Defining polynomial
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\(x^{15} + 10 x^{13} + 45 x^{12} + 40 x^{11} + 393 x^{10} + 890 x^{9} + 750 x^{8} - 3970 x^{7} + 9610 x^{6} + 13085 x^{5} + 151045 x^{4} + 26525 x^{3} + 116170 x^{2} - 53795 x + 67662\)
|
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $5$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{11}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 11 }) }$: | $15$ |
| This field is Galois and abelian over $\Q_{11}.$ | |
| Visible slopes: | None |
Intermediate fields
| 11.3.0.1, 11.5.4.4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 11.3.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of
\( x^{3} + 2 x + 9 \)
|
| Relative Eisenstein polynomial: |
\( x^{5} + 11 \)
$\ \in\Q_{11}(t)[x]$
|
Ramification polygon
Not computedInvariants of the Galois closure
| Galois group: | $C_{15}$ (as 15T1) |
| Inertia group: | Intransitive group isomorphic to $C_5$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $3$ |
| Tame degree: | $5$ |
| Wild slopes: | None |
| Galois mean slope: | $4/5$ |
| Galois splitting model: | Not computed |