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Properties

Label 22.0.88508087973...3408.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{33}\cdot 3^{10}\cdot 11^{30}$
Root discriminant $122.60$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T12

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11520000, -12672000, 11721600, -25597440, 35863344, -19434096, 28619052, -24020040, 12429780, -6176808, 5575317, -2114704, 416306, -327360, 175219, -32296, 8096, -5544, 1419, 0, -22, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 22*x^20 + 1419*x^18 - 5544*x^17 + 8096*x^16 - 32296*x^15 + 175219*x^14 - 327360*x^13 + 416306*x^12 - 2114704*x^11 + 5575317*x^10 - 6176808*x^9 + 12429780*x^8 - 24020040*x^7 + 28619052*x^6 - 19434096*x^5 + 35863344*x^4 - 25597440*x^3 + 11721600*x^2 - 12672000*x + 11520000)
 
gp: K = bnfinit(x^22 - 22*x^20 + 1419*x^18 - 5544*x^17 + 8096*x^16 - 32296*x^15 + 175219*x^14 - 327360*x^13 + 416306*x^12 - 2114704*x^11 + 5575317*x^10 - 6176808*x^9 + 12429780*x^8 - 24020040*x^7 + 28619052*x^6 - 19434096*x^5 + 35863344*x^4 - 25597440*x^3 + 11721600*x^2 - 12672000*x + 11520000, 1)
 

Normalized defining polynomial

\( x^{22} - 22 x^{20} + 1419 x^{18} - 5544 x^{17} + 8096 x^{16} - 32296 x^{15} + 175219 x^{14} - 327360 x^{13} + 416306 x^{12} - 2114704 x^{11} + 5575317 x^{10} - 6176808 x^{9} + 12429780 x^{8} - 24020040 x^{7} + 28619052 x^{6} - 19434096 x^{5} + 35863344 x^{4} - 25597440 x^{3} + 11721600 x^{2} - 12672000 x + 11520000 \)

