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Properties

Label 22.0.25355436660...4304.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{32}\cdot 3^{26}\cdot 11^{33}$
Root discriminant $366.29$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T40

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![301353699564, 824171054652, 819274473963, 417345567804, 245071592040, 97343192496, 12184182186, 5675962688, 1786415928, 660477972, 417479403, 3593532, 19467756, -2157848, -3347223, 99528, -8404, -3432, 6006, 0, -132, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 132*x^20 + 6006*x^18 - 3432*x^17 - 8404*x^16 + 99528*x^15 - 3347223*x^14 - 2157848*x^13 + 19467756*x^12 + 3593532*x^11 + 417479403*x^10 + 660477972*x^9 + 1786415928*x^8 + 5675962688*x^7 + 12184182186*x^6 + 97343192496*x^5 + 245071592040*x^4 + 417345567804*x^3 + 819274473963*x^2 + 824171054652*x + 301353699564)
 
gp: K = bnfinit(x^22 - 132*x^20 + 6006*x^18 - 3432*x^17 - 8404*x^16 + 99528*x^15 - 3347223*x^14 - 2157848*x^13 + 19467756*x^12 + 3593532*x^11 + 417479403*x^10 + 660477972*x^9 + 1786415928*x^8 + 5675962688*x^7 + 12184182186*x^6 + 97343192496*x^5 + 245071592040*x^4 + 417345567804*x^3 + 819274473963*x^2 + 824171054652*x + 301353699564, 1)
 

Normalized defining polynomial

\( x^{22} - 132 x^{20} + 6006 x^{18} - 3432 x^{17} - 8404 x^{16} + 99528 x^{15} - 3347223 x^{14} - 2157848 x^{13} + 19467756 x^{12} + 3593532 x^{11} + 417479403 x^{10} + 660477972 x^{9} + 1786415928 x^{8} + 5675962688 x^{7} + 12184182186 x^{6} + 97343192496 x^{5} + 245071592040 x^{4} + 417345567804 x^{3} + 819274473963 x^{2} + 824171054652 x + 301353699564 \)

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:  \(-253554366603650015745082936335958480508679892151832674304=-\,2^{32}\cdot 3^{26}\cdot 11^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $366.29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{3}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{4}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{5}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{6}$, $\frac{1}{12} a^{13} + \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{36} a^{14} - \frac{1}{18} a^{11} - \frac{1}{9} a^{8} + \frac{1}{3} a^{6} + \frac{1}{18} a^{5} + \frac{1}{3} a^{3} + \frac{1}{12} a^{2}$, $\frac{1}{72} a^{15} - \frac{1}{24} a^{13} - \frac{1}{36} a^{12} - \frac{1}{18} a^{9} - \frac{1}{6} a^{7} - \frac{2}{9} a^{6} + \frac{1}{6} a^{4} + \frac{1}{24} a^{3} - \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{144} a^{16} - \frac{1}{144} a^{15} + \frac{1}{144} a^{14} + \frac{1}{144} a^{13} - \frac{5}{72} a^{12} + \frac{1}{36} a^{11} + \frac{1}{18} a^{10} - \frac{1}{18} a^{9} + \frac{1}{18} a^{8} + \frac{2}{9} a^{7} + \frac{1}{36} a^{6} + \frac{1}{18} a^{5} - \frac{7}{48} a^{4} - \frac{5}{48} a^{3} - \frac{11}{48} a^{2} + \frac{3}{16} a - \frac{3}{8}$, $\frac{1}{432} a^{17} - \frac{1}{432} a^{16} + \frac{1}{432} a^{15} + \frac{1}{432} a^{14} + \frac{7}{216} a^{13} + \frac{1}{108} a^{12} - \frac{1}{27} a^{11} - \frac{2}{27} a^{10} + \frac{1}{54} a^{9} + \frac{2}{27} a^{8} - \frac{5}{108} a^{7} - \frac{17}{54} a^{6} - \frac{47}{144} a^{5} + \frac{1}{48} a^{4} + \frac{37}{144} a^{3} - \frac{13}{48} a^{2} - \frac{11}{24} a$, $\frac{1}{864} a^{18} + \frac{1}{432} a^{15} + \frac{1}{288} a^{14} - \frac{1}{48} a^{13} - \frac{1}{72} a^{12} + \frac{1}{18} a^{11} - \frac{1}{36} a^{10} - \frac{1}{27} a^{9} + \frac{5}{72} a^{8} - \frac{1}{72} a^{7} - \frac{205}{864} a^{6} - \frac{19}{72} a^{5} - \frac{13}{36} a^{4} + \frac{59}{144} a^{3} + \frac{3}{32} a^{2} - \frac{17}{48} a + \frac{1}{4}$, $\frac{1}{2592} a^{19} - \frac{1}{648} a^{16} - \frac{1}{864} a^{15} + \frac{1}{144} a^{13} + \frac{11}{216} a^{12} + \frac{1}{54} a^{11} + \frac{2}{81} a^{10} - \frac{11}{216} a^{9} - \frac{13}{216} a^{8} - \frac{253}{2592} a^{7} + \frac{91}{216} a^{6} - \frac{19}{108} a^{5} - \frac{7}{27} a^{4} - \frac{17}{288} a^{3} - \frac{25}{72} a^{2} + \frac{19}{48} a - \frac{1}{8}$, $\frac{1}{5184} a^{20} - \frac{1}{1728} a^{18} - \frac{1}{1296} a^{17} + \frac{5}{1728} a^{16} + \frac{1}{432} a^{15} - \frac{5}{576} a^{14} + \frac{1}{54} a^{13} + \frac{11}{216} a^{12} + \frac{17}{648} a^{11} + \frac{7}{432} a^{10} - \frac{29}{432} a^{9} - \frac{1}{5184} a^{8} - \frac{1}{216} a^{7} - \frac{547}{1728} a^{6} - \frac{215}{432} a^{5} + \frac{31}{192} a^{4} + \frac{61}{144} a^{3} - \frac{1}{192} a^{2} + \frac{19}{48} a + \frac{1}{16}$, $\frac{1}{15972858357285436038959685657153443895682450385276082892105111142698299734210171643475619614918512682273198464} a^{21} + \frac{848316229916132983007150976594264724769286870183436698382921328436811035349525164799949698767299423065691}{15972858357285436038959685657153443895682450385276082892105111142698299734210171643475619614918512682273198464} a^{20} + \frac{3049256646690828829634421027945358575382212417690693674962534380270530896209081427777391034297244904723509}{15972858357285436038959685657153443895682450385276082892105111142698299734210171643475619614918512682273198464} a^{19} + \frac{2011228029857062098936866209313468184629676568727484237270125208734342669271728615415963470947911829179935}{15972858357285436038959685657153443895682450385276082892105111142698299734210171643475619614918512682273198464} a^{18} - \frac{13188625524888366559331738818412686033978259697300276117515736108740233734937353668311318592728861685032133}{15972858357285436038959685657153443895682450385276082892105111142698299734210171643475619614918512682273198464} a^{17} - \frac{44879086342952062144132788793469454151049476516974413646412064020723511300042652210382066979081179928421151}{15972858357285436038959685657153443895682450385276082892105111142698299734210171643475619614918512682273198464} a^{16} + \frac{2466376929445439410332274849652750634458515453689778309874813139277476835435992600799401286614273223280957}{5324286119095145346319895219051147965227483461758694297368370380899433244736723881158539871639504227424399488} a^{15} - \frac{57232896373212615982305864080583599433498972900679543613049809628716326522285325307528411333718805005159249}{5324286119095145346319895219051147965227483461758694297368370380899433244736723881158539871639504227424399488} a^{14