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Properties

Label 22.6.84360700005...0896.1
Degree $22$
Signature $[6, 8]$
Discriminant $2^{24}\cdot 3^{20}\cdot 11^{28}$
Root discriminant $122.34$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $M_{22}$ (as 22T38)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![122880, 219648, -811008, -1545984, 2760384, 4046856, -6508656, -2591160, 5932872, -55044, -2370456, 755828, 258896, -250492, 50820, 20856, -14652, 3993, -330, -132, 55, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 55*x^20 - 132*x^19 - 330*x^18 + 3993*x^17 - 14652*x^16 + 20856*x^15 + 50820*x^14 - 250492*x^13 + 258896*x^12 + 755828*x^11 - 2370456*x^10 - 55044*x^9 + 5932872*x^8 - 2591160*x^7 - 6508656*x^6 + 4046856*x^5 + 2760384*x^4 - 1545984*x^3 - 811008*x^2 + 219648*x + 122880)
 
gp: K = bnfinit(x^22 - 11*x^21 + 55*x^20 - 132*x^19 - 330*x^18 + 3993*x^17 - 14652*x^16 + 20856*x^15 + 50820*x^14 - 250492*x^13 + 258896*x^12 + 755828*x^11 - 2370456*x^10 - 55044*x^9 + 5932872*x^8 - 2591160*x^7 - 6508656*x^6 + 4046856*x^5 + 2760384*x^4 - 1545984*x^3 - 811008*x^2 + 219648*x + 122880, 1)
 

