Defining polynomial
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\(x^{14} + 7 x^{11} + 7 x^{10} + 35 x^{9} + 14 x^{7} + 7\)
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Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $22$ |
| Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 7 }) }$: | $1$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible slopes: | $[7/4]$ |
Intermediate fields
| $\Q_{7}(\sqrt{7\cdot 3})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}$ |
| Relative Eisenstein polynomial: |
\( x^{14} + 7 x^{11} + 7 x^{10} + 35 x^{9} + 14 x^{7} + 7 \)
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Ramification polygon
| Residual polynomials: | $2z^{3} + 4$,$z^{7} + 2$ |
| Associated inertia: | $3$,$1$ |
| Indices of inseparability: | $[9, 0]$ |
Invariants of the Galois closure
| Galois group: | $D_7^2:C_6$ (as 14T32) |
| Inertia group: | $C_7^2:C_4$ (as 14T12) |
| Wild inertia group: | $C_7^2$ |
| Unramified degree: | $6$ |
| Tame degree: | $4$ |
| Wild slopes: | $[7/4, 7/4]$ |
| Galois mean slope: | $339/196$ |
| Galois splitting model: |
$x^{14} - 2604967932973255611007995950134415113289 x^{11} + 47977887561476695708664293208073408848867734349761497 x^{10} - 501180279369097675920068676715134684815309023275709280951667798960 x^{9} - 8863916451459039331072009359817154845410079993717736991910744851747278587147966 x^{8} + 5530848388807917605937376437187852025188966162568578453757677720950904004199432651683365551 x^{7} + 4450752188177395455726841230161765288184644210752510973963214909385364384007859594323842870310525827354970 x^{6} + 58359250002277345324762592962635441361191792460786507593453146955646587559879856047453684939979878966158097228994558749 x^{5} + 336874462867585111129292961776815732582389334222393941773961228949216139087348461044993517520803582639308367611119249675744550113217 x^{4} + 1837848373100751095808263075643601734953864059104150115281589373992810428289950416876051397175904594227810891481519185193719694240794514376577632 x^{3} + 20168442383254745574259233915989662451921481687123949733602455102260734471175689275710809509819183344304509462471534577784772271187725959192675278506639153877 x^{2} + 144017726423707561949823789437667784018394971663104198931828114168369166052942759185812525381925433233812849672010358292424643634646431966191398275905179649629312631543288 x + 418893310691966029936375698062220172767987858911188259660254357634167677782045503554685164411875369369743585424926349151335789750591365256693895529901469042613713933863028720639083904$
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