Normalized defining polynomial
\( x^{22} - 9 x^{21} + 48 x^{20} - 183 x^{19} + 655 x^{18} - 2122 x^{17} + 6491 x^{16} - 17468 x^{15} + 45646 x^{14} - 108112 x^{13} + 251537 x^{12} - 521318 x^{11} + 1081334 x^{10} - 1963079 x^{9} + 3653567 x^{8} - 5688311 x^{7} + 9483883 x^{6} - 12150619 x^{5} + 18112705 x^{4} - 17363285 x^{3} + 23145134 x^{2} - 12720749 x + 15354571 \)
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: | \(-489639287266880371204241478228795637811=-\,11^{11}\cdot 23^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $57.36$ | 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: | $11, 23$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(253=11\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{253}(1,·)$, $\chi_{253}(131,·)$, $\chi_{253}(133,·)$, $\chi_{253}(12,·)$, $\chi_{253}(142,·)$, $\chi_{253}(144,·)$, $\chi_{253}(210,·)$, $\chi_{253}(78,·)$, $\chi_{253}(87,·)$, $\chi_{253}(100,·)$, $\chi_{253}(219,·)$, $\chi_{253}(197,·)$, $\chi_{253}(32,·)$, $\chi_{253}(208,·)$, $\chi_{253}(98,·)$, $\chi_{253}(164,·)$, $\chi_{253}(232,·)$, $\chi_{253}(177,·)$, $\chi_{253}(243,·)$, $\chi_{253}(54,·)$, $\chi_{253}(186,·)$, $\chi_{253}(188,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{47} a^{20} + \frac{12}{47} a^{19} - \frac{1}{47} a^{18} - \frac{9}{47} a^{17} + \frac{15}{47} a^{16} + \frac{9}{47} a^{15} + \frac{3}{47} a^{14} + \frac{2}{47} a^{13} - \frac{6}{47} a^{12} + \frac{12}{47} a^{11} - \frac{17}{47} a^{10} - \frac{15}{47} a^{9} + \frac{13}{47} a^{8} + \frac{11}{47} a^{7} + \frac{6}{47} a^{6} + \frac{16}{47} a^{5} + \frac{22}{47} a^{4} - \frac{21}{47} a^{3} + \frac{20}{47} a^{2} - \frac{8}{47} a$, $\frac{1}{278445176511616137369420557399385127158749614155228131210266193} a^{21} - \frac{812574882666301842875453513237860597563402573755774013521513}{278445176511616137369420557399385127158749614155228131210266193} a^{20} - \frac{44692086554015659925213412266593927142224055711373561839620961}{278445176511616137369420557399385127158749614155228131210266193} a^{19} - \frac{92633959219051079103853436582030148424604970355093860830447579}{278445176511616137369420557399385127158749614155228131210266193} a^{18} + \frac{40766112382702123348458843784934372596914229210180805369713542}{278445176511616137369420557399385127158749614155228131210266193} a^{17} + \frac{120927258930627461097226071955391182497922244687810325418155592}{278445176511616137369420557399385127158749614155228131210266193} a^{16} - \frac{133257257144392901608447865362863753078137300103411126868339729}{278445176511616137369420557399385127158749614155228131210266193} a^{15} + \frac{3747829414627951674238827337392311418592470561871180718424247}{278445176511616137369420557399385127158749614155228131210266193} a^{14} - \frac{60029194682876377399486416381929475743963691769493848326035927}{278445176511616137369420557399385127158749614155228131210266193} a^{13} - \frac{95497039831896199446650399265540789874888495023127633634651317}{278445176511616137369420557399385127158749614155228131210266193} a^{12} - \frac{47050886648815620964273333208424406999440302092002096935937129}{278445176511616137369420557399385127158749614155228131210266193} a^{11} - \frac{2542303752210093611206014961255489796591810624553398134285676}{5924365457693960369562139519135853769335098173515492153409919} a^{10} + \frac{34183026399017368163557916031690636003584151490390597066050387}{278445176511616137369420557399385127158749614155228131210266193} a^{9} + \frac{98470092901517247893207132332207621268577921774318651952687698}{278445176511616137369420557399385127158749614155228131210266193} a^{8} + \frac{104964504068599140194797652981133688245683800540048531666465064}{278445176511616137369420557399385127158749614155228131210266193} a^{7} - \frac{35207826817178339082778214968505446431228153373240065656582873}{278445176511616137369420557399385127158749614155228131210266193} a^{6} + \frac{68803719446911969458964862343922651184255264643182930040596828}{278445176511616137369420557399385127158749614155228131210266193} a^{5} - \frac{77766755469594544130720223276095637129239550676151968753524559}{278445176511616137369420557399385127158749614155228131210266193} a^{4} + \frac{125757222537154693969902511005611523988538735190187707984735221}{278445176511616137369420557399385127158749614155228131210266193} a^{3} - \frac{94909693620128499206895699911010375758525424555517985458567342}{278445176511616137369420557399385127158749614155228131210266193} a^{2} - \frac{27210150844985322027547372061602879989844398779435964490591438}{278445176511616137369420557399385127158749614155228131210266193} a - \frac{1174707593182358074679031058298038968283456945279851419067286}{5924365457693960369562139519135853769335098173515492153409919}$
Class group and class number
$C_{15499}$, which has order $15499$ (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
| \(\Q(\sqrt{-11}) \), \(\Q(\zeta_{23})^+\) |
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 | $22$ | ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ | $22$ | R | $22$ | $22$ | $22$ | R | $22$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ | $22$ | $22$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ | ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 11 | Data not computed | ||||||
| $23$ | 23.11.10.10 | $x^{11} - 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |
| 23.11.10.10 | $x^{11} - 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ | |