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Properties

Label 22.2.33628005738...9888.1
Degree $22$
Signature $[2, 10]$
Discriminant $2^{22}\cdot 3^{11}\cdot 11^{40}$
Root discriminant $271.04$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{11}^2$ (as 22T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1476749756992, -7712130115008, 11516770365584, -9782751374528, 6196361225120, -3093851273920, 1171364754340, -343375574872, 84322007704, -18050078772, 3437427653, -659105044, 119139548, -15102780, 2442352, -585508, 75790, -12892, 1232, -44, 44, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 44*x^20 - 44*x^19 + 1232*x^18 - 12892*x^17 + 75790*x^16 - 585508*x^15 + 2442352*x^14 - 15102780*x^13 + 119139548*x^12 - 659105044*x^11 + 3437427653*x^10 - 18050078772*x^9 + 84322007704*x^8 - 343375574872*x^7 + 1171364754340*x^6 - 3093851273920*x^5 + 6196361225120*x^4 - 9782751374528*x^3 + 11516770365584*x^2 - 7712130115008*x + 1476749756992)
 
gp: K = bnfinit(x^22 + 44*x^20 - 44*x^19 + 1232*x^18 - 12892*x^17 + 75790*x^16 - 585508*x^15 + 2442352*x^14 - 15102780*x^13 + 119139548*x^12 - 659105044*x^11 + 3437427653*x^10 - 18050078772*x^9 + 84322007704*x^8 - 343375574872*x^7 + 1171364754340*x^6 - 3093851273920*x^5 + 6196361225120*x^4 - 9782751374528*x^3 + 11516770365584*x^2 - 7712130115008*x + 1476749756992, 1)
 

Normalized defining polynomial

\( x^{22} + 44 x^{20} - 44 x^{19} + 1232 x^{18} - 12892 x^{17} + 75790 x^{16} - 585508 x^{15} + 2442352 x^{14} - 15102780 x^{13} + 119139548 x^{12} - 659105044 x^{11} + 3437427653 x^{10} - 18050078772 x^{9} + 84322007704 x^{8} - 343375574872 x^{7} + 1171364754340 x^{6} - 3093851273920 x^{5} + 6196361225120 x^{4} - 9782751374528 x^{3} + 11516770365584 x^{2} - 7712130115008 x + 1476749756992 \)

