Defining polynomial
| $x^{6} + \left(b_{23} \pi^{4} + b_{17} \pi^{3}\right) x^{5} + \left(b_{21} \pi^{4} + b_{15} \pi^{3}\right) x^{3} + \left(b_{19} \pi^{4} + b_{13} \pi^{3}\right) x + c_{24} \pi^{5} + \pi$ |
Invariants
| Residue field characteristic: | $2$ |
| Degree: | $6$ |
| Base field: | $\Q_{2}(\sqrt{-2\cdot 5})$ |
| Ramification index $e$: | $6$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $17$ |
| Absolute Artin slopes: | $[3,4]$ |
| Swan slopes: | $[4]$ |
| Means: | $\langle2\rangle$ |
| Rams: | $(12)$ |
| Field count: | $64$ (complete) |
| Ambiguity: | $2$ |
| Mass: | $64$ |
| Absolute Mass: | $32$ |
Diagrams
Varying
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $D_{12}$ (show 2), $S_3\times D_4$ (show 2), $C_4:S_4$ (show 6), $D_4\times S_4$ (show 6), $C_2^4:D_{12}$ (show 24), $C_2\wr D_6$ (show 24) |
| Hidden Artin slopes: | $[2,\frac{8}{3},\frac{8}{3},\frac{23}{6},\frac{23}{6}]^{2}$ (show 16), $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6},\frac{19}{6}]^{2}$ (show 8), $[\frac{4}{3},\frac{4}{3},\frac{19}{6},\frac{19}{6}]^{2}$ (show 8), $[2]^{2}$ (show 2), $[\frac{8}{3},\frac{8}{3},\frac{23}{6},\frac{23}{6}]^{2}$ (show 16), $[\ ]^{2}$ (show 2), $[\frac{4}{3},\frac{4}{3},2]^{2}$ (show 2), $[2,\frac{8}{3},\frac{8}{3}]^{2}$ (show 4), $[\frac{8}{3},\frac{8}{3}]^{2}$ (show 4), $[\frac{4}{3},\frac{4}{3}]^{2}$ (show 2) |
| Indices of inseparability: | $[24,12,0]$ |
| Associated inertia: | $[2,1,1]$ |
| Jump Set: | $[3,9,21]$ |
Fields
Showing all 2
Download displayed columns for results| Label | Polynomial $/ \Q_p$ | Galois group $/ \Q_p$ | Galois degree $/ \Q_p$ | $\#\Aut(K/\Q_p)$ | Hidden Artin slopes $/ \Q_p$ | Ind. of Insep. $/ \Q_p$ | Assoc. Inertia $/ \Q_p$ | Jump Set |
|---|---|---|---|---|---|---|---|---|
| 2.1.12.35a1.67 | $x^{12} + 8 x^{11} + 10$ | $D_4\times S_4$ (as 12T86) | $192$ | $2$ | $[\frac{4}{3},\frac{4}{3},2]^{2}$ | $[24, 12, 0]$ | $[2, 1, 1]$ | $[3, 9, 21]$ |
| 2.1.12.35a1.68 | $x^{12} + 8 x^{11} + 26$ | $D_4\times S_4$ (as 12T86) | $192$ | $2$ | $[\frac{4}{3},\frac{4}{3},2]^{2}$ | $[24, 12, 0]$ | $[2, 1, 1]$ | $[3, 9, 21]$ |