Normalized defining polynomial
\( x^{22} - 2 x^{21} + 91 x^{20} - 162 x^{19} + 4154 x^{18} - 6608 x^{17} + 123265 x^{16} - 174450 x^{15} + 2616392 x^{14} - 3263164 x^{13} + 41456020 x^{12} - 44873920 x^{11} + 498358184 x^{10} - 457441488 x^{9} + 4533809731 x^{8} - 3404345062 x^{7} + 30545809231 x^{6} - 17679217442 x^{5} + 145074301925 x^{4} - 57845366150 x^{3} + 437292880806 x^{2} - 90650889312 x + 634556225161 \)
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: | \(-719807926798998083647106533713510400000000000=-\,2^{33}\cdot 5^{11}\cdot 23^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $109.39$ | 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: | $2, 5, 23$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(920=2^{3}\cdot 5\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{920}(1,·)$, $\chi_{920}(259,·)$, $\chi_{920}(841,·)$, $\chi_{920}(139,·)$, $\chi_{920}(499,·)$, $\chi_{920}(721,·)$, $\chi_{920}(899,·)$, $\chi_{920}(121,·)$, $\chi_{920}(601,·)$, $\chi_{920}(859,·)$, $\chi_{920}(361,·)$, $\chi_{920}(219,·)$, $\chi_{920}(81,·)$, $\chi_{920}(41,·)$, $\chi_{920}(561,·)$, $\chi_{920}(579,·)$, $\chi_{920}(179,·)$, $\chi_{920}(59,·)$, $\chi_{920}(739,·)$, $\chi_{920}(761,·)$, $\chi_{920}(441,·)$, $\chi_{920}(699,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{47} a^{20} - \frac{12}{47} a^{19} + \frac{6}{47} a^{18} - \frac{18}{47} a^{17} + \frac{2}{47} a^{16} + \frac{23}{47} a^{15} + \frac{2}{47} a^{14} - \frac{21}{47} a^{13} - \frac{16}{47} a^{12} - \frac{1}{47} a^{11} - \frac{1}{47} a^{10} + \frac{15}{47} a^{9} - \frac{10}{47} a^{8} + \frac{4}{47} a^{7} - \frac{19}{47} a^{6} - \frac{3}{47} a^{5} + \frac{10}{47} a^{4} + \frac{17}{47} a^{3} - \frac{11}{47} a^{2} - \frac{4}{47} a - \frac{11}{47}$, $\frac{1}{51197899833579289860167221847669336114555214005356210297146851641976303269201634663138407} a^{21} - \frac{296491381721015873797890745057498804532165777238023340552943571449519454538144719366805}{51197899833579289860167221847669336114555214005356210297146851641976303269201634663138407} a^{20} + \frac{19554908851490182321530143584937367046219623757841943098662858692909698686307647088165848}{51197899833579289860167221847669336114555214005356210297146851641976303269201634663138407} a^{19} - \frac{15091307421535785085675357101971114646260556709184157110581215209210177076638453890355857}{51197899833579289860167221847669336114555214005356210297146851641976303269201634663138407} a^{18} - \frac{9410541746094897712774243432657612434710873972188204933367780346290777782173305871787470}{51197899833579289860167221847669336114555214005356210297146851641976303269201634663138407} a^{17} + \frac{15559414016218835120128549114460190002537770295415025861311559577608805696385607635979901}{51197899833579289860167221847669336114555214005356210297146851641976303269201634663138407} a^{16} - \frac{10684733073511142452638522630025855627857225563236975186399287620113272221090084540125213}{51197899833579289860167221847669336114555214005356210297146851641976303269201634663138407} a^{15} - \frac{18267390642635339220528803084495618172343500638439148270603209645500226841096571746367640}{51197899833579289860167221847669336114555214005356210297146851641976303269201634663138407} a^{14} + \frac{19603767620137090950896317490440622848082403124179184362474927879695171778910139220662106}{51197899833579289860167221847669336114555214005356210297146851641976303269201634663138407} a^{13} + \frac{19639149402397160471733207190932289165152716923832955152748327904771662608702345140484256}{51197899833579289860167221847669336114555214005356210297146851641976303269201634663138407} a^{12} - \frac{24454698113477287826544198019657652948671355961792980291314680155413503158395594016229924}{51197899833579289860167221847669336114555214005356210297146851641976303269201634663138407} a^{11} - \frac{8260275309260742060674952592829003155076142322073373376697339012708264763575676679116096}{51197899833579289860167221847669336114555214005356210297146851641976303269201634663138407} a^{10} + \frac{13986787484060769706905788692816202283437908367445258916575067350025348815784633006142725}{51197899833579289860167221847669336114555214005356210297146851641976303269201634663138407} a^{9} - \frac{12717369826761489140997661021746869663863145560938413319723751493609377492614375707618360}{51197899833579289860167221847669336114555214005356210297146851641976303269201634663138407} a^{8} + \frac{6579085148083514416849989437951959274745707498197881652603385281603234633400155081387027}{51197899833579289860167221847669336114555214005356210297146851641976303269201634663138407} a^{7} - \frac{16328647547256607316609311226549334756690550161657286350996706897541708734521521058085927}{51197899833579289860167221847669336114555214005356210297146851641976303269201634663138407} a^{6} + \frac{22209327570144471326728645743472829735447935662161735387282167678613498754312724150799264}{51197899833579289860167221847669336114555214005356210297146851641976303269201634663138407} a^{5} - \frac{21721497923990959801744978702648335706285506914073534128212223898445628998013072509369086}{51197899833579289860167221847669336114555214005356210297146851641976303269201634663138407} a^{4} - \frac{13259710016856193536063339014224604982965485016805053820806242480179000562869372250719075}{51197899833579289860167221847669336114555214005356210297146851641976303269201634663138407} a^{3} + \frac{1120061569509873086666491948539774163299907573202182098115434715577674665014055295030546}{51197899833579289860167221847669336114555214005356210297146851641976303269201634663138407} a^{2} - \frac{3260132370604462183835929729069927347531569338607692313255147393634655825282384650356319}{51197899833579289860167221847669336114555214005356210297146851641976303269201634663138407} a - \frac{1685591928305849363534915438991263095658234039475269376324820536014601631918321164739745}{51197899833579289860167221847669336114555214005356210297146851641976303269201634663138407}$
Class group and class number
$C_{17310766}$, which has order $17310766$ (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
| \(\Q(\sqrt{-10}) \), \(\Q(\zeta_{23})^+\) |
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 | $22$ | R | ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ | R | $22$ | $22$ | ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ | ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$ |
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 | ||||||
| 5 | Data not computed | ||||||
| $23$ | 23.11.10.10 | $x^{11} - 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |
| 23.11.10.10 | $x^{11} - 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ | |