Defining polynomial
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\(x^{13} + 19\)
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Invariants
| Base field: | $\Q_{19}$ |
| Degree $d$: | $13$ |
| Ramification exponent $e$: | $13$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{19}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 19 }) }$: | $1$ |
| This field is not Galois over $\Q_{19}.$ | |
| Visible slopes: | None |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 19 }$. |
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{19}$ |
| Relative Eisenstein polynomial: |
\( x^{13} + 19 \)
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Ramification polygon
| Residual polynomials: | $z^{12} + 13z^{11} + 2z^{10} + z^{9} + 12z^{8} + 14z^{7} + 6z^{6} + 6z^{5} + 14z^{4} + 12z^{3} + z^{2} + 2z + 13$ |
| Associated inertia: | $12$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $F_{13}$ (as 13T6) |
| Inertia group: | $C_{13}$ (as 13T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $12$ |
| Tame degree: | $13$ |
| Wild slopes: | None |
| Galois mean slope: | $12/13$ |
| Galois splitting model: |
$x^{13} - 52 x^{12} + 1248 x^{11} - 18304 x^{10} + 183040 x^{9} - 1317888 x^{8} + 7028736 x^{7} - 28114944 x^{6} + 84344832 x^{5} - 187432960 x^{4} + 299892736 x^{3} - 327155712 x^{2} + 218103808 x - 67108883$
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