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:  \(336280057382622056553974616702903009453525746621349888=2^{22}\cdot 3^{11}\cdot 11^{40}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $271.04$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{484} a^{11} + \frac{1}{22} a^{9} - \frac{1}{22} a^{8} - \frac{5}{22} a^{7} - \frac{7}{22} a^{6} - \frac{9}{44} a^{5} - \frac{4}{11} a^{4} + \frac{1}{11} a^{3} - \frac{1}{22} a^{2} - \frac{5}{11} a - \frac{17}{121}$, $\frac{1}{2904} a^{12} - \frac{31}{264} a^{10} + \frac{3}{88} a^{9} - \frac{7}{88} a^{8} + \frac{21}{88} a^{7} + \frac{23}{132} a^{6} - \frac{9}{88} a^{5} + \frac{2}{11} a^{4} - \frac{15}{44} a^{3} + \frac{17}{66} a^{2} + \frac{75}{242} a - \frac{1}{3}$, $\frac{1}{2904} a^{13} + \frac{1}{2904} a^{11} + \frac{3}{88} a^{10} + \frac{1}{88} a^{9} + \frac{13}{88} a^{8} + \frac{29}{132} a^{7} - \frac{21}{88} a^{6} - \frac{21}{44} a^{5} - \frac{3}{44} a^{4} - \frac{2}{33} a^{3} + \frac{53}{242} a^{2} - \frac{8}{33} a - \frac{1}{121}$, $\frac{1}{8712} a^{14} + \frac{1}{8712} a^{13} - \frac{1}{8712} a^{12} - \frac{1}{4356} a^{11} - \frac{29}{396} a^{10} - \frac{2}{33} a^{9} - \frac{53}{792} a^{8} - \frac{35}{792} a^{7} - \frac{305}{792} a^{6} - \frac{4}{33} a^{5} + \frac{91}{396} a^{4} + \frac{107}{1089} a^{3} + \frac{85}{1089} a^{2} + \frac{311}{1089} a + \frac{380}{1089}$, $\frac{1}{8712} a^{15} + \frac{1}{8712} a^{13} - \frac{1}{8712} a^{12} - \frac{1}{2904} a^{11} + \frac{37}{792} a^{10} + \frac{19}{198} a^{9} + \frac{7}{88} a^{8} - \frac{5}{66} a^{7} - \frac{1}{9} a^{6} - \frac{113}{396} a^{5} + \frac{53}{726} a^{4} - \frac{79}{198} a^{3} + \frac{731}{2178} a^{2} - \frac{32}{363} a - \frac{299}{1089}$, $\frac{1}{17424} a^{16} - \frac{1}{17424} a^{14} - \frac{1}{17424} a^{12} - \frac{59}{528} a^{10} + \frac{1}{12} a^{9} + \frac{199}{1584} a^{8} - \frac{5}{132} a^{7} + \frac{17}{528} a^{6} + \frac{31}{1452} a^{5} - \frac{223}{792} a^{4} + \frac{59}{242} a^{3} - \frac{95}{396} a^{2} + \frac{133}{726} a + \frac{26}{99}$, $\frac{1}{52272} a^{17} - \frac{1}{17424} a^{15} + \frac{1}{26136} a^{14} - \frac{1}{52272} a^{13} - \frac{1}{52272} a^{11} - \frac{29}{2376} a^{10} - \frac{11}{432} a^{9} + \frac{13}{108} a^{8} + \frac{461}{4752} a^{7} + \frac{571}{26136} a^{6} - \frac{175}{792} a^{5} + \frac{1151}{13068} a^{4} + \frac{725}{13068} a^{3} - \frac{83}{242} a^{2} + \frac{1}{33} a + \frac{1129}{3267}$, $\frac{1}{104544} a^{18} - \frac{1}{34848} a^{16} + \frac{1}{52272} a^{15} - \frac{1}{104544} a^{14} - \frac{1}{104544} a^{12} - \frac{49}{52272} a^{11} + \frac{97}{864} a^{10} + \frac{29}{594} a^{9} + \frac{569}{9504} a^{8} - \frac{9527}{52272} a^{7} + \frac{347}{1584} a^{6} - \frac{2413}{26136} a^{5} + \frac{3101}{26136} a^{4} + \frac{37}{121} a^{3} + \frac{5}{33} a^{2} + \frac{119}{3267} a + \frac{18}{121}$, $\frac{1}{209088} a^{19} - \frac{1}{209088} a^{17} + \frac{1}{104544} a^{16} + \frac{5}{209088} a^{15} + \frac{1}{52272} a^{14} + \frac{1}{23232} a^{13} - \frac{1}{104544} a^{12} - \frac{181}{209088} a^{11} + \frac{61}{528} a^{10} + \frac{67}{576} a^{9} - \frac{11551}{104544} a^{8} - \frac{13}{594} a^{7} + \frac{743}{4356} a^{6} - \frac{24685}{52272} a^{5} + \frac{161}{26136} a^{4} + \frac{5131}{13068} a^{3} + \frac{493}{3267} a^{2} + \frac{5}{484} a - \frac{659}{6534}$, $\frac{1}{627264} a^{20} - \frac{1}{627264} a^{19} - \frac{1}{627264} a^{18} + \frac{1}{209088} a^{17} + \frac{5}{209088} a^{16} + \frac{23}{627264} a^{15} + \frac{17}{627264} a^{14} + \frac{37}{627264} a^{13} + \frac{49}{627264} a^{12} - \frac{37}{57024} a^{11} + \frac{905}{19008} a^{10} - \frac{65375}{627264} a^{9} + \frac{26093}{313632} a^{8} + \frac{5063}{78408} a^{7} + \frac{6227}{39204} a^{6} - \frac{21265}{156816} a^{5} - \frac{50}{1089} a^{4} + \frac{2857}{13068} a^{3} + \frac{7543}{19602} a^{2} + \frac{5465}{39204} a + \frac{661}{1782}$, $\frac{1}{323076652590217562985101568419059062461584628975199897500906869451231399862638883167467642368} a^{21} - \frac{254386711861246915588774966984749632113705543951286497710962719821485911720623168627287}{323076652590217562985101568419059062461584628975199897500906869451231399862638883167467642368} a^{20} + \frac{220333592388513020820225903135586337988993573891000092744278106989778418492310260617383}{107692217530072520995033856139686354153861542991733299166968956483743799954212961055822547456} a^{19} - \frac{440941402690976556804317655112087975573308793041123564562639750729839572812803068694935}{323076652590217562985101568419059062461584628975199897500906869451231399862638883167467642368} a^{18} - \frac{611022350726278589454417922363555114942337865098601743129634771702382947876657095986061}{107692217530072520995033856139686354153861542991733299166968956483743799954212961055822547456} a^{17} + \frac{6624642872693201251093630162486114500577646561367786538231772578507746700268900050900349}{323076652590217562985101568419059062461584628975199897500906869451231399862638883167467642368} a^{16} + \frac{4784118699666955090719250970758102006462065604449524770355296081088472106236512430012857}{107692217530072520995033856139686354153861542991733299166968956483743799954212961055822547456} a^{15} + \frac{5527757006826984635339177491603365019311391978001863368005634843039786522530300254370999}{323076652590217562985101568419059062461584628975199897500906869451231399862638883167467642368} a^{14} - \frac{606792400965678301239859583333424702242749862006986444511310509687963259574258596275}{3263400531214318818031328973929889519813986151264645429302089590416478786491301850176440832} a^{13} + \frac{73554150669785347850365501919624784561578700389461494551473202646167990891373254408201}{1329533549753981740679430322712177211776068431996707397123073536836343209311271124145957376} a^{12} - \frac{73363955753806039436543164657235946570213176665799772317111097799955297420828448242178361}{323076652590217562985101568419059062461584628975199897500906869451231399862638883167467642368} a^{11} - \frac{21620912167632585137488315199631978173017779415850607140366432961886758042750766071407296685}{323076652590217562985101568419059062461584628975199897500906869451231399862638883167467642368} a^{10} + \frac{165863107295845793963886098950380378618707612725308924986034842990712889658443664260967317}{1495725243473229458264359113051199363248076985996295821763457728940886110475180014664202048} a^{9} - \frac{6181428493587469138026479438690439995177331252275200737904142256266693646228011125754378191}{26923054382518130248758464034921588538465385747933324791742239120935949988553240263955636864} a^{8} - \frac{4768958958501082429518634864222054065449386986515238737185172991345474518199045082356510119}{80769163147554390746275392104764765615396157243799974375226717362807849965659720791866910592} a^{7} - \frac{4204872618387898543565212238490018829757986318383659453071530423828736779512975392108521077}{8974351460839376749586154678307196179488461915977774930580746373645316662851080087985212288} a^{6} + \frac{2915497590926076322535496423584299885258716625669515586564781794428955071644587272229553131}{20192290786888597686568848026191191403849039310949993593806679340701962491414930197966727648} a^{5} + \frac{554836536433692795119427923291067098754267317414031876679126245077226425364418708921302007}{6730763595629532562189616008730397134616346436983331197935559780233987497138310065988909216} a^{4} + \frac{3902155911288929216916513197146259514760035740149767655117704896014867037674065434097147203}{20192290786888597686568848026191191403849039310949993593806679340701962491414930197966727648} a^{3} - \frac{83711954872429907095811616973943052781338882332636639701924463418608863500393321403050481}{611887599602684778380874182611854284965122403362121017994141798203089772467119096908082656} a^{2} - \frac{9519962517370397730955590520399203497219955360112762731329718067754193311865309514182787}{1262018174180537355410553001636949462740564956934374599612917458793872655713433137372920478} a - \frac{1049671614747951419960231024214141298158798259597808076508371359099850304283246032723793293}{5048072696722149421642212006547797850962259827737498398451669835175490622853732549491681912}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 13731470738200000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{11}^2$ (as 22T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 484
The 49 conjugacy class representatives for $D_{11}^2$
Character table for $D_{11}^2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \)

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 R R $22$ $22$ R ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{11}$ $22$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.11.20.16$x^{11} + 11 x^{10} + 11$$11$$1$$20$$D_{11}$$[2]^{2}$
11.11.20.16$x^{11} + 11 x^{10} + 11$$11$$1$$20$$D_{11}$$[2]^{2}$