Defining polynomial
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\(x^{14} + 2 x^{13} + 2 x^{11} + 2 x^{7} + 2\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $14$ |
| Ramification index $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $20$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
| Root number: | $-i$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[2]$ |
| Visible Swan slopes: | $[1]$ |
| Means: | $\langle\frac{1}{2}\rangle$ |
| Rams: | $(7)$ |
| Jump set: | $[7, 25]$ |
| Roots of unity: | $2$ |
Intermediate fields
| 2.1.7.6a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: |
\( x^{14} + 2 x^{13} + 2 x^{11} + 2 x^{7} + 2 \)
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Ramification polygon
| Residual polynomials: | $z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1$,$z + 1$ |
| Associated inertia: | $3$,$1$ |
| Indices of inseparability: | $[7, 0]$ |
Invariants of the Galois closure
| Galois degree: | $2688$ |
| Galois group: | $C_2\wr C_7:C_3$ (as 14T44) |
| Inertia group: | $C_2\wr C_7$ (as 14T29) |
| Wild inertia group: | $C_2^7$ |
| Galois unramified degree: | $3$ |
| Galois tame degree: | $7$ |
| Galois Artin slopes: | $[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}, \frac{10}{7}, \frac{10}{7}, \frac{10}{7}, 2]$ |
| Galois Swan slopes: | $[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{3}{7},\frac{3}{7},\frac{3}{7},1]$ |
| Galois mean slope: | $1.6941964285714286$ |
| Galois splitting model: |
$x^{14} - 21 x^{12} - 7 x^{10} + 707 x^{8} + 2415 x^{6} + 2961 x^{4} + 1323 x^{2} + 81$
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