Defining polynomial
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\(x^{15} + 3 x^{2} + 6 x + 3\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $15$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $15$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3})$ |
| Root number: | $-i$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[11/10]$ |
Intermediate fields
| 3.5.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{3}$ |
| Relative Eisenstein polynomial: |
\( x^{15} + 3 x^{2} + 6 x + 3 \)
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Ramification polygon
| Residual polynomials: | $2z + 1$,$z^{12} + 2z^{9} + z^{6} + z^{3} + 2$ |
| Associated inertia: | $1$,$4$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_3^4:(C_2\times F_5)$ (as 15T52) |
| Inertia group: | $C_3^4:C_{10}$ (as 15T33) |
| Wild inertia group: | $C_3^4$ |
| Unramified degree: | $4$ |
| Tame degree: | $10$ |
| Wild slopes: | $[11/10, 11/10, 11/10, 11/10]$ |
| Galois mean slope: | $889/810$ |
| Galois splitting model: |
$x^{15} - 12 x^{14} - 24 x^{13} + 685 x^{12} - 840 x^{11} - 11532 x^{10} + 42694 x^{9} + 27738 x^{8} - 582210 x^{7} + 1878920 x^{6} - 3095874 x^{5} + 2807568 x^{4} - 1382389 x^{3} + 614850 x^{2} - 552630 x + 256955$
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