Normalized defining polynomial
\( x^{22} - 9 x^{21} + 59 x^{20} - 273 x^{19} + 1195 x^{18} - 4387 x^{17} + 15557 x^{16} - 47572 x^{15} + 143444 x^{14} - 378050 x^{13} + 996721 x^{12} - 2278940 x^{11} + 5312470 x^{10} - 10499975 x^{9} + 21744445 x^{8} - 36444035 x^{7} + 67020055 x^{6} - 91292990 x^{5} + 148564990 x^{4} - 149610555 x^{3} + 214516775 x^{2} - 122653854 x + 155003209 \)
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: | \(-14844328957686912445797815584548193359375=-\,3^{11}\cdot 5^{11}\cdot 23^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $66.99$ | 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: | $3, 5, 23$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(345=3\cdot 5\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{345}(256,·)$, $\chi_{345}(1,·)$, $\chi_{345}(196,·)$, $\chi_{345}(331,·)$, $\chi_{345}(269,·)$, $\chi_{345}(271,·)$, $\chi_{345}(16,·)$, $\chi_{345}(209,·)$, $\chi_{345}(211,·)$, $\chi_{345}(151,·)$, $\chi_{345}(284,·)$, $\chi_{345}(29,·)$, $\chi_{345}(31,·)$, $\chi_{345}(164,·)$, $\chi_{345}(104,·)$, $\chi_{345}(301,·)$, $\chi_{345}(239,·)$, $\chi_{345}(179,·)$, $\chi_{345}(119,·)$, $\chi_{345}(121,·)$, $\chi_{345}(59,·)$, $\chi_{345}(254,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{47} a^{20} + \frac{2}{47} a^{19} - \frac{18}{47} a^{18} - \frac{11}{47} a^{17} - \frac{11}{47} a^{16} + \frac{12}{47} a^{15} - \frac{1}{47} a^{14} + \frac{15}{47} a^{13} - \frac{18}{47} a^{12} - \frac{20}{47} a^{11} + \frac{3}{47} a^{10} - \frac{12}{47} a^{9} + \frac{7}{47} a^{8} + \frac{9}{47} a^{7} + \frac{6}{47} a^{6} + \frac{21}{47} a^{5} - \frac{5}{47} a^{4} - \frac{21}{47} a^{3} - \frac{7}{47} a^{2} + \frac{14}{47} a - \frac{13}{47}$, $\frac{1}{220759451457059865401733349878002586455028708159137325679417421458757} a^{21} - \frac{768146294021795421607585869477261846742708606827165953281985842522}{220759451457059865401733349878002586455028708159137325679417421458757} a^{20} - \frac{76751661336825641458301679458205487264024669860343596961524416771918}{220759451457059865401733349878002586455028708159137325679417421458757} a^{19} + \frac{54851261992514987139244105605437028380800507593052834581540717964107}{220759451457059865401733349878002586455028708159137325679417421458757} a^{18} - \frac{69158044908569245008915814238522083120149483856242051865439340434083}{220759451457059865401733349878002586455028708159137325679417421458757} a^{17} + \frac{44299065884824659419507505253206872163392715887404295783135246954421}{220759451457059865401733349878002586455028708159137325679417421458757} a^{16} + \frac{108153651313471445993817477048380292374708886155039274988238517227321}{220759451457059865401733349878002586455028708159137325679417421458757} a^{15} - \frac{77998975832635461026296351970532001639065995121805558738815123369127}{220759451457059865401733349878002586455028708159137325679417421458757} a^{14} + \frac{91442462536094201166302423611002886286838767803304869328529815376948}{220759451457059865401733349878002586455028708159137325679417421458757} a^{13} + \frac{85335211911212638219784189068503089030646414813940118993905202003197}{220759451457059865401733349878002586455028708159137325679417421458757} a^{12} - \frac{64417172401983380016956246277692809455047643620405974285045493383144}{220759451457059865401733349878002586455028708159137325679417421458757} a^{11} + \frac{54228862664889775068275243571918924381988148253431178955903815698584}{220759451457059865401733349878002586455028708159137325679417421458757} a^{10} - \frac{46442563837049991981257319072521025476098908423912450001148416659}{220759451457059865401733349878002586455028708159137325679417421458757} a^{9} - \frac{61936914873908278581720059267634614821550358542939511460037302597419}{220759451457059865401733349878002586455028708159137325679417421458757} a^{8} + \frac{97875303504589317720591866492025072336691040789559803795323211274900}{220759451457059865401733349878002586455028708159137325679417421458757} a^{7} - \frac{64059524833695797039885255158642635191304694106193898847818826713153}{220759451457059865401733349878002586455028708159137325679417421458757} a^{6} + \frac{77077788848503231549721166453386216697829379374929619157814415460434}{220759451457059865401733349878002586455028708159137325679417421458757} a^{5} + \frac{88746053339423870476417796123197413821711321114770699893575134487039}{220759451457059865401733349878002586455028708159137325679417421458757} a^{4} - \frac{86279378965322686310773972359463337462438411870735898168238267166797}{220759451457059865401733349878002586455028708159137325679417421458757} a^{3} - \frac{48205223376687886721675938271308653468524317383610380967653274573221}{220759451457059865401733349878002586455028708159137325679417421458757} a^{2} - \frac{70389403703225790004782463145012179026986994202801054522348679197591}{220759451457059865401733349878002586455028708159137325679417421458757} a + \frac{69209418321451723518985339945042030256757284596854285144655656742208}{220759451457059865401733349878002586455028708159137325679417421458757}$
Class group and class number
$C_{91498}$, which has order $91498$ (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: | \( 1038656.82438 \) (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
| \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{23})^+\) |
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 | ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ | R | R | $22$ | $22$ | $22$ | ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ | R | $22$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | $22$ | $22$ | $22$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ | ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ | $22$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| $23$ | 23.11.10.10 | $x^{11} - 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |
| 23.11.10.10 | $x^{11} - 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ | |