Defining polynomial
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\(x^{15} - 9 x^{14} + 228 x^{13} + 2040 x^{12} + 2547 x^{11} - 22914 x^{10} - 63036 x^{9} + 193752 x^{8} - 76950 x^{7} - 3051 x^{6} + 77517 x^{5} - 19116 x^{4} - 7857 x^{3} + 7533 x^{2} - 1944 x + 243\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $15$ |
| 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: | $[3/2]$ |
Intermediate fields
| 3.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 3.5.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{5} + 2 x + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{3} + \left(3 t^{4} + 3 t^{3} + 6 t^{2} + 3\right) x^{2} + \left(3 t^{4} + 6 t\right) x + 3 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + 2t^{4} + t$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_7^3:C_6$ (as 15T44) |
| Inertia group: | Intransitive group isomorphic to $C_3^4:S_3$ |
| Wild inertia group: | $C_3^5$ |
| Unramified degree: | $5$ |
| Tame degree: | $2$ |
| Wild slopes: | $[3/2, 3/2, 3/2, 3/2, 3/2]$ |
| Galois mean slope: | $727/486$ |
| Galois splitting model: |
$x^{15} - 15 x^{12} - 75 x^{9} + 1875 x^{6} - 6250 x^{3} - 3125$
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