Normalized defining polynomial
\( x^{22} - 2 x^{21} - 85 x^{20} + 158 x^{19} + 3034 x^{18} - 5168 x^{17} - 59231 x^{16} + 90766 x^{15} + 691144 x^{14} - 929756 x^{13} - 4958124 x^{12} + 5663424 x^{11} + 21755176 x^{10} - 20144336 x^{9} - 56643133 x^{8} + 39793946 x^{7} + 82921103 x^{6} - 40082434 x^{5} - 62983979 x^{4} + 17930170 x^{3} + 21937542 x^{2} - 2672256 x - 2672279 \)
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: | \(2611441967281400084968119933496263205453824=2^{33}\cdot 3^{11}\cdot 23^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $84.73$ | 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: | $2, 3, 23$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(552=2^{3}\cdot 3\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{552}(1,·)$, $\chi_{552}(515,·)$, $\chi_{552}(193,·)$, $\chi_{552}(265,·)$, $\chi_{552}(395,·)$, $\chi_{552}(131,·)$, $\chi_{552}(25,·)$, $\chi_{552}(409,·)$, $\chi_{552}(347,·)$, $\chi_{552}(361,·)$, $\chi_{552}(289,·)$, $\chi_{552}(35,·)$, $\chi_{552}(169,·)$, $\chi_{552}(323,·)$, $\chi_{552}(491,·)$, $\chi_{552}(49,·)$, $\chi_{552}(179,·)$, $\chi_{552}(73,·)$, $\chi_{552}(371,·)$, $\chi_{552}(121,·)$, $\chi_{552}(59,·)$, $\chi_{552}(443,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{47} a^{19} - \frac{15}{47} a^{18} + \frac{7}{47} a^{17} - \frac{5}{47} a^{16} - \frac{16}{47} a^{15} - \frac{19}{47} a^{14} + \frac{5}{47} a^{13} - \frac{4}{47} a^{12} + \frac{1}{47} a^{11} + \frac{20}{47} a^{10} - \frac{5}{47} a^{9} - \frac{22}{47} a^{8} + \frac{2}{47} a^{7} - \frac{10}{47} a^{6} + \frac{11}{47} a^{5} - \frac{11}{47} a^{4} - \frac{11}{47} a^{3} - \frac{2}{47} a^{2} + \frac{21}{47} a$, $\frac{1}{47} a^{20} + \frac{17}{47} a^{18} + \frac{6}{47} a^{17} + \frac{3}{47} a^{16} + \frac{23}{47} a^{15} + \frac{2}{47} a^{14} - \frac{23}{47} a^{13} - \frac{12}{47} a^{12} - \frac{12}{47} a^{11} + \frac{13}{47} a^{10} - \frac{3}{47} a^{9} + \frac{1}{47} a^{8} + \frac{20}{47} a^{7} + \frac{2}{47} a^{6} + \frac{13}{47} a^{5} + \frac{12}{47} a^{4} + \frac{21}{47} a^{3} - \frac{9}{47} a^{2} - \frac{14}{47} a$, $\frac{1}{19572085544165529777021928002751619850500993511527301387791220644471} a^{21} - \frac{67793601747447861088880423420588670733369122869008105819234234755}{19572085544165529777021928002751619850500993511527301387791220644471} a^{20} + \frac{53089816003393565996875752300452429266925519684945564339797886618}{19572085544165529777021928002751619850500993511527301387791220644471} a^{19} - \frac{3762057825092916016377187928591480454350818103343783514345678096603}{19572085544165529777021928002751619850500993511527301387791220644471} a^{18} + \frac{3459096080033799592862897069596690302333528121709167113859946688373}{19572085544165529777021928002751619850500993511527301387791220644471} a^{17} - \frac{6643917614197155786578326667173034878671101558140042884712376241115}{19572085544165529777021928002751619850500993511527301387791220644471} a^{16} + \frac{7666512773576002435170677658191319586048328323791235432129661865373}{19572085544165529777021928002751619850500993511527301387791220644471} a^{15} + \frac{2282990875373252310263478961525986270160157394678032928309436518388}{19572085544165529777021928002751619850500993511527301387791220644471} a^{14} - \frac{5023614081115098952489132647363830532732040060216181846408133346251}{19572085544165529777021928002751619850500993511527301387791220644471} a^{13} - \frac{393990230989210667652358231349262646206043696983532381533095563952}{19572085544165529777021928002751619850500993511527301387791220644471} a^{12} - \frac{81862249769852156216976784371022833142716972225891891795814621845}{416427352003521910149402723462800422351084968330368114633855758393} a^{11} + \frac{4168143954464711934367988693934727854515605552154358757448233376921}{19572085544165529777021928002751619850500993511527301387791220644471} a^{10} + \frac{2223694664958826317779879094196868786044593559023501793913672468221}{19572085544165529777021928002751619850500993511527301387791220644471} a^{9} + \frac{4782187920213044276160300939967367298820408859030215833304522037442}{19572085544165529777021928002751619850500993511527301387791220644471} a^{8} + \frac{6233801708900744617640213528852580817450248552860476192356137744030}{19572085544165529777021928002751619850500993511527301387791220644471} a^{7} + \frac{1924460985193653934080194700010740121796849609569231122526628252053}{19572085544165529777021928002751619850500993511527301387791220644471} a^{6} + \frac{6904681388630653596690624277564248644167685464714941122372205759765}{19572085544165529777021928002751619850500993511527301387791220644471} a^{5} + \frac{6360008972129012115934659436453301873717319185600562733810657878007}{19572085544165529777021928002751619850500993511527301387791220644471} a^{4} - \frac{2808422575996915737064103050310635196690252842132212898724334654195}{19572085544165529777021928002751619850500993511527301387791220644471} a^{3} - \frac{1275503326789758077359789220222818751868505341827080579896188937033}{19572085544165529777021928002751619850500993511527301387791220644471} a^{2} + \frac{3882115907487725213320022837959833021764370746987301016479866890963}{19572085544165529777021928002751619850500993511527301387791220644471} a + \frac{2616936754742675558512343687640188860444864508374906955303141}{7324117558146260093733449240424229599716569082617234722793249}$
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: | \( 152044041063000 \) (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
| \(\Q(\sqrt{6}) \), \(\Q(\zeta_{23})^+\) |
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 | R | ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ | $22$ | $22$ | $22$ | $22$ | ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ | R | ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ | $22$ | $22$ | $22$ | ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ | ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ | $22$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 3 | Data not computed | ||||||
| $23$ | 23.11.10.10 | $x^{11} - 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |
| 23.11.10.10 | $x^{11} - 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ | |