Normalized defining polynomial
\( x^{22} - 9 x^{21} + 70 x^{20} - 363 x^{19} + 1845 x^{18} - 7462 x^{17} + 30023 x^{16} - 100740 x^{15} + 341082 x^{14} - 977548 x^{13} + 2866861 x^{12} - 7075946 x^{11} + 18221714 x^{10} - 38558891 x^{9} + 87678393 x^{8} - 156007735 x^{7} + 313286973 x^{6} - 448933295 x^{5} + 793158991 x^{4} - 832219241 x^{3} + 1285528692 x^{2} - 758821285 x + 1019110541 \)
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: | \(-199915437101854222076001836684210081547019=-\,19^{11}\cdot 23^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $75.39$ | 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: | $19, 23$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(437=19\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{437}(1,·)$, $\chi_{437}(324,·)$, $\chi_{437}(265,·)$, $\chi_{437}(75,·)$, $\chi_{437}(77,·)$, $\chi_{437}(400,·)$, $\chi_{437}(18,·)$, $\chi_{437}(151,·)$, $\chi_{437}(284,·)$, $\chi_{437}(94,·)$, $\chi_{437}(96,·)$, $\chi_{437}(417,·)$, $\chi_{437}(39,·)$, $\chi_{437}(170,·)$, $\chi_{437}(208,·)$, $\chi_{437}(210,·)$, $\chi_{437}(303,·)$, $\chi_{437}(305,·)$, $\chi_{437}(246,·)$, $\chi_{437}(248,·)$, $\chi_{437}(58,·)$, $\chi_{437}(381,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{6533} a^{20} - \frac{120}{6533} a^{19} - \frac{226}{6533} a^{18} - \frac{739}{6533} a^{17} - \frac{882}{6533} a^{16} + \frac{382}{6533} a^{15} + \frac{2345}{6533} a^{14} - \frac{2764}{6533} a^{13} - \frac{1630}{6533} a^{12} - \frac{1581}{6533} a^{11} - \frac{547}{6533} a^{10} + \frac{1874}{6533} a^{9} + \frac{2551}{6533} a^{8} - \frac{1853}{6533} a^{7} - \frac{2828}{6533} a^{6} + \frac{631}{6533} a^{5} - \frac{674}{6533} a^{4} - \frac{2984}{6533} a^{3} - \frac{3113}{6533} a^{2} + \frac{741}{6533} a + \frac{17}{139}$, $\frac{1}{78014444873648371129857004141805757431165320095252192103160561737761739} a^{21} - \frac{2482297610145391536519785704328511083682885488752285307524297296839}{78014444873648371129857004141805757431165320095252192103160561737761739} a^{20} - \frac{544032616710879941248288916931376089083879760955149934118183946537931}{78014444873648371129857004141805757431165320095252192103160561737761739} a^{19} + \frac{31612886375228399331495422297011135503501693893589201911160444695969483}{78014444873648371129857004141805757431165320095252192103160561737761739} a^{18} - \frac{1051087400189911282901017296399270040026931169234266448216983179105666}{78014444873648371129857004141805757431165320095252192103160561737761739} a^{17} - \frac{11945239358277619683425561928802836283853195886674796115156425273737021}{78014444873648371129857004141805757431165320095252192103160561737761739} a^{16} + \frac{15107036619482693960615692524699647556793135940831222677316784877388725}{78014444873648371129857004141805757431165320095252192103160561737761739} a^{15} - \frac{13720773607941173669122963881476344713350833805668191700975976061412166}{78014444873648371129857004141805757431165320095252192103160561737761739} a^{14} + \frac{17252790569935225463658182845670829542609910410231505524737323333374303}{78014444873648371129857004141805757431165320095252192103160561737761739} a^{13} - \frac{32546558419727105302006395830202857814096685447813263968129368113103561}{78014444873648371129857004141805757431165320095252192103160561737761739} a^{12} - \frac{18216245670559972486531794440810709041214304951839170171079247279314395}{78014444873648371129857004141805757431165320095252192103160561737761739} a^{11} - \frac{6963695204902350228974747430090059401830870025155042396213974319483487}{78014444873648371129857004141805757431165320095252192103160561737761739} a^{10} - \frac{5496477259658420934246509669609604340269243136106592235463429216143951}{78014444873648371129857004141805757431165320095252192103160561737761739} a^{9} - \frac{8913451945078285897409710940379678862897493184371847089137802427775509}{78014444873648371129857004141805757431165320095252192103160561737761739} a^{8} + \frac{14356359317540486873924536892930871504283282986827966909147769844186340}{78014444873648371129857004141805757431165320095252192103160561737761739} a^{7} - \frac{32760307148282238147610655109611676403396408948664528537044354419282951}{78014444873648371129857004141805757431165320095252192103160561737761739} a^{6} - \frac{29512417579792767553366387155915160786238937278930014990396260361758697}{78014444873648371129857004141805757431165320095252192103160561737761739} a^{5} - \frac{33485949815520673272138601356142722274037335751067303941850177547214358}{78014444873648371129857004141805757431165320095252192103160561737761739} a^{4} + \frac{13207649232893515163670616471786953277209567502134513274675286720026494}{78014444873648371129857004141805757431165320095252192103160561737761739} a^{3} - \frac{35615136423549660614628612304605286241554137570756293958769766974798650}{78014444873648371129857004141805757431165320095252192103160561737761739} a^{2} - \frac{9190451067686745809323351434545107714401073048963591190840787416242266}{78014444873648371129857004141805757431165320095252192103160561737761739} a - \frac{146548323671027574533993321890191351425847847603108507878931951081702}{1659881805822305768720361790251186328322666385005365789428948122080037}$
Class group and class number
$C_{408991}$, which has order $408991$ (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
| \(\Q(\sqrt{-19}) \), \(\Q(\zeta_{23})^+\) |
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 | $22$ | $22$ | ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ | R | R | $22$ | $22$ | $22$ | $22$ | ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ | $22$ | $22$ |
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 |
|---|---|---|---|---|---|---|---|
| 19 | Data not computed | ||||||
| $23$ | 23.11.10.10 | $x^{11} - 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |
| 23.11.10.10 | $x^{11} - 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ | |