↑

Properties

Label 22.0.18876201495...4199.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,199^{21}$
Root discriminant $156.44$
Ramified prime $199$
Class number $6543$ (GRH)
Class group $[6543]$ (GRH)
Galois group $C_{22}$ (as 22T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![181383703, 621199012, 836966655, 630276336, 297595974, 47956306, -17889931, -16592675, 3820034, 5135296, 1689686, 271590, 200007, 67013, 95687, 6372, 1852, 1411, 89, 101, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 5*x^20 + 101*x^19 + 89*x^18 + 1411*x^17 + 1852*x^16 + 6372*x^15 + 95687*x^14 + 67013*x^13 + 200007*x^12 + 271590*x^11 + 1689686*x^10 + 5135296*x^9 + 3820034*x^8 - 16592675*x^7 - 17889931*x^6 + 47956306*x^5 + 297595974*x^4 + 630276336*x^3 + 836966655*x^2 + 621199012*x + 181383703)
 
gp: K = bnfinit(x^22 - x^21 + 5*x^20 + 101*x^19 + 89*x^18 + 1411*x^17 + 1852*x^16 + 6372*x^15 + 95687*x^14 + 67013*x^13 + 200007*x^12 + 271590*x^11 + 1689686*x^10 + 5135296*x^9 + 3820034*x^8 - 16592675*x^7 - 17889931*x^6 + 47956306*x^5 + 297595974*x^4 + 630276336*x^3 + 836966655*x^2 + 621199012*x + 181383703, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 5 x^{20} + 101 x^{19} + 89 x^{18} + 1411 x^{17} + 1852 x^{16} + 6372 x^{15} + 95687 x^{14} + 67013 x^{13} + 200007 x^{12} + 271590 x^{11} + 1689686 x^{10} + 5135296 x^{9} + 3820034 x^{8} - 16592675 x^{7} - 17889931 x^{6} + 47956306 x^{5} + 297595974 x^{4} + 630276336 x^{3} + 836966655 x^{2} + 621199012 x + 181383703 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

