Defining polynomial
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\(x^{14} + 4 x^{9} + 4 x^{7} + 10\)
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Invariants
| Base field: | $\Q_{2}$ |
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| Degree $d$: | $14$ |
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| Ramification index $e$: | $14$ |
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| Residue field degree $f$: | $1$ |
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| Discriminant exponent $c$: | $27$ |
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| Discriminant root field: | $\Q_{2}(\sqrt{2\cdot 5})$ | |
| Root number: | $-1$ | |
| $\Aut(K/\Q_{2})$: | $C_2$ | |
| This field is not Galois over $\Q_{2}.$ | ||
| Visible Artin slopes: | $[3]$ | |
| Visible Swan slopes: | $[2]$ | |
| Means: | $\langle1\rangle$ | |
| Rams: | $(14)$ | |
| Jump set: | $[7, 21]$ | |
| Roots of unity: | $2$ |
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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: | $\Q_{2}$ |
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| Relative Eisenstein polynomial: |
\( x^{14} + 4 x^{9} + 4 x^{7} + 10 \)
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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: | $[14, 0]$ |
Invariants of the Galois closure
| Galois degree: | $336$ |
| Galois group: | $F_8:C_6$ (as 14T18) |
| Inertia group: | $C_2\times F_8$ (as 14T9) |
| Wild inertia group: | $C_2^4$ |
| Galois unramified degree: | $3$ |
| Galois tame degree: | $7$ |
| Galois Artin slopes: | $[\frac{12}{7}, \frac{12}{7}, \frac{12}{7}, 3]$ |
| Galois Swan slopes: | $[\frac{5}{7},\frac{5}{7},\frac{5}{7},2]$ |
| Galois mean slope: | $2.3035714285714284$ |
| Galois splitting model: |
$x^{14} - 756 x^{10} - 4158 x^{8} + 63504 x^{6} - 20412 x^{2} + 4374$
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