Normalized defining polynomial
\( x^{22} + 181 x^{20} + 13028 x^{18} + 490585 x^{16} + 10635319 x^{14} + 137334967 x^{12} + 1056115843 x^{10} + 4729178714 x^{8} + 11925395238 x^{6} + 15697884460 x^{4} + 8830679464 x^{2} + 939238609 \)
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: | \(-39785189666798958828616586618395684610035295457378304=-\,2^{22}\cdot 199^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $245.98$ | 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: | $2, 199$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(796=2^{2}\cdot 199\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{796}(1,·)$, $\chi_{796}(387,·)$, $\chi_{796}(261,·)$, $\chi_{796}(519,·)$, $\chi_{796}(523,·)$, $\chi_{796}(461,·)$, $\chi_{796}(399,·)$, $\chi_{796}(785,·)$, $\chi_{796}(659,·)$, $\chi_{796}(139,·)$, $\chi_{796}(217,·)$, $\chi_{796}(615,·)$, $\chi_{796}(537,·)$, $\chi_{796}(459,·)$, $\chi_{796}(103,·)$, $\chi_{796}(711,·)$, $\chi_{796}(125,·)$, $\chi_{796}(501,·)$, $\chi_{796}(121,·)$, $\chi_{796}(313,·)$, $\chi_{796}(61,·)$, $\chi_{796}(63,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{19} a^{13} - \frac{3}{19} a^{11} - \frac{7}{19} a^{9} + \frac{8}{19} a^{7} + \frac{5}{19} a^{5} - \frac{1}{19} a^{3} + \frac{5}{19} a$, $\frac{1}{19} a^{14} - \frac{3}{19} a^{12} - \frac{7}{19} a^{10} + \frac{8}{19} a^{8} + \frac{5}{19} a^{6} - \frac{1}{19} a^{4} + \frac{5}{19} a^{2}$, $\frac{1}{19} a^{15} + \frac{3}{19} a^{11} + \frac{6}{19} a^{9} - \frac{9}{19} a^{7} - \frac{5}{19} a^{5} + \frac{2}{19} a^{3} - \frac{4}{19} a$, $\frac{1}{19} a^{16} + \frac{3}{19} a^{12} + \frac{6}{19} a^{10} - \frac{9}{19} a^{8} - \frac{5}{19} a^{6} + \frac{2}{19} a^{4} - \frac{4}{19} a^{2}$, $\frac{1}{19} a^{17} - \frac{4}{19} a^{11} - \frac{7}{19} a^{9} + \frac{9}{19} a^{7} + \frac{6}{19} a^{5} - \frac{1}{19} a^{3} + \frac{4}{19} a$, $\frac{1}{30229} a^{18} - \frac{119}{30229} a^{16} - \frac{72}{30229} a^{14} + \frac{12661}{30229} a^{12} - \frac{6791}{30229} a^{10} + \frac{10308}{30229} a^{8} - \frac{14636}{30229} a^{6} + \frac{9656}{30229} a^{4} - \frac{14548}{30229} a^{2} + \frac{758}{1591}$, $\frac{1}{30229} a^{19} - \frac{119}{30229} a^{17} - \frac{72}{30229} a^{15} - \frac{67}{30229} a^{13} + \frac{1164}{30229} a^{11} + \frac{8717}{30229} a^{9} + \frac{4456}{30229} a^{7} + \frac{6474}{30229} a^{5} - \frac{1820}{30229} a^{3} + \frac{11220}{30229} a$, $\frac{1}{144962718953821828604741512500095363878169} a^{20} + \frac{2001085695908902881294638468326971551}{144962718953821828604741512500095363878169} a^{18} - \frac{1637259994659215226253925073877756066500}{144962718953821828604741512500095363878169} a^{16} + \frac{3239332405892760010937036993538320999121}{144962718953821828604741512500095363878169} a^{14} + \frac{71322171979936849684610822560325154087079}{144962718953821828604741512500095363878169} a^{12} - \frac{5940905981487450297871855035746534210352}{144962718953821828604741512500095363878169} a^{10} + \frac{33169547443406761249090231233629037365138}{144962718953821828604741512500095363878169} a^{8} - \frac{44464924774687716163843582126022532265324}{144962718953821828604741512500095363878169} a^{6} - \frac{64071029431765007204673360039756385613675}{144962718953821828604741512500095363878169} a^{4} - \frac{618254270084504433242928624074215122301}{3917911323076265637965986824326901726437} a^{2} + \frac{1049823113751544042125688378539409854495}{7629616787043254137091658552636598098851}$, $\frac{1}{233824865672514609539448059662653821935486597} a^{21} - \frac{1513372204502499838363811648541833304525}{233824865672514609539448059662653821935486597} a^{19} + \frac{2284466394762889566819211483608032392069420}{233824865672514609539448059662653821935486597} a^{17} - \frac{2367279246488879405174204339948624521249982}{233824865672514609539448059662653821935486597} a^{15} - \frac{5953730097572502836736568873977182380793182}{233824865672514609539448059662653821935486597} a^{13} + \frac{45106530123821318193183974574283685623777728}{233824865672514609539448059662653821935486597} a^{11} + \frac{5482884019910324718232752575024714593117783}{12306571877500768923128845245402832733446663} a^{9} + \frac{105633064667970595897030643234609612862208356}{233824865672514609539448059662653821935486597} a^{7} + \frac{57006623045044902567719987410268884921598167}{233824865672514609539448059662653821935486597} a^{5} - \frac{21397373917249835590584810918721814043732530}{233824865672514609539448059662653821935486597} a^{3} - \frac{9703887262025602293184262201846434924033461}{233824865672514609539448059662653821935486597} a$
Class group and class number
$C_{2097569}$, which has order $2097569$ (assuming GRH)
Unit group
| Rank: | $10$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{13767192571160672117825}{62592952517930880991653130526797} a^{21} - \frac{2511116595514434321760975}{62592952517930880991653130526797} a^{19} - \frac{182614766266257589795511039}{62592952517930880991653130526797} a^{17} - \frac{6966325857148435326796818779}{62592952517930880991653130526797} a^{15} - \frac{153382317283888556993186901789}{62592952517930880991653130526797} a^{13} - \frac{2015658623128782336477083919198}{62592952517930880991653130526797} a^{11} - \frac{15799487323603083942147385549458}{62592952517930880991653130526797} a^{9} - \frac{72321732745355854662228243579933}{62592952517930880991653130526797} a^{7} - \frac{5089698488907659019765276910857}{1691701419403537324098733257481} a^{5} - \frac{13969012061773337225175091327504}{3294365921996362157455427922463} a^{3} - \frac{202503777944008737755955457433043}{62592952517930880991653130526797} a \) (order $4$) | 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: | \( 117135822355.96071 \) (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
| \(\Q(\sqrt{-1}) \), 11.11.97393677359695041798001.1 |
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 | $22$ | ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ | $22$ | $22$ | ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{11}$ | $22$ | ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{22}$ | ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}$ | $22$ | ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ | $22$ |
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 | ||||||
| 199 | Data not computed | ||||||