 5)$ undefined
7.2.7.20a22.1 $( x^{2} + 6 x + 3 )^{7} + \left(42 x + 28\right) ( x^{2} + 6 x + 3 )^{4} + 7$ $C_7:F_7$ (as 14T14) $294$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{2}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{2}$ $[\frac{5}{3}]_{3}$ $[\frac{2}{3}]_{3}$ $[4, 0]$ $[1]$ $z^2 + (6 t + 4)$ undefined
7.2.7.20a23.1 $( x^{2} + 6 x + 3 )^{7} + 28 x ( x^{2} + 6 x + 3 )^{4} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{4}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{4}$ $[\frac{5}{3}]^{2}_{3}$ $[\frac{2}{3}]^{2}_{3}$ $[4, 0]$ $[2]$ $z^2 + (4 t + 3)$ undefined
7.2.7.20a24.1 $( x^{2} + 6 x + 3 )^{7} + \left(28 x + 42\right) ( x^{2} + 6 x + 3 )^{4} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{4}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{4}$ $[\frac{5}{3}]^{2}_{3}$ $[\frac{2}{3}]^{2}_{3}$ $[4, 0]$ $[2]$ $z^2 + (4 t + 4)$ undefined
7.2.7.20a25.1 $( x^{2} + 6 x + 3 )^{7} + 14 x ( x^{2} + 6 x + 3 )^{4} + 7$ $C_7^2:C_{12}$ (as 14T23) $588$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{4}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{4}$ $[\frac{5}{3}]^{2}_{3}$ $[\frac{2}{3}]^{2}_{3}$ $[4, 0]$ $[2]$ $z^2 + (2 t + 5)$ undefined
7.2.7.20a26.1 $( x^{2} + 6 x + 3 )^{7} + \left(14 x + 7\right) ( x^{2} + 6 x + 3 )^{4} + 7$ $C_7:F_7$ (as 14T14) $294$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{2}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{2}$ $[\frac{5}{3}]_{3}$ $[\frac{2}{3}]_{3}$ $[4, 0]$ $[1]$ $z^2 + (2 t + 4)$ undefined
7.2.7.20a27.1 $( x^{2} + 6 x + 3 )^{7} + \left(14 x + 42\right) ( x^{2} + 6 x + 3 )^{4} + 7$ $C_7:F_7$ (as 14T14) $294$ $1$ $[\frac{5}{3}, \frac{5}{3}]_{3}^{2}$ $[\frac{2}{3},\frac{2}{3}]_{3}^{2}$ $[\frac{5}{3}]_{3}$ $[\frac{2}{3}]_{3}$ $[4, 0]$ $[1]$ $z^2 + (2 t + 6)$ undefined
  displayed columns for results