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Properties

Label 22.2.69060143384...0625.1
Degree $22$
Signature $[2, 10]$
Discriminant $5^{16}\cdot 11^{40}$
Root discriminant $252.22$
Ramified primes $5, 11$
Class number $11$ (GRH)
Class group $[11]$ (GRH)
Galois group $C_{11}:D_{11}.C_2$ (as 22T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10858117145, 107908541730, -20879387045, 45202657280, -308868066991, 222554589730, 68368497131, -18265694150, 7709668824, 590825576, 268019961, 24054074, -11311036, 1107876, -321409, -181016, 5489, -1694, -704, 0, 44, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 44*x^20 - 704*x^18 - 1694*x^17 + 5489*x^16 - 181016*x^15 - 321409*x^14 + 1107876*x^13 - 11311036*x^12 + 24054074*x^11 + 268019961*x^10 + 590825576*x^9 + 7709668824*x^8 - 18265694150*x^7 + 68368497131*x^6 + 222554589730*x^5 - 308868066991*x^4 + 45202657280*x^3 - 20879387045*x^2 + 107908541730*x + 10858117145)
 
gp: K = bnfinit(x^22 + 44*x^20 - 704*x^18 - 1694*x^17 + 5489*x^16 - 181016*x^15 - 321409*x^14 + 1107876*x^13 - 11311036*x^12 + 24054074*x^11 + 268019961*x^10 + 590825576*x^9 + 7709668824*x^8 - 18265694150*x^7 + 68368497131*x^6 + 222554589730*x^5 - 308868066991*x^4 + 45202657280*x^3 - 20879387045*x^2 + 107908541730*x + 10858117145, 1)
 

