Defining polynomial
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$( x^{2} + 6 x + 3 )^{7} + 21 x ( x^{2} + 6 x + 3 )^{5} + 7$
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Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $14$ |
| Ramification index $e$: | $7$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $22$ |
| Discriminant root field: | $\Q_{7}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{7})$: | $C_1$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible Artin slopes: | $[\frac{11}{6}]$ |
| Visible Swan slopes: | $[\frac{5}{6}]$ |
| Means: | $\langle\frac{5}{7}\rangle$ |
| Rams: | $(\frac{5}{6})$ |
| Jump set: | undefined |
| Roots of unity: | $48 = (7^{ 2 } - 1)$ |
Intermediate fields
| $\Q_{7}(\sqrt{3})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical 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: |
\( x^{7} + 21 t x^{5} + 7 \)
$\ \in\Q_{7}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + (2 t + 3)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[5, 0]$ |
Invariants of the Galois closure
| Galois degree: | $588$ |
| Galois group: | $C_7^2:C_{12}$ (as 14T23) |
| Inertia group: | Intransitive group isomorphic to $C_7:F_7$ |
| Wild inertia group: | $C_7^2$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $6$ |
| Galois Artin slopes: | $[\frac{11}{6}, \frac{11}{6}]$ |
| Galois Swan slopes: | $[\frac{5}{6},\frac{5}{6}]$ |
| Galois mean slope: | $1.8129251700680271$ |
| Galois splitting model: |
$x^{14} - 510886853187457491230770760422357362 x^{12} - 49789902213773537269276688023115725763825828252923536 x^{11} + 73856945303473997444320284004367331826362594871704613071001360066219503 x^{10} + 11350578243175480286554939771000267973452587976456766242108958850771925802672113837817136 x^{9} - 3054690122841014715534064283316286209147845751879716936221289548752080571402682610135833405269460600948400 x^{8} - 495369201195278039375935522251067932316093542505307566195927579274263405523322348817248175698598557232783699354755006532128 x^{7} + 42832668439740822722107250855055830720983419664599828698116889518834963511034181822739671319483198527546485099785544565709891276198151902639 x^{6} + 6413346383382361248553113067518358424376265833592166752448739174912117290211096889332918692147924736919921952549096245258386550899726570448906693323961543392 x^{5} - 178669729212606560541530916891848669313023282537364113769878677949632425676916240407266403634882830394787244180404675770020424245440546638766798561923392133956442053332455082 x^{4} - 20949427082353570922837578822961443071656182727108162068262793020322228467602777790604212348230561263563797093160313158195595775786468461897409232255307521482123710703250649551499979854519120 x^{3} + 89377491381949149937008101098546018658163559538450985505801262069705755180186573441788140032243243560597500307178392307672467968222662056165852200778246961791314052745873106465410259217820863824604725484033 x^{2} + 13325612740011870844202512669235284165447093041656986039794888125059135190737668078258999465906800100997767588101081450156456856117424577674382218565391563783549900939270809237971684008386888724584613472335431416274297394160 x + 13648960277331125531264492299774369923740440671738035686603500399722624700942123675708321665458137100617966224292637873346997700310486397933885616537229177999981106996243441073350793520414401338296471933818302788849790300013203002139066668$
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