Normalized defining polynomial
\( x^{22} - 9 x^{21} + 114 x^{20} - 723 x^{19} + 5545 x^{18} - 27862 x^{17} + 161687 x^{16} - 673652 x^{15} + 3199234 x^{14} - 11272660 x^{13} + 45446261 x^{12} - 136089890 x^{11} + 474573050 x^{10} - 1198919315 x^{9} + 3649189205 x^{8} - 7608385175 x^{7} + 20266939465 x^{6} - 33305104255 x^{5} + 77468338075 x^{4} - 90946009745 x^{3} + 183517941560 x^{2} - 117944870669 x + 204320499769 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 11]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-165693399436195066245409511458333024560546875=-\,5^{11}\cdot 7^{11}\cdot 23^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $102.32$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $5, 7, 23$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(805=5\cdot 7\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{805}(384,·)$, $\chi_{805}(1,·)$, $\chi_{805}(386,·)$, $\chi_{805}(771,·)$, $\chi_{805}(71,·)$, $\chi_{805}(139,·)$, $\chi_{805}(524,·)$, $\chi_{805}(141,·)$, $\chi_{805}(209,·)$, $\chi_{805}(211,·)$, $\chi_{805}(279,·)$, $\chi_{805}(349,·)$, $\chi_{805}(351,·)$, $\chi_{805}(36,·)$, $\chi_{805}(104,·)$, $\chi_{805}(489,·)$, $\chi_{805}(491,·)$, $\chi_{805}(174,·)$, $\chi_{805}(561,·)$, $\chi_{805}(629,·)$, $\chi_{805}(246,·)$, $\chi_{805}(699,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{47} a^{20} - \frac{15}{47} a^{19} + \frac{11}{47} a^{18} - \frac{9}{47} a^{17} - \frac{2}{47} a^{16} + \frac{19}{47} a^{15} - \frac{3}{47} a^{14} + \frac{16}{47} a^{13} + \frac{4}{47} a^{12} - \frac{2}{47} a^{11} - \frac{21}{47} a^{10} + \frac{15}{47} a^{9} + \frac{12}{47} a^{8} + \frac{13}{47} a^{7} - \frac{20}{47} a^{6} - \frac{22}{47} a^{5} - \frac{4}{47} a^{4} + \frac{18}{47} a^{3} + \frac{17}{47} a^{2} - \frac{1}{47} a - \frac{10}{47}$, $\frac{1}{64893935449716736173634097917333940226158282208678157064731692276712374036692710170377} a^{21} - \frac{504719484143524130793458750174716795155875203626991805684271675390203038747697049564}{64893935449716736173634097917333940226158282208678157064731692276712374036692710170377} a^{20} - \frac{3516711681804593981034518271745989760538201950556342418966482135420864931408793294284}{64893935449716736173634097917333940226158282208678157064731692276712374036692710170377} a^{19} + \frac{28303573694669860889464554168542882425920389019358935547990316013611515349034193396681}{64893935449716736173634097917333940226158282208678157064731692276712374036692710170377} a^{18} - \frac{20167303951153147941649735048168005284457454468935151236456445964350942995837744750093}{64893935449716736173634097917333940226158282208678157064731692276712374036692710170377} a^{17} + \frac{9349198700677328754258450030979565983042720178096202593752247536268917024922706664288}{64893935449716736173634097917333940226158282208678157064731692276712374036692710170377} a^{16} + \frac{1485173985359093326773746158643094746556065593773773221047658929028894889977693992162}{64893935449716736173634097917333940226158282208678157064731692276712374036692710170377} a^{15} - \frac{17753581573790699993975280316825441669549508670348332680197028389586689378572771500658}{64893935449716736173634097917333940226158282208678157064731692276712374036692710170377} a^{14} - \frac{24267944004101510468662760122484073577063569100409232878146646695385853747967453364100}{64893935449716736173634097917333940226158282208678157064731692276712374036692710170377} a^{13} - \frac{29747227598663170715468541515008111840212988984773193092139327008724081932913236159657}{64893935449716736173634097917333940226158282208678157064731692276712374036692710170377} a^{12} + \frac{1388316227525765262795719032697062899584673308737699697837341433399766541083622256350}{64893935449716736173634097917333940226158282208678157064731692276712374036692710170377} a^{11} + \frac{20429534966612975243758942386830561427177122906217753151037464233971878980768497807482}{64893935449716736173634097917333940226158282208678157064731692276712374036692710170377} a^{10} + \frac{29440465245868816758232951252172191401494179750876720733530468444675168332032221634205}{64893935449716736173634097917333940226158282208678157064731692276712374036692710170377} a^{9} - \frac{10076417538846305902458840332900696185397924819985680242631738367264718105008047624930}{64893935449716736173634097917333940226158282208678157064731692276712374036692710170377} a^{8} + \frac{7614529775657746935002256862986926524004081050263338440761838960971300501383425112856}{64893935449716736173634097917333940226158282208678157064731692276712374036692710170377} a^{7} - \frac{12033419274357002459399055768755965047802165409254878016919718742996061878406542790182}{64893935449716736173634097917333940226158282208678157064731692276712374036692710170377} a^{6} - \frac{32274084403393654373938142978573061560329155160976231177491312732926633404153917542813}{64893935449716736173634097917333940226158282208678157064731692276712374036692710170377} a^{5} - \frac{18985095653877296598957926112866349019451632517726730939160798993560153667027405092286}{64893935449716736173634097917333940226158282208678157064731692276712374036692710170377} a^{4} - \frac{28242326505843773019803817048909758561467553605831869251699031967153217292441141362744}{64893935449716736173634097917333940226158282208678157064731692276712374036692710170377} a^{3} - \frac{27111380369331564424053956421967777410954793348813547662068074124789204860580211074589}{64893935449716736173634097917333940226158282208678157064731692276712374036692710170377} a^{2} + \frac{31173320245272103766166927481645579744842853000591130712740968183636776751720508383375}{64893935449716736173634097917333940226158282208678157064731692276712374036692710170377} a - \frac{3088970198370827180855830684157965310035507419261261476746391058297721030352705968229}{64893935449716736173634097917333940226158282208678157064731692276712374036692710170377}$
Class group and class number
$C_{6263182}$, which has order $6263182$ (assuming GRH)
Unit group
| Rank: | $10$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 1038656.8243805699 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 22 |
| The 22 conjugacy class representatives for $C_{22}$ |
| Character table for $C_{22}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-35}) \), \(\Q(\zeta_{23})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $22$ | ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ | R | R | ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ | $22$ | R | ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ | $22$ | $22$ | $22$ | $22$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ | $22$ | $22$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| 7 | Data not computed | ||||||
| 23 | Data not computed | ||||||