Normalized defining polynomial

\( x^{22} - 11 x^{21} + 55 x^{20} - 132 x^{19} - 330 x^{18} + 3993 x^{17} - 14652 x^{16} + 20856 x^{15} + 50820 x^{14} - 250492 x^{13} + 258896 x^{12} + 755828 x^{11} - 2370456 x^{10} - 55044 x^{9} + 5932872 x^{8} - 2591160 x^{7} - 6508656 x^{6} + 4046856 x^{5} + 2760384 x^{4} - 1545984 x^{3} - 811008 x^{2} + 219648 x + 122880 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8436070000587514853066040247015770903225040896=2^{24}\cdot 3^{20}\cdot 11^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $122.34$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7}$, $\frac{1}{12} a^{15} - \frac{1}{12} a^{14} - \frac{1}{12} a^{12} + \frac{1}{12} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{12} a^{16} - \frac{1}{12} a^{14} - \frac{1}{12} a^{13} - \frac{1}{6} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{24} a^{17} - \frac{1}{24} a^{16} - \frac{1}{24} a^{15} - \frac{1}{12} a^{13} + \frac{1}{24} a^{12} - \frac{1}{6} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{48} a^{18} - \frac{1}{48} a^{17} + \frac{1}{48} a^{16} - \frac{1}{24} a^{15} - \frac{1}{24} a^{14} + \frac{5}{48} a^{13} - \frac{1}{24} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{576} a^{19} + \frac{1}{576} a^{18} - \frac{5}{576} a^{17} - \frac{1}{72} a^{16} - \frac{1}{288} a^{15} + \frac{1}{576} a^{14} + \frac{1}{36} a^{13} - \frac{1}{18} a^{12} + \frac{13}{144} a^{11} - \frac{3}{16} a^{10} + \frac{1}{6} a^{9} - \frac{5}{48} a^{8} + \frac{3}{8} a^{7} - \frac{19}{48} a^{6} - \frac{7}{24} a^{5} + \frac{11}{24} a^{4} + \frac{5}{12} a^{3} + \frac{1}{8} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{2304} a^{20} + \frac{1}{2304} a^{19} - \frac{5}{2304} a^{18} - \frac{1}{288} a^{17} - \frac{1}{1152} a^{16} + \frac{1}{2304} a^{15} + \frac{5}{72} a^{14} + \frac{7}{144} a^{13} + \frac{49}{576} a^{12} + \frac{5}{64} a^{11} + \frac{1}{24} a^{10} - \frac{41}{192} a^{9} + \frac{3}{32} a^{8} - \frac{43}{192} a^{7} + \frac{41}{96} a^{6} - \frac{37}{96} a^{5} - \frac{7}{48} a^{4} - \frac{7}{32} a^{3} + \frac{7}{24} a^{2} - \frac{1}{12} a$, $\frac{1}{8830198109564222618741274995501283864344937630927257918976} a^{21} + \frac{1262195070188571277300634496180201181440870884314258691}{8830198109564222618741274995501283864344937630927257918976} a^{20} + \frac{40491096423957300530549885751072614695466495628736165}{981133123284913624304586110611253762704993070103028657664} a^{19} + \frac{8232161427280579408341336337568486829706070208606414133}{1471699684927370436456879165916880644057489605154542986496} a^{18} + \frac{8295525965243332124448536161674577249428984796677888887}{490566561642456812152293055305626881352496535051514328832} a^{17} - \frac{86047129194336426979207011690682147825472044192813860097}{2943399369854740872913758331833761288114979210309085972992} a^{16} + \frac{4456103597954165216494728877782994390365302111889073529}{490566561642456812152293055305626881352496535051514328832} a^{15} + \frac{3778399148425973364147041485093297805543181695932641301}{45990615153980326139277473934902520126796550161079468328} a^{14} - \frac{27777449277961394267068504380360523040627365651533504511}{245283280821228406076146527652813440676248267525757164416} a^{13} + \frac{74842268821781383404368993470078941462757575421754020607}{2207549527391055654685318748875320966086234407731814479744} a^{12} - \frac{40744825744459989156005155022917824218869188115030515463}{1103774763695527827342659374437660483043117203865907239872} a^{11} - \frac{18625768324197886537105044223868415874642532556987873843}{245283280821228406076146527652813440676248267525757164416} a^{10} - \frac{106587605700350990886691318203704053025638633940602203}{3832551262831693844939789494575210010566379180089955694} a^{9} - \frac{48192706193427917997981176662120141002419289208663251261}{245283280821228406076146527652813440676248267525757164416} a^{8} - \frac{25011137247858471231628167417775673088270925095573432843}{61320820205307101519036631913203360169062066881439291104} a^{7} - \frac{69801044898200342076501707787426983550768704584621661803}{367924921231842609114219791479220161014372401288635746624} a^{6} - \frac{17272607921789266678007774352156742812476090206066098539}{45990615153980326139277473934902520126796550161079468328} a^{5} - \frac{2329557183986482635843341526285498449798871806568509387}{122641640410614203038073263826406720338124133762878582208} a^{4} + \frac{9227117707836614252600383918772503469402551002137269465}{183962460615921304557109895739610080507186200644317873312} a^{3} - \frac{1492698687571947577670138723088395781257217554860943083}{5748826894247540767409684241862815015849568770134933541} a^{2} + \frac{624203455497416454525950994018923461717121585827864379}{22995307576990163069638736967451260063398275080539734164} a + \frac{2754866821501231619383764531670397429635634870242423156}{5748826894247540767409684241862815015849568770134933541}$

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:  $13$
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:  \( 24861330893900000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$M_{22}$ (as 22T38):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 443520
The 12 conjugacy class representatives for $M_{22}$
Character table for $M_{22}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

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 ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
2.12.18.59$x^{12} - 2 x^{11} + 6 x^{10} + 4 x^{9} + 6 x^{8} + 12 x^{7} - 4 x^{6} - 8 x^{3} + 16 x^{2} - 8$$4$$3$$18$$A_4$$[2, 2]^{3}$
$3$3.11.10.1$x^{11} - 3$$11$$1$$10$$C_{11}:C_5$$[\ ]_{11}^{5}$
3.11.10.1$x^{11} - 3$$11$$1$$10$$C_{11}:C_5$$[\ ]_{11}^{5}$
$11$11.11.14.1$x^{11} + 33 x^{4} + 11$$11$$1$$14$$C_{11}:C_5$$[7/5]_{5}$
11.11.14.1$x^{11} + 33 x^{4} + 11$$11$$1$$14$$C_{11}:C_5$$[7/5]_{5}$