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:  \(-1887620149539230539058375534310517606114631604199=-\,199^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $156.44$
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:  $199$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(199\)
Dirichlet character group:    $\lbrace$$\chi_{199}(1,·)$, $\chi_{199}(11,·)$, $\chi_{199}(198,·)$, $\chi_{199}(136,·)$, $\chi_{199}(137,·)$, $\chi_{199}(138,·)$, $\chi_{199}(139,·)$, $\chi_{199}(78,·)$, $\chi_{199}(18,·)$, $\chi_{199}(85,·)$, $\chi_{199}(96,·)$, $\chi_{199}(103,·)$, $\chi_{199}(60,·)$, $\chi_{199}(125,·)$, $\chi_{199}(114,·)$, $\chi_{199}(74,·)$, $\chi_{199}(181,·)$, $\chi_{199}(121,·)$, $\chi_{199}(188,·)$, $\chi_{199}(61,·)$, $\chi_{199}(62,·)$, $\chi_{199}(63,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1591} a^{19} - \frac{673}{1591} a^{18} + \frac{287}{1591} a^{17} + \frac{626}{1591} a^{16} + \frac{724}{1591} a^{15} + \frac{39}{1591} a^{14} - \frac{165}{1591} a^{13} - \frac{193}{1591} a^{12} + \frac{53}{1591} a^{11} - \frac{171}{1591} a^{10} + \frac{445}{1591} a^{9} + \frac{743}{1591} a^{8} - \frac{219}{1591} a^{7} + \frac{440}{1591} a^{6} + \frac{173}{1591} a^{5} - \frac{47}{1591} a^{4} + \frac{355}{1591} a^{3} + \frac{382}{1591} a^{2} - \frac{154}{1591} a + \frac{235}{1591}$, $\frac{1}{106512999973} a^{20} + \frac{22318269}{106512999973} a^{19} + \frac{33132313307}{106512999973} a^{18} - \frac{22014274871}{106512999973} a^{17} - \frac{6908323294}{106512999973} a^{16} + \frac{1991682719}{106512999973} a^{15} + \frac{39282419327}{106512999973} a^{14} - \frac{32154011330}{106512999973} a^{13} - \frac{1331989594}{106512999973} a^{12} - \frac{3934366371}{106512999973} a^{11} + \frac{48912338815}{106512999973} a^{10} + \frac{39840693453}{106512999973} a^{9} - \frac{21519140725}{106512999973} a^{8} + \frac{4923656631}{106512999973} a^{7} + \frac{33089280822}{106512999973} a^{6} + \frac{41341856277}{106512999973} a^{5} - \frac{42863931935}{106512999973} a^{4} - \frac{51439502170}{106512999973} a^{3} - \frac{39609210636}{106512999973} a^{2} - \frac{51948261240}{106512999973} a - \frac{32317086955}{106512999973}$, $\frac{1}{85647296399425251347836686663766794912272638811144394222397011234176198152097455051} a^{21} + \frac{162822000456843876462857518744941368998981495046966398584370468462250311}{85647296399425251347836686663766794912272638811144394222397011234176198152097455051} a^{20} + \frac{10622242638220265868601881430097391503920079222353380776141306411842584618371029}{85647296399425251347836686663766794912272638811144394222397011234176198152097455051} a^{19} + \frac{15212152248546878256568448260798908432709693193151671887276608209457050958544378515}{85647296399425251347836686663766794912272638811144394222397011234176198152097455051} a^{18} - \frac{33329257651247435335305488358405617929756888034387671359684262065507339415145529774}{85647296399425251347836686663766794912272638811144394222397011234176198152097455051} a^{17} - \frac{3300068633217283332391517068039253606558409059361352899685841265858777728635227196}{85647296399425251347836686663766794912272638811144394222397011234176198152097455051} a^{16} - \frac{20883274950982910383899485447211685206513616205298230389521076756581586787111399853}{85647296399425251347836686663766794912272638811144394222397011234176198152097455051} a^{15} + \frac{15963245862354564131038634431345851287790765264012221014990334945860020931011938536}{85647296399425251347836686663766794912272638811144394222397011234176198152097455051} a^{14} - \frac{34016666149278401142036621547774644175349648540306735061019109703413991878968337057}{85647296399425251347836686663766794912272638811144394222397011234176198152097455051} a^{13} - \frac{38091823515919176469284388950737026723142043321789879799566909325098995403969009311}{85647296399425251347836686663766794912272638811144394222397011234176198152097455051} a^{12} + \frac{30756322694609618178014469144632350276005667093890435616146609890649167902081157921}{85647296399425251347836686663766794912272638811144394222397011234176198152097455051} a^{11} + \frac{716110559845565955475763329139692354669947680169877631217360479039726869373954433}{1991797590684308170879922945668995230517968344445218470288302586841306933769708257} a^{10} - \frac{21976654052125860312317704440717507795365786623133987958826134004948759691241404579}{85647296399425251347836686663766794912272638811144394222397011234176198152097455051} a^{9} - \frac{9489333552525689841848900961542194088182932014356016865524132321902270300113836196}{85647296399425251347836686663766794912272638811144394222397011234176198152097455051} a^{8} - \frac{42721987603542758922228275344994865407946945589593894837970630187892968767584180681}{85647296399425251347836686663766794912272638811144394222397011234176198152097455051} a^{7} + \frac{26368036566331410590069846272568904072181142416572994025993318741064946645578347590}{85647296399425251347836686663766794912272638811144394222397011234176198152097455051} a^{6} - \frac{119781923392593889767672562358496036113055906570310154807693551555364955174013012}{85647296399425251347836686663766794912272638811144394222397011234176198152097455051} a^{5} + \frac{6961266430839144611236339027946144561845446435892301984513107875786051759221684850}{85647296399425251347836686663766794912272638811144394222397011234176198152097455051} a^{4} + \frac{26710107214742725467697773952622997416969163832284211965441603486886766517546234042}{85647296399425251347836686663766794912272638811144394222397011234176198152097455051} a^{3} - \frac{34925445936866825047003062379759451253556931060272518355654178755972994632190790354}{85647296399425251347836686663766794912272638811144394222397011234176198152097455051} a^{2} + \frac{40011786519054234244920780273302755114941975789171084771861941952599833907144460528}{85647296399425251347836686663766794912272638811144394222397011234176198152097455051} a - \frac{12398401746011092137523221577484351686676759205375820204331404984658199965055232742}{85647296399425251347836686663766794912272638811144394222397011234176198152097455051}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{6543}$, which has order $6543$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.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:  \( 117135822355.96071 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{22}$ (as 22T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 22
The 22 conjugacy class representatives for $C_{22}$
Character table for $C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{-199}) \), 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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{11}$ $22$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{22}$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ $22$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
199Data not computed