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Properties

Label 22.6.96855122250...0000.1
Degree $22$
Signature $[6, 8]$
Discriminant $2^{36}\cdot 3^{18}\cdot 5^{38}$
Root discriminant $123.11$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $M_{11}$ (as 22T22)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31476600, 99773100, -210194100, -211059000, 384246000, 122933160, -170945940, -106934760, 20598000, 47982300, 10865440, -5970310, -3206290, -68625, 139525, -61268, -10388, 2338, -830, 90, 14, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 14*x^20 + 90*x^19 - 830*x^18 + 2338*x^17 - 10388*x^16 - 61268*x^15 + 139525*x^14 - 68625*x^13 - 3206290*x^12 - 5970310*x^11 + 10865440*x^10 + 47982300*x^9 + 20598000*x^8 - 106934760*x^7 - 170945940*x^6 + 122933160*x^5 + 384246000*x^4 - 211059000*x^3 - 210194100*x^2 + 99773100*x + 31476600)
 
gp: K = bnfinit(x^22 - x^21 + 14*x^20 + 90*x^19 - 830*x^18 + 2338*x^17 - 10388*x^16 - 61268*x^15 + 139525*x^14 - 68625*x^13 - 3206290*x^12 - 5970310*x^11 + 10865440*x^10 + 47982300*x^9 + 20598000*x^8 - 106934760*x^7 - 170945940*x^6 + 122933160*x^5 + 384246000*x^4 - 211059000*x^3 - 210194100*x^2 + 99773100*x + 31476600, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 14 x^{20} + 90 x^{19} - 830 x^{18} + 2338 x^{17} - 10388 x^{16} - 61268 x^{15} + 139525 x^{14} - 68625 x^{13} - 3206290 x^{12} - 5970310 x^{11} + 10865440 x^{10} + 47982300 x^{9} + 20598000 x^{8} - 106934760 x^{7} - 170945940 x^{6} + 122933160 x^{5} + 384246000 x^{4} - 211059000 x^{3} - 210194100 x^{2} + 99773100 x + 31476600 \)

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:  \(9685512225000000000000000000000000000000000000=2^{36}\cdot 3^{18}\cdot 5^{38}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $123.11$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7}$, $\frac{1}{6} a^{10} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{4}$, $\frac{1}{12} a^{12} - \frac{1}{6} a^{9} - \frac{1}{12} a^{8} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{13} + \frac{1}{12} a^{9} + \frac{1}{6} a^{8} - \frac{1}{2} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{14} - \frac{1}{12} a^{10} + \frac{1}{6} a^{8} - \frac{1}{6} a^{4}$, $\frac{1}{12} a^{15} - \frac{1}{12} a^{11} + \frac{1}{6} a^{9} - \frac{1}{6} a^{5}$, $\frac{1}{60} a^{16} + \frac{1}{30} a^{15} - \frac{1}{30} a^{11} - \frac{1}{15} a^{10} - \frac{1}{6} a^{9} + \frac{1}{12} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{60} a^{17} + \frac{1}{60} a^{15} - \frac{1}{30} a^{12} - \frac{1}{12} a^{11} - \frac{1}{30} a^{10} + \frac{1}{12} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{120} a^{18} - \frac{1}{120} a^{17} + \frac{1}{60} a^{15} - \frac{1}{24} a^{14} + \frac{1}{40} a^{13} + \frac{1}{60} a^{12} - \frac{1}{12} a^{11} - \frac{1}{30} a^{10} + \frac{1}{12} a^{9} - \frac{1}{12} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{240} a^{19} + \frac{1}{240} a^{17} + \frac{1}{48} a^{15} + \frac{1}{30} a^{14} - \frac{1}{48} a^{13} + \frac{1}{30} a^{12} + \frac{1}{24} a^{11} + \frac{1}{6} a^{9} - \frac{1}{12} a^{8} + \frac{1}{6} a^{7} + \frac{5}{12} a^{6} - \frac{1}{12} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{720} a^{20} - \frac{1}{720} a^{19} - \frac{1}{720} a^{18} - \frac{1}{240} a^{17} + \frac{1}{720} a^{16} + \frac{7}{720} a^{15} - \frac{23}{720} a^{14} + \frac{7}{720} a^{13} - \frac{7}{360} a^{12} + \frac{1}{40} a^{11} - \frac{7}{180} a^{10} - \frac{5}{36} a^{9} + \frac{2}{9} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{12} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{158375787926544186227195085399459301875755194726427826888452598861165558603065742376800} a^{21} + \frac{1251980049866802242392624488894220842381588087810340957656541458797000859805563617}{2083891946401897187199935334203411866786252562189839827479639458699546823724549241800} a^{20} - \frac{30230464542121189286167245142517412055900051191842257337171930734944136365557547727}{15837578792654418622719508539945930187575519472642782688845259886116555860306574237680} a^{19} + \frac{1106364159699180927199097968241939160815911895717677404892620599680867415573493077}{659899116360600775946646189164413757815646644693449278701885828588189827512773926570} a^{18} + \frac{70273071059002248997475501207991322251790942820362160605065746352515526928904490277}{15837578792654418622719508539945930187575519472642782688845259886116555860306574237680} a^{17} + \frac{23720019984869528417604232256806948947256538986934167891076582925211666742032835923}{9898486745409011639199692837466206367234699670401739180528287428822847412691608898550} a^{16} + \frac{1125042354164696500736931817776764188903540582974780069577436844028016529163993322479}{39593946981636046556798771349864825468938798681606956722113149715291389650766435594200} a^{15} + \frac{172199600133662003846381809616420525855932147682423692372594861980111455714705464889}{7918789396327209311359754269972965093787759736321391344422629943058277930153287118840} a^{14} + \frac{1157332886324806632052972947317547512077777622974417067480976453763215574558149708293}{31675157585308837245439017079891860375151038945285565377690519772233111720613148475360} a^{13} - \frac{421078991172490907775616780396792057842808615412117377375253857932248376963722447}{263959646544240310378658475665765503126258657877379711480754331435275931005109570628} a^{12} - \frac{122984416775564352256434180304561240487635691047728703026846277012787538256030548913}{15837578792654418622719508539945930187575519472642782688845259886116555860306574237680} a^{11} - \frac{284816146535097214670614162660700995476386976285030478070294102518324871458441271539}{3959394698163604655679877134986482546893879868160695672211314971529138965076643559420} a^{10} + \frac{115943322165918939760141534294484606197575623665441179988957306717022078025491558909}{791878939632720931135975426997296509378775973632139134442262994305827793015328711884} a^{9} + \frac{88481638710504672127278344750065414896515874179080804269291315835698808354561116337}{527919293088480620757316951331531006252517315754759422961508662870551862010219141256} a^{8} + \frac{135031039324772170930625474470925190898101501044367643243170540474372193486502442041}{527919293088480620757316951331531006252517315754759422961508662870551862010219141256} a^{7} + \frac{293553795897518663056014215588112378580965343041546763728413756969938278382737714133}{879865488480801034595528252219218343754195526257932371602514438117586436683698568760} a^{6} - \frac{8083807642333879489891226392957746641174129884683206366549588574346816721526445924}{329949558180300387973323094582206878907823322346724639350942914294094913756386963285} a^{5} - \frac{43446048007483282058679517623393844621669935470042793296049462418703173666026085897}{87986548848080103459552825221921834375419552625793237160251443811758643668369856876} a^{4} - \frac{36679617265921514825130870141451246391451703232541151005194621683558635268280567299}{87986548848080103459552825221921834375419552625793237160251443811758643668369856876} a^{3} - \frac{374285128221060713973367291887641942560501088217361909617252580320227378115381125}{4888141602671116858864045845662324131967752923655179842236191322875480203798325382} a^{2} + \frac{39348395511148011878822154085443130409746046134944906459880135073908852715236645567}{175973097696160206919105650443843668750839105251586474320502887623517287336739713752} a + \frac{7199294056577085572612276378473916032094370246515069285935634109497638507529767703}{29328849616026701153184275073973944791806517541931079053417147937252881222789952292}$

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:  \( 59783437776300000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$M_{11}$ (as 22T22):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7920
The 10 conjugacy class representatives for $M_{11}$
Character table for $M_{11}$

Intermediate fields

11.3.6561000000000000000000.1

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

Sibling fields

Degree 11 sibling: data not computed
Degree 12 sibling: 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 R ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.4.8.7$x^{4} + 4 x^{2} + 4 x + 2$$4$$1$$8$$S_4$$[8/3, 8/3]_{3}^{2}$
2.4.8.7$x^{4} + 4 x^{2} + 4 x + 2$$4$$1$$8$$S_4$$[8/3, 8/3]_{3}^{2}$
2.12.20.34$x^{12} + 14 x^{10} + 16 x^{8} - 8 x^{6} - 8 x^{4} + 16 x^{2} + 16$$6$$2$$20$$S_4$$[8/3, 8/3]_{3}^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.5.9.4$x^{5} + 30$$5$$1$$9$$F_5$$[9/4]_{4}$
5.5.9.4$x^{5} + 30$$5$$1$$9$$F_5$$[9/4]_{4}$
5.10.19.3$x^{10} + 30$$10$$1$$19$$F_5$$[9/4]_{4}$