Normalized defining polynomial
\( x^{22} - 9 x^{21} - 95 x^{20} + 987 x^{19} + 3645 x^{18} - 46387 x^{17} - 70417 x^{16} + 1223460 x^{15} + 652512 x^{14} - 19938598 x^{13} - 856739 x^{12} + 208664464 x^{11} - 37826626 x^{10} - 1414727471 x^{9} + 330778443 x^{8} + 6132077765 x^{7} - 984959307 x^{6} - 16336349510 x^{5} - 133094234 x^{4} + 24406337029 x^{3} + 5872500477 x^{2} - 15736819750 x - 7810183279 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[22, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(944450376932556277593597370211011550363631641=23^{20}\cdot 41^{11}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $110.75$ | 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: | $23, 41$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(943=23\cdot 41\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{943}(1,·)$, $\chi_{943}(450,·)$, $\chi_{943}(901,·)$, $\chi_{943}(903,·)$, $\chi_{943}(657,·)$, $\chi_{943}(739,·)$, $\chi_{943}(532,·)$, $\chi_{943}(409,·)$, $\chi_{943}(860,·)$, $\chi_{943}(288,·)$, $\chi_{943}(737,·)$, $\chi_{943}(163,·)$, $\chi_{943}(165,·)$, $\chi_{943}(614,·)$, $\chi_{943}(81,·)$, $\chi_{943}(616,·)$, $\chi_{943}(491,·)$, $\chi_{943}(370,·)$, $\chi_{943}(821,·)$, $\chi_{943}(696,·)$, $\chi_{943}(698,·)$, $\chi_{943}(124,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{89967755418110535440536222329317879928703206293171840837767696181706939106895891374333} a^{21} + \frac{1799649451245896190524641768663909314792997699720498648866582113618667283205648038261}{89967755418110535440536222329317879928703206293171840837767696181706939106895891374333} a^{20} - \frac{8944493867375418382861200803293050653511706811414295642795921232698781937417268480238}{89967755418110535440536222329317879928703206293171840837767696181706939106895891374333} a^{19} + \frac{11271082173884508032358167389817467835172669604472462976449495019920960700367528990478}{89967755418110535440536222329317879928703206293171840837767696181706939106895891374333} a^{18} + \frac{12540580121339229465091534442357907383341304981054582277686724421596222967372786421380}{89967755418110535440536222329317879928703206293171840837767696181706939106895891374333} a^{17} - \frac{29183967497094068774631381971731108803660468391401107058818447917200437822668419783124}{89967755418110535440536222329317879928703206293171840837767696181706939106895891374333} a^{16} + \frac{17253077343668263534196652522018344563113527465916736933433318035629711200585011822207}{89967755418110535440536222329317879928703206293171840837767696181706939106895891374333} a^{15} - \frac{44346949368000748029234872819301040584904832300350093803133991454724833793415377450545}{89967755418110535440536222329317879928703206293171840837767696181706939106895891374333} a^{14} + \frac{39264021175651455596993300732515370880831446482087469966093950613120186024056467561395}{89967755418110535440536222329317879928703206293171840837767696181706939106895891374333} a^{13} + \frac{14232830476953175810330324829063953735101209331798177716856723463584197313735144308877}{89967755418110535440536222329317879928703206293171840837767696181706939106895891374333} a^{12} + \frac{10341041241171863318278018919261709267049286622984084363729808848971904003294940809926}{89967755418110535440536222329317879928703206293171840837767696181706939106895891374333} a^{11} - \frac{14647939535584822423262241113240814002961686028035969006100977322610317750019080899790}{89967755418110535440536222329317879928703206293171840837767696181706939106895891374333} a^{10} - \frac{16161078743720364015252879535763473425287759028669230703764728226941401498822112822246}{89967755418110535440536222329317879928703206293171840837767696181706939106895891374333} a^{9} + \frac{34560706627228902327310436819041973796132117148833833452363021744726557302619994957733}{89967755418110535440536222329317879928703206293171840837767696181706939106895891374333} a^{8} + \frac{6187771092681418467541614713857182839462923880947937957857370819911227431616813391207}{89967755418110535440536222329317879928703206293171840837767696181706939106895891374333} a^{7} - \frac{14046891528701901188085546796894867020383098395629073633495689254757056714603341528727}{89967755418110535440536222329317879928703206293171840837767696181706939106895891374333} a^{6} + \frac{37574710143281464551361509724231278772293070792543120162394203647350687237333302511976}{89967755418110535440536222329317879928703206293171840837767696181706939106895891374333} a^{5} - \frac{12906597706232369527334923340825274841187690671761981353630949179744066784993055725178}{89967755418110535440536222329317879928703206293171840837767696181706939106895891374333} a^{4} + \frac{23726823463499895780356379523467428784529370841596123594990933331078845931322836880859}{89967755418110535440536222329317879928703206293171840837767696181706939106895891374333} a^{3} - \frac{38691859878415757893748194003364547435898165885381097101023270713282903074176361675966}{89967755418110535440536222329317879928703206293171840837767696181706939106895891374333} a^{2} + \frac{2956999975619328779998556273851405336111134911120990218457780848000441254563424809434}{89967755418110535440536222329317879928703206293171840837767696181706939106895891374333} a + \frac{29486207817613943741901290982345380756022217271903683472454171077075075668652850275921}{89967755418110535440536222329317879928703206293171840837767696181706939106895891374333}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $21$ | 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: | \( 1977369244418017.5 \) (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
| \(\Q(\sqrt{41}) \), \(\Q(\zeta_{23})^+\) |
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 | ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ | $22$ | $22$ | $22$ | $22$ | $22$ | R | $22$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ | R | ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$ | $22$ | ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $23$ | 23.11.10.10 | $x^{11} - 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |
| 23.11.10.10 | $x^{11} - 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ | |
| 41 | Data not computed | ||||||