Normalized defining polynomial

\( x^{22} + 44 x^{20} - 704 x^{18} - 1694 x^{17} + 5489 x^{16} - 181016 x^{15} - 321409 x^{14} + 1107876 x^{13} - 11311036 x^{12} + 24054074 x^{11} + 268019961 x^{10} + 590825576 x^{9} + 7709668824 x^{8} - 18265694150 x^{7} + 68368497131 x^{6} + 222554589730 x^{5} - 308868066991 x^{4} + 45202657280 x^{3} - 20879387045 x^{2} + 107908541730 x + 10858117145 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(69060143384057543649123162895045349831909332275390625=5^{16}\cdot 11^{40}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $252.22$
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:  $5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{12} + \frac{2}{5} a^{10} - \frac{1}{2} a^{9} + \frac{1}{10} a^{8} - \frac{1}{10} a^{6} + \frac{1}{10} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{11} - \frac{1}{2} a^{10} + \frac{1}{10} a^{9} - \frac{1}{2} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{50} a^{14} + \frac{1}{50} a^{13} - \frac{1}{25} a^{12} - \frac{3}{25} a^{11} + \frac{1}{25} a^{10} + \frac{11}{50} a^{9} - \frac{1}{25} a^{8} - \frac{8}{25} a^{7} + \frac{17}{50} a^{6} - \frac{19}{50} a^{5} + \frac{12}{25} a^{4} - \frac{1}{10} a^{3} - \frac{2}{5} a^{2} - \frac{3}{10} a - \frac{1}{10}$, $\frac{1}{150} a^{15} - \frac{1}{150} a^{14} - \frac{2}{75} a^{13} - \frac{7}{150} a^{12} - \frac{6}{25} a^{11} - \frac{21}{50} a^{10} - \frac{49}{150} a^{9} - \frac{17}{150} a^{8} - \frac{1}{150} a^{7} + \frac{26}{75} a^{6} - \frac{19}{75} a^{5} - \frac{29}{75} a^{4} + \frac{1}{10} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{15}$, $\frac{1}{150} a^{16} + \frac{1}{150} a^{14} - \frac{1}{30} a^{13} + \frac{1}{30} a^{12} + \frac{1}{10} a^{11} - \frac{1}{15} a^{10} + \frac{1}{5} a^{8} - \frac{3}{10} a^{7} + \frac{28}{75} a^{6} - \frac{2}{5} a^{5} + \frac{11}{150} a^{4} + \frac{7}{30} a^{3} - \frac{2}{15} a^{2} - \frac{1}{5} a - \frac{2}{15}$, $\frac{1}{750} a^{17} - \frac{1}{750} a^{15} + \frac{31}{750} a^{13} - \frac{7}{750} a^{12} - \frac{23}{375} a^{11} - \frac{121}{250} a^{10} + \frac{13}{375} a^{9} - \frac{347}{750} a^{8} + \frac{59}{150} a^{7} - \frac{233}{750} a^{6} + \frac{13}{50} a^{5} - \frac{16}{375} a^{4} + \frac{1}{75} a^{3} + \frac{67}{150} a^{2} + \frac{19}{50} a - \frac{67}{150}$, $\frac{1}{5250} a^{18} + \frac{1}{1750} a^{17} + \frac{1}{375} a^{16} + \frac{2}{875} a^{15} - \frac{1}{375} a^{14} - \frac{47}{2625} a^{13} - \frac{1}{750} a^{12} - \frac{13}{125} a^{11} - \frac{374}{2625} a^{10} - \frac{107}{750} a^{9} - \frac{461}{5250} a^{8} - \frac{409}{2625} a^{7} - \frac{383}{1750} a^{6} - \frac{31}{2625} a^{5} + \frac{379}{5250} a^{4} - \frac{13}{75} a^{3} - \frac{62}{175} a^{2} - \frac{43}{150} a - \frac{81}{175}$, $\frac{1}{5250} a^{19} - \frac{1}{2625} a^{17} + \frac{1}{1050} a^{16} - \frac{4}{2625} a^{15} - \frac{26}{2625} a^{14} - \frac{257}{5250} a^{13} - \frac{1}{250} a^{12} + \frac{159}{1750} a^{11} - \frac{1669}{5250} a^{10} + \frac{419}{875} a^{9} - \frac{638}{2625} a^{8} - \frac{79}{175} a^{7} - \frac{743}{1750} a^{6} + \frac{34}{175} a^{5} + \frac{769}{1750} a^{4} - \frac{1}{70} a^{3} + \frac{31}{1050} a^{2} - \frac{367}{1050} a - \frac{81}{350}$, $\frac{1}{78750} a^{20} - \frac{1}{78750} a^{19} - \frac{2}{39375} a^{18} + \frac{4}{39375} a^{17} - \frac{37}{26250} a^{16} + \frac{13}{15750} a^{15} - \frac{82}{13125} a^{14} - \frac{1937}{39375} a^{13} - \frac{73}{8750} a^{12} - \frac{11}{78750} a^{11} - \frac{10471}{39375} a^{10} - \frac{17}{350} a^{9} - \frac{4748}{39375} a^{8} - \frac{5381}{26250} a^{7} - \frac{20759}{78750} a^{6} + \frac{4721}{13125} a^{5} + \frac{10988}{39375} a^{4} + \frac{113}{875} a^{3} - \frac{391}{1575} a^{2} + \frac{967}{3150} a - \frac{1132}{7875}$, $\frac{1}{2046712346328751613666939593964330164010088860205171893226784226498715445813267929790044197698833935993154020287500} a^{21} + \frac{12789607625136925359959771423052536886628024240888851227819562992972230715568038459430727725564117809977291357}{2046712346328751613666939593964330164010088860205171893226784226498715445813267929790044197698833935993154020287500} a^{20} - \frac{3460278066109807161630402407750183849921619244828434680820859516512995252577395516891004156313780085487623029}{682237448776250537888979864654776721336696286735057297742261408832905148604422643263348065899611311997718006762500} a^{19} + \frac{8203243166619695065409811046681514133612385426285780211034915118700782469161220843638390038233543659881393257}{682237448776250537888979864654776721336696286735057297742261408832905148604422643263348065899611311997718006762500} a^{18} - \frac{11440373464062277906993649762584844155251657704070655792665354757842807762663739167004120050155262480715104737}{2046712346328751613666939593964330164010088860205171893226784226498715445813267929790044197698833935993154020287500} a^{17} - \frac{2516018840389293347805904277616916947155522484852288005458052494548967094775345297353675463493009520