Defining polynomial
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\(x^{15} + 3 x^{14} + 6 x^{13} + 3 x^{12} + 6 x^{11} + 3 x^{6} + 3 x^{3} + 18 x^{2} + 18 x + 12\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $15$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $25$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3\cdot 2})$ |
| Root number: | $i$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[21/10]$ |
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: |
\( x^{15} + 3 x^{14} + 6 x^{13} + 3 x^{12} + 6 x^{11} + 3 x^{6} + 3 x^{3} + 18 x^{2} + 18 x + 12 \)
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Ramification polygon
| Residual polynomials: | $2z + 2$,$z^{12} + 2z^{9} + z^{6} + z^{3} + 2$ |
| Associated inertia: | $1$,$4$ |
| Indices of inseparability: | $[11, 0]$ |
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: | $[21/10, 21/10, 21/10, 21/10]$ |
| Galois mean slope: | $563/270$ |
| Galois splitting model: |
$x^{15} - 21 x^{13} - 22 x^{12} + 144 x^{11} + 312 x^{10} - 56 x^{9} - 864 x^{8} - 1890 x^{7} - 3464 x^{6} - 4482 x^{5} - 4476 x^{4} - 4846 x^{3} - 4608 x^{2} - 2550 x - 596$
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