Normalized defining polynomial
\( x^{22} - 2 x^{21} - 96 x^{20} + 178 x^{19} + 3899 x^{18} - 6608 x^{17} - 87467 x^{16} + 133046 x^{15} + 1188426 x^{14} - 1584074 x^{13} - 10109060 x^{12} + 11435962 x^{11} + 53939118 x^{10} - 49578134 x^{9} - 176910567 x^{8} + 124307164 x^{7} + 342485752 x^{6} - 168333340 x^{5} - 366604062 x^{4} + 109236440 x^{3} + 193879860 x^{2} - 25736464 x - 38604719 \)
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: | \(14232954634830452964011747236817552143286272=2^{22}\cdot 7^{11}\cdot 23^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $91.52$ | 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: | $2, 7, 23$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(644=2^{2}\cdot 7\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{644}(1,·)$, $\chi_{644}(587,·)$, $\chi_{644}(197,·)$, $\chi_{644}(449,·)$, $\chi_{644}(393,·)$, $\chi_{644}(139,·)$, $\chi_{644}(141,·)$, $\chi_{644}(335,·)$, $\chi_{644}(531,·)$, $\chi_{644}(533,·)$, $\chi_{644}(279,·)$, $\chi_{644}(27,·)$, $\chi_{644}(29,·)$, $\chi_{644}(223,·)$, $\chi_{644}(225,·)$, $\chi_{644}(167,·)$, $\chi_{644}(169,·)$, $\chi_{644}(363,·)$, $\chi_{644}(561,·)$, $\chi_{644}(307,·)$, $\chi_{644}(55,·)$, $\chi_{644}(85,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{47} a^{20} + \frac{16}{47} a^{19} + \frac{21}{47} a^{18} - \frac{18}{47} a^{17} - \frac{16}{47} a^{16} - \frac{11}{47} a^{15} - \frac{10}{47} a^{13} - \frac{8}{47} a^{12} - \frac{18}{47} a^{11} - \frac{8}{47} a^{10} - \frac{11}{47} a^{9} - \frac{14}{47} a^{8} - \frac{12}{47} a^{7} - \frac{13}{47} a^{6} - \frac{2}{47} a^{5} + \frac{20}{47} a^{4} - \frac{23}{47} a^{3} + \frac{12}{47} a$, $\frac{1}{39474541244284750238048417010399988899251006556975568600094486590861068493} a^{21} - \frac{188942093996520254299964123729844262662711019555291101847908769074888500}{39474541244284750238048417010399988899251006556975568600094486590861068493} a^{20} - \frac{963461149491892907750855601846681563345942308837081020002160912870997592}{39474541244284750238048417010399988899251006556975568600094486590861068493} a^{19} + \frac{15437252604335860365159011081199311848675557277693732742027057145822152780}{39474541244284750238048417010399988899251006556975568600094486590861068493} a^{18} - \frac{10509291766506788762728682474476286723954269157587604850874763850362041125}{39474541244284750238048417010399988899251006556975568600094486590861068493} a^{17} - \frac{16913393986677229516897030039937856387364656659805486666827513252660815320}{39474541244284750238048417010399988899251006556975568600094486590861068493} a^{16} + \frac{11080090772555870133157863444450595698944382773631174006821484277487197217}{39474541244284750238048417010399988899251006556975568600094486590861068493} a^{15} - \frac{10603853896860192095405489511895332145139573153089398690430321879385837257}{39474541244284750238048417010399988899251006556975568600094486590861068493} a^{14} + \frac{18179981593193306605570800149255991122234030600461505310705674400519460955}{39474541244284750238048417010399988899251006556975568600094486590861068493} a^{13} + \frac{7377446962153355713150192515808063793656817463121292183686179498505167284}{39474541244284750238048417010399988899251006556975568600094486590861068493} a^{12} + \frac{5773571825865208654160144067954168127047133764242841784198994663777824262}{39474541244284750238048417010399988899251006556975568600094486590861068493} a^{11} - \frac{399102950627993957609706931992021824496060741115520442288365342659326256}{839883856261377664639328021497872104239383118233522736172223118954490819} a^{10} - \frac{804111443415785296006522763260962025747074117210906622647289256427070011}{39474541244284750238048417010399988899251006556975568600094486590861068493} a^{9} - \frac{1964919661376334240206971704893608136221294541317255014139431294299279695}{39474541244284750238048417010399988899251006556975568600094486590861068493} a^{8} - \frac{10912212391742207900422776643342565224580634721817599563509641983374908962}{39474541244284750238048417010399988899251006556975568600094486590861068493} a^{7} - \frac{440668907743168548060131330212056334354632640523322042806247377894686836}{39474541244284750238048417010399988899251006556975568600094486590861068493} a^{6} - \frac{10013898009507110686350467977499234808754230513202559925801976257546061208}{39474541244284750238048417010399988899251006556975568600094486590861068493} a^{5} + \frac{14415794145268654424040409015261097910395619065830092626880837038192261912}{39474541244284750238048417010399988899251006556975568600094486590861068493} a^{4} + \frac{8806417799771849189195842499269190935974565435644058310193294436796768530}{39474541244284750238048417010399988899251006556975568600094486590861068493} a^{3} - \frac{18416605957846594471657725867703124600093865708834024033910615602632906932}{39474541244284750238048417010399988899251006556975568600094486590861068493} a^{2} + \frac{2913823239245113604742527142306254735167213312235417749840058195172611579}{39474541244284750238048417010399988899251006556975568600094486590861068493} a - \frac{56256038854223441175754342715106654690790101918551632796609997818116259}{839883856261377664639328021497872104239383118233522736172223118954490819}$
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: | \( 393171463879000 \) (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
| \(\Q(\sqrt{7}) \), \(\Q(\zeta_{23})^+\) |
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 | R | ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ | $22$ | R | $22$ | $22$ | $22$ | ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ | R | ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ | $22$ | $22$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ | ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 7 | Data not computed | ||||||
| 23 | Data not computed | ||||||