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:  \(-8850808797378409798254793906469559840075153408=-\,2^{33}\cdot 3^{10}\cdot 11^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $122.60$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{40} a^{11} - \frac{1}{20} a^{9} + \frac{1}{20} a^{8} + \frac{3}{40} a^{7} - \frac{1}{20} a^{5} - \frac{1}{4} a^{4} + \frac{1}{10} a^{3} - \frac{1}{10} a^{2} - \frac{2}{5} a$, $\frac{1}{80} a^{12} + \frac{3}{80} a^{10} + \frac{1}{40} a^{9} + \frac{3}{80} a^{8} - \frac{7}{80} a^{6} - \frac{1}{8} a^{5} + \frac{1}{20} a^{4} + \frac{1}{5} a^{3} - \frac{9}{20} a^{2} - \frac{1}{2} a$, $\frac{1}{80} a^{13} - \frac{1}{80} a^{11} + \frac{1}{40} a^{10} + \frac{1}{80} a^{9} + \frac{1}{40} a^{8} + \frac{1}{80} a^{7} - \frac{1}{8} a^{6} + \frac{1}{40} a^{5} - \frac{7}{40} a^{4} + \frac{7}{20} a^{3} - \frac{1}{20} a^{2} + \frac{3}{10} a$, $\frac{1}{160} a^{14} - \frac{1}{160} a^{13} - \frac{1}{160} a^{12} - \frac{1}{160} a^{11} + \frac{9}{160} a^{10} + \frac{9}{160} a^{9} + \frac{1}{160} a^{8} - \frac{3}{160} a^{7} - \frac{9}{80} a^{6} + \frac{1}{5} a^{5} - \frac{1}{20} a^{4} + \frac{3}{40} a^{3} - \frac{1}{10} a^{2} - \frac{1}{2} a$, $\frac{1}{160} a^{15} - \frac{1}{80} a^{11} + \frac{1}{20} a^{10} - \frac{1}{20} a^{9} - \frac{1}{20} a^{8} - \frac{3}{160} a^{7} - \frac{1}{10} a^{5} - \frac{1}{10} a^{4} - \frac{17}{40} a^{3} + \frac{1}{10} a^{2} - \frac{2}{5} a$, $\frac{1}{640} a^{16} - \frac{1}{160} a^{13} - \frac{1}{320} a^{12} + \frac{1}{160} a^{11} - \frac{9}{160} a^{10} - \frac{9}{160} a^{9} - \frac{27}{640} a^{8} - \frac{17}{160} a^{7} + \frac{1}{160} a^{6} + \frac{9}{80} a^{5} - \frac{13}{160} a^{4} - \frac{9}{20} a^{3} - \frac{1}{40} a^{2} + \frac{3}{10} a$, $\frac{1}{1920} a^{17} - \frac{1}{480} a^{15} - \frac{1}{320} a^{13} - \frac{1}{120} a^{11} - \frac{7}{120} a^{10} - \frac{71}{1920} a^{9} + \frac{53}{480} a^{7} + \frac{1}{24} a^{6} - \frac{11}{160} a^{5} - \frac{1}{20} a^{4} + \frac{11}{40} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{38400} a^{18} - \frac{1}{38400} a^{17} + \frac{1}{4800} a^{16} + \frac{7}{9600} a^{15} - \frac{17}{6400} a^{14} + \frac{1}{1280} a^{13} + \frac{1}{300} a^{12} + \frac{19}{1600} a^{11} + \frac{61}{7680} a^{10} + \frac{1679}{38400} a^{9} + \frac{259}{4800} a^{8} + \frac{41}{640} a^{7} + \frac{25}{384} a^{6} - \frac{389}{3200} a^{5} - \frac{69}{800} a^{4} + \frac{3}{32} a^{3} - \frac{13}{200} a^{2} + \frac{1}{20} a$, $\frac{1}{76800} a^{19} + \frac{7}{76800} a^{17} + \frac{3}{6400} a^{16} + \frac{83}{38400} a^{15} - \frac{3}{3200} a^{14} + \frac{79}{38400} a^{13} - \frac{47}{9600} a^{12} - \frac{679}{76800} a^{11} + \frac{1}{75} a^{10} + \frac{1831}{76800} a^{9} - \frac{547}{19200} a^{8} - \frac{103}{960} a^{7} - \frac{631}{9600} a^{6} - \frac{33}{256} a^{5} - \frac{77}{800} a^{4} - \frac{717}{1600} a^{3} - \frac{183}{400} a^{2} - \frac{19}{40} a - \frac{1}{2}$, $\frac{1}{8140800} a^{20} - \frac{7}{1356800} a^{19} + \frac{3}{542720} a^{18} + \frac{193}{1017600} a^{17} + \frac{1759}{4070400} a^{16} + \frac{653}{407040} a^{15} + \frac{8293}{4070400} a^{14} - \frac{157}{508800} a^{13} + \frac{419}{2713600} a^{12} + \frac{1093}{271360} a^{11} + \frac{435373}{8140800} a^{10} + \frac{41159}{1017600} a^{9} + \frac{37309}{1017600} a^{8} - \frac{7651}{169600} a^{7} + \frac{149239}{2035200} a^{6} - \frac{4287}{169600} a^{5} - \frac{18291}{169600} a^{4} - \frac{1629}{5300} a^{3} + \frac{2831}{21200} a^{2} + \frac{1}{106} a + \frac{22}{53}$, $\frac{1}{6574185038221687915367658573118984325696820676279630848000} a^{21} - \frac{18897348414737041662043781477335846893222692034527}{438279002548112527691177238207932288379788045085308723200} a^{20} + \frac{13646165252151517815279065827499298609141502611728961}{2191395012740562638455886191039661441898940225426543616000} a^{19} + \frac{148891201033632954030865497659430646426549188398499}{438279002548112527691177238207932288379788045085308723200} a^{18} - \frac{177883894691880384228262101413134602767424598218387791}{1095697506370281319227943095519830720949470112713271808000} a^{17} - \frac{389913773852635172913072656814935342589599681527089589}{1095697506370281319227943095519830720949470112713271808000} a^{16} + \frac{4055237513110437740637205597446280883549128587327764423}{3287092519110843957683829286559492162848410338139815424000} a^{15} - \frac{6075928421556255397325192749674553361007734268226204043}{3287092519110843957683829286559492162848410338139815424000} a^{14} - \frac{11662320036392307963575321068317240927274559630297325837}{2191395012740562638455886191039661441898940225426543616000} a^{13} + \frac{719249444803296092878544958424428261426905583008477323}{1314837007644337583073531714623796865139364135255926169600} a^{12} - \frac{47070518110970720684999978952304649742146590010752705789}{6574185038221687915367658573118984325696820676279630848000} a^{11} - \frac{307175856255323747495200013031231302617274833224154847939}{6574185038221687915367658573118984325696820676279630848000} a^{10} - \frac{15054828339540081866166897109919552373835074281501187}{17120273537035645612936610867497355014835470511144872000} a^{9} - \frac{27473611858318043560139186479981234631464820088933778151}{821773129777710989420957321639873040712102584534953856000} a^{8} + \frac{23239250043325353281762495902909530426887031473574700231}{328709251911084395768382928655949216284841033813981542400} a^{7} + \frac{28063215826710999458641223594122976063724943655666399071}{328709251911084395768382928655949216284841033813981542400} a^{6} + \frac{14955617421840664427570061700992317225855187614335760787}{68481094148142582451746443469989420059341882044579488000} a^{5} - \frac{2389693492369201877870303476949199269470318767116784367}{136962188296285164903492886939978840118683764089158976000} a^{4} - \frac{2876520371381624751176679622509047113600007736747404259}{17120273537035645612936610867497355014835470511144872000} a^{3} - \frac{18426322084884168547535407154303238485177424577532927}{3424054707407129122587322173499471002967094102228974400} a^{2} - \frac{30841386824168758921902682296692449374051949819283401}{85601367685178228064683054337486775074177352555724360} a - \frac{2982824835174687300091392830400090628582773326734973}{8560136768517822806468305433748677507417735255572436}$

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:  \( 493113388733000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T12:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1210
The 31 conjugacy class representatives for t22n12
Character table for t22n12 is not computed

Intermediate fields

\(\Q(\sqrt{-2}) \)

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

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{11}$ $22$ R $22$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $22$ $22$ $22$ $22$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.5.0.1$x^{5} - x + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
3.5.0.1$x^{5} - x + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
3.11.10.1$x^{11} - 3$$11$$1$$10$$C_{11}:C_5$$[\ ]_{11}^{5}$
$11$11.11.12.1$x^{11} + 88 x^{2} + 11$$11$$1$$12$$C_{11}:C_5$$[6/5]_{5}$
11.11.18.4$x^{11} + 88 x^{8} + 11$$11$$1$$18$$C_{11}:C_5$$[9/5]_{5}$