} + \frac{27626087080712116871042404869217848376934040557242959891948178257709189762772009500183614818090321907377965}{665535764886893168289986902381393495653435432719836787171046297612429155592090485144817483954938028428049936} a^{13} + \frac{17663610228073548037958599881079209531164064355296312442488008125366160333970478167295428674691168377039239}{249575911832584938108745088393022560870038287269938795189142361604660933347033931929306556483101760660518726} a^{12} - \frac{105123723231777067377219592968439480362471553826555944718377254264421189209477296424726939751903314335238205}{3993214589321359009739921414288360973920612596319020723026277785674574933552542910868904903729628170568299616} a^{11} + \frac{8344882774771323821879146171132723201703353232864006561958750464577215935510812004701713911890803465499523}{249575911832584938108745088393022560870038287269938795189142361604660933347033931929306556483101760660518726} a^{10} + \frac{295369639079487935247714082224238981748568680582857154639560437472082405261830810743364073892499209383887787}{15972858357285436038959685657153443895682450385276082892105111142698299734210171643475619614918512682273198464} a^{9} + \frac{3430112414825256086125133009685893731009157065581377538213303192780405221465316577613331394428247752098838397}{15972858357285436038959685657153443895682450385276082892105111142698299734210171643475619614918512682273198464} a^{8} - \frac{753716558174283249237112405920562494980717647168176022847896315005370370153730406955089114941046596886912153}{15972858357285436038959685657153443895682450385276082892105111142698299734210171643475619614918512682273198464} a^{7} + \frac{1999650184049978447637212980365560954725066888944314328962437422028367924718567694751548635898512130481730343}{5324286119095145346319895219051147965227483461758694297368370380899433244736723881158539871639504227424399488} a^{6} + \frac{693949993119380957269304606499199610054111961451852297874519591700343937303674318886972226615031133168522329}{1774762039698381782106631739683715988409161153919564765789456793633144414912241293719513290546501409141466496} a^{5} + \frac{1756692726063331558881358463553832343884914602681970566607165351906996321637705321400146918323643700267877193}{5324286119095145346319895219051147965227483461758694297368370380899433244736723881158539871639504227424399488} a^{4} + \frac{243924887388625303071683998797396130122607631642490265317785711085095704422293433783418242490741723996613705}{1774762039698381782106631739683715988409161153919564765789456793633144414912241293719513290546501409141466496} a^{3} + \frac{78305323641249153085376380774586287614785217924058439909729488869059549638349697869013739198256671625719223}{1774762039698381782106631739683715988409161153919564765789456793633144414912241293719513290546501409141466496} a^{2} - \frac{10892782713393703821292397763046712288772613195219910240119322429963088539003883169246318103960817894368737}{36974209160382953793888161243410749758524190706657599287280349867357175310671693619156526886385446023780552} a + \frac{14387855369380761208713865906079056785682533089883562179088030115591692885470951737595436716423287902253927}{49298945547177271725184214991214333011365587608876799049707133156476233747562258158875369181847261365040736}$

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:  $10$
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:  \( 1688335612100000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T40:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 871200
The 44 conjugacy class representatives for t22n40
Character table for t22n40 is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \)

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

Sibling fields

Degree 24 sibling: data not computed
Arithmetically equvalently 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 ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ $22$ R ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ $22$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.11.0.1}{11} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $22$ ${\href{/LocalNumberField/31.11.0.1}{11} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.11.0.1}{11} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.6.5$x^{4} + 2 x^{2} - 4$$2$$2$$6$$D_{4}$$[2, 3]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.8.16.17$x^{8} + 2 x^{6} + 4 x^{5} + 6 x^{4} + 20$$4$$2$$16$$C_2^2:C_4$$[2, 2, 3]^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.6.9.2$x^{6} + 3 x^{4} + 6$$6$$1$$9$$C_6$$[2]_{2}$
11Data not computed