838431522593}{2046712346328751613666939593964330164010088860205171893226784226498715445813267929790044197698833935993154020287500} a^{16} - \frac{276381875738752066194921308645317462417757493295150772312056929104410647609917174291834116212668259840203902808}{511678086582187903416734898491082541002522215051292973306696056624678861453316982447511049424708483998288505071875} a^{15} + \frac{1489108771292569253551047702504974933615066216973216857881074202508776519103220238133886427522435311082848865361}{204671234632875161366693959396433016401008886020517189322678422649871544581326792979004419769883393599315402028750} a^{14} + \frac{95762320944616680838379943081034235629539838740602252828260432800108317600821384501557305825042335795049885037871}{2046712346328751613666939593964330164010088860205171893226784226498715445813267929790044197698833935993154020287500} a^{13} - \frac{1208369472930054529045953021054932850029424538915985164294532009107004186338288448705006042804771448111501627101}{292387478046964516238134227709190023430012694315024556175254889499816492259038275684292028242690562284736288612500} a^{12} - \frac{206095354540518970299796988650635498206312108207121424666393285193366881006749859461365082571216074285483417719}{3898499707292860216508456369455866979066835924200327415670065193330886563453843675790560376569207497129817181500} a^{11} - \frac{731843326465890617922638443877209077183130514090280405640924066445466244219017711651159584076916299680911289073421}{2046712346328751613666939593964330164010088860205171893226784226498715445813267929790044197698833935993154020287500} a^{10} + \frac{171525012656987362524052522938386597636387576808270756200980850719387138572543357094144067245238463505237199140406}{511678086582187903416734898491082541002522215051292973306696056624678861453316982447511049424708483998288505071875} a^{9} + \frac{149293331984330639757584105960547906756323984785336685863326476288010393005443482443738808819217023852427964578697}{1023356173164375806833469796982165082005044430102585946613392113249357722906633964895022098849416967996577010143750} a^{8} - \frac{91267505565365734684995788273638927238621941619805460584703692491805594417385966554587582147984481235093344846769}{1023356173164375806833469796982165082005044430102585946613392113249357722906633964895022098849416967996577010143750} a^{7} + \frac{181092929209661997425186991411023263793228026373865272346727945134741189697809510211712741292624602954217644928506}{511678086582187903416734898491082541002522215051292973306696056624678861453316982447511049424708483998288505071875} a^{6} - \frac{342528176873786482192020257463484677072074519126366080835041698901228679866608723688116833297201666867535246274631}{2046712346328751613666939593964330164010088860205171893226784226498715445813267929790044197698833935993154020287500} a^{5} - \frac{96582358447453360385125794906224850649631891577953793782915577669144218913881115434269368654317905547286476627167}{2046712346328751613666939593964330164010088860205171893226784226498715445813267929790044197698833935993154020287500} a^{4} - \frac{30016798399836449483245360752043281071513174168561742524728538577389326616654184316436078405664030366486494190307}{102335617316437580683346979698216508200504443010258594661339211324935772290663396489502209884941696799657701014375} a^{3} + \frac{3294197222410684497462615164623587533024766210584288396340317683429653099950621279734836130924711908446566867453}{13644748975525010757779597293095534426733925734701145954845228176658102972088452865266961317992226239954360135250} a^{2} + \frac{202519842416750954508786852474233147202145419341107738108071351902796460131299509365168507646111640791438982653271}{409342469265750322733387918792866032802017772041034378645356845299743089162653585958008839539766787198630804057500} a + \frac{168183046632592497628672138482153058010456977684075326677068170015375404541037350659731165699264811738120198178443}{409342469265750322733387918792866032802017772041034378645356845299743089162653585958008839539766787198630804057500}$

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

Class group and class number

$C_{11}$, which has order $11$ (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:  $11$
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:  \( 345685913257000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{11}:D_{11}.C_2$ (as 22T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 484
The 34 conjugacy class representatives for $C_{11}:D_{11}.C_2$
Character table for $C_{11}:D_{11}.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 22 siblings: data not computed
Degree 44 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. 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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$11$11.11.20.10$x^{11} - 11 x^{10} + 132$$11$$1$$20$$C_{11}$$[2]$
11.11.20.8$x^{11} - 11 x^{10} + 1221$$11$$1$$20$$C_{11}$$[2]$