Defining polynomial
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\(x^{14} + 42 x^{9} + 42 x^{8} + 14 x^{7} + 441 x^{4} + 882 x^{3} + 441 x^{2} + 294 x + 49\)
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Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $7$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $14$ |
| Discriminant root field: | $\Q_{7}$ |
| Root number: | $-1$ |
| $\card{ \Aut(K/\Q_{ 7 }) }$: | $1$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible slopes: | $[7/6]$ |
Intermediate fields
| $\Q_{7}(\sqrt{3})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}(\sqrt{3})$ $\cong \Q_{7}(t)$ where $t$ is a root of
\( x^{2} + 6 x + 3 \)
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| Relative Eisenstein polynomial: |
\( x^{7} + 21 x^{2} + \left(7 t + 42\right) x + 7 \)
$\ \in\Q_{7}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + 6t + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_7^2:C_{12}$ (as 14T23) |
| Inertia group: | Intransitive group isomorphic to $C_7:F_7$ |
| Wild inertia group: | $C_7^2$ |
| Unramified degree: | $2$ |
| Tame degree: | $6$ |
| Wild slopes: | $[7/6, 7/6]$ |
| Galois mean slope: | $341/294$ |
| Galois splitting model: |
$x^{14} - 564052017496651404892855580211604422 x^{12} - 154865641108811054995009263860974583302189621494546283 x^{11} + 68274493059286391931952662785126184806268034083698188288919870656005193 x^{10} + 43847589619610288791764730575958309535305546399471039007303416720226413425171311857176085 x^{9} + 9148355250370158987218946199764221034455885689967512406635041602757026317291968557918421422254976762784826 x^{8} + 794479778500099572292327210076367104352751202560205733855902933278345854118642269122134869278822866545664936046356404027544 x^{7} + 5761513882929886990680537639806027314200648939646226236793632641509838968648205018822920628021170653206906306159493412875103066639203431855 x^{6} - 3125472782046491957350201539596069103278782427172319082751881579749405623040308642803107121879514458224004927346965069834514879902106136396185725087120039987 x^{5} - 91521547870604903085709374725135913076381858272931381577430075109889993712449215537354862199844234498087304983925508643867754681059381869850717251560441887482945072937990417 x^{4} + 5603036721451203330015559608281941817568015607448589218114237392626863694398643920288288291092033389294137829402301553859306081762435954744883476148325237523665981920807477413019439577537072 x^{3} + 109020836519736346319459629667823398432516572802485985957811105117205961861811304660384352734054094141502906071762773241696462635710179761538792913021946099931584029571440777583042899407321831880756693967562 x^{2} - 5967498622224037431865658871468901980953759601132816707360001312227773233473002490513606845581729208599942882071000618393043614548112322226970103348453898377258843415415887263148085464980913119958848084287985013478495823847 x + 51250465851001016825268129989367400133728170308779441838490740889876330928544118818264922776427236230887724359549174121904872469357732745273692270474242452280316080126646141179785965621564275949718114744620273626266780717395534203262681967$
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