Normalized defining polynomial
\( x^{22} - 9 x^{21} - 73 x^{20} + 807 x^{19} + 1975 x^{18} - 30667 x^{17} - 19975 x^{16} + 644732 x^{15} - 103492 x^{14} - 8223098 x^{13} + 4607977 x^{12} + 65804524 x^{11} - 48200954 x^{10} - 331663703 x^{9} + 247797313 x^{8} + 1036117069 x^{7} - 652601717 x^{6} - 1936994870 x^{5} + 759946006 x^{4} + 2008563717 x^{3} - 128681365 x^{2} - 908178246 x - 257391023 \)
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: | \(86738130821466057117717765143796460851305217=3^{11}\cdot 11^{11}\cdot 23^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $99.36$ | 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: | $3, 11, 23$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(759=3\cdot 11\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{759}(1,·)$, $\chi_{759}(131,·)$, $\chi_{759}(133,·)$, $\chi_{759}(265,·)$, $\chi_{759}(395,·)$, $\chi_{759}(397,·)$, $\chi_{759}(461,·)$, $\chi_{759}(593,·)$, $\chi_{759}(725,·)$, $\chi_{759}(100,·)$, $\chi_{759}(331,·)$, $\chi_{759}(463,·)$, $\chi_{759}(197,·)$, $\chi_{759}(32,·)$, $\chi_{759}(496,·)$, $\chi_{759}(98,·)$, $\chi_{759}(164,·)$, $\chi_{759}(232,·)$, $\chi_{759}(430,·)$, $\chi_{759}(560,·)$, $\chi_{759}(692,·)$, $\chi_{759}(694,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{95147470122769387936108902109394982118164803410249222576103593713286442054705381} a^{21} + \frac{30141826783869466372849131550512499504191045006325856186136645975063976971470454}{95147470122769387936108902109394982118164803410249222576103593713286442054705381} a^{20} + \frac{853027126686047709321778795877759669164378396779593472075052844033343103783164}{95147470122769387936108902109394982118164803410249222576103593713286442054705381} a^{19} - \frac{23234721316024505519704260406298970916435595676738180638644465101863312018623624}{95147470122769387936108902109394982118164803410249222576103593713286442054705381} a^{18} - \frac{32939756671501276864177154818392058059349992499801575869413544957052459387188619}{95147470122769387936108902109394982118164803410249222576103593713286442054705381} a^{17} - \frac{47543072305238780468463084596514957107736356870266504196730736720357340501940654}{95147470122769387936108902109394982118164803410249222576103593713286442054705381} a^{16} - \frac{28588488647506013710235322525247785219405268055425927236684294119626992708547791}{95147470122769387936108902109394982118164803410249222576103593713286442054705381} a^{15} - \frac{42828467958058082720423711566224208268060744366000869101676631321629763080504841}{95147470122769387936108902109394982118164803410249222576103593713286442054705381} a^{14} + \frac{36885579979858650657477803895715008769837223250051742794256818834200375751235061}{95147470122769387936108902109394982118164803410249222576103593713286442054705381} a^{13} + \frac{13976602647035945777259882064196447394361232153831941347037988603769524635508713}{95147470122769387936108902109394982118164803410249222576103593713286442054705381} a^{12} + \frac{7563545425831484073741858586601916839161192649045037036515637798937647331959061}{95147470122769387936108902109394982118164803410249222576103593713286442054705381} a^{11} + \frac{35773904544199070686109035841595206856134077122007889452446625850050423280249614}{95147470122769387936108902109394982118164803410249222576103593713286442054705381} a^{10} - \frac{3140069473050060810379085126291053395942132961260559510833248464417047036238164}{95147470122769387936108902109394982118164803410249222576103593713286442054705381} a^{9} + \frac{19710753285482635682838375997767013586362691386782493190767335082777137011007454}{95147470122769387936108902109394982118164803410249222576103593713286442054705381} a^{8} - \frac{13425187917370167833653752847206299380845963902925736841461763374523977841342064}{95147470122769387936108902109394982118164803410249222576103593713286442054705381} a^{7} + \frac{33263374928849193592403466546204125169147407912352022054383504202995336293666095}{95147470122769387936108902109394982118164803410249222576103593713286442054705381} a^{6} - \frac{19955335524508015359988043894015227549102502436857895188866553626463880664667431}{95147470122769387936108902109394982118164803410249222576103593713286442054705381} a^{5} + \frac{30789393129383691926112261257359076461751365476065336276618418802363648301021521}{95147470122769387936108902109394982118164803410249222576103593713286442054705381} a^{4} + \frac{42023890364600914891843342836817599344785701234832551814701308569168211047846745}{95147470122769387936108902109394982118164803410249222576103593713286442054705381} a^{3} + \frac{35175650668611543490973846463932207236315903328199372082755590701858618221697226}{95147470122769387936108902109394982118164803410249222576103593713286442054705381} a^{2} + \frac{30597627074782951741479820036739653790807671624927426339064645786971884040604229}{95147470122769387936108902109394982118164803410249222576103593713286442054705381} a + \frac{25033268028654547915887905098192465660218127175880814743732812080344241286482760}{95147470122769387936108902109394982118164803410249222576103593713286442054705381}$
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: | \( 498239383130000 \) (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
| \(\Q(\sqrt{33}) \), \(\Q(\zeta_{23})^+\) |
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 | ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ | R | $22$ | $22$ | R | $22$ | ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ | $22$ | R | ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$ | $22$ | $22$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 11 | Data not computed | ||||||
| 23 | Data not computed | ||||||