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Properties

Label 22.0.11914309055...6368.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{24}\cdot 3^{29}\cdot 7^{4}\cdot 23^{4}\cdot 137^{16}$
Root discriminant $817.61$
Ramified primes $2, 3, 7, 23, 137$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T46

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2916, 1944, 972, 11880, 13644, 6156, 15876, 31986, 32238, 25902, 36558, 71280, 42921, 18555, 42993, 12294, 11013, -891, 1263, -12, 45, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 3*x^21 + 45*x^20 - 12*x^19 + 1263*x^18 - 891*x^17 + 11013*x^16 + 12294*x^15 + 42993*x^14 + 18555*x^13 + 42921*x^12 + 71280*x^11 + 36558*x^10 + 25902*x^9 + 32238*x^8 + 31986*x^7 + 15876*x^6 + 6156*x^5 + 13644*x^4 + 11880*x^3 + 972*x^2 + 1944*x + 2916)
 
gp: K = bnfinit(x^22 - 3*x^21 + 45*x^20 - 12*x^19 + 1263*x^18 - 891*x^17 + 11013*x^16 + 12294*x^15 + 42993*x^14 + 18555*x^13 + 42921*x^12 + 71280*x^11 + 36558*x^10 + 25902*x^9 + 32238*x^8 + 31986*x^7 + 15876*x^6 + 6156*x^5 + 13644*x^4 + 11880*x^3 + 972*x^2 + 1944*x + 2916, 1)
 

Normalized defining polynomial

\( x^{22} - 3 x^{21} + 45 x^{20} - 12 x^{19} + 1263 x^{18} - 891 x^{17} + 11013 x^{16} + 12294 x^{15} + 42993 x^{14} + 18555 x^{13} + 42921 x^{12} + 71280 x^{11} + 36558 x^{10} + 25902 x^{9} + 32238 x^{8} + 31986 x^{7} + 15876 x^{6} + 6156 x^{5} + 13644 x^{4} + 11880 x^{3} + 972 x^{2} + 1944 x + 2916 \)

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:  \(-11914309055197948986020479547159042393930184375104869730372026368=-\,2^{24}\cdot 3^{29}\cdot 7^{4}\cdot 23^{4}\cdot 137^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $817.61$
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:  $2, 3, 7, 23, 137$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{18} a^{10} - \frac{1}{2} a^{8} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{18} a^{11} - \frac{1}{6} a^{9} - \frac{1}{3} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{18} a^{12} + \frac{1}{3} a^{8} + \frac{1}{6} a^{6} - \frac{1}{3} a^{3}$, $\frac{1}{18} a^{13} + \frac{1}{6} a^{7} - \frac{1}{3} a^{4}$, $\frac{1}{18} a^{14} + \frac{1}{6} a^{8} - \frac{1}{3} a^{5}$, $\frac{1}{18} a^{15} - \frac{1}{6} a^{9} - \frac{1}{3} a^{6}$, $\frac{1}{18} a^{16} - \frac{1}{2} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{36} a^{17} - \frac{1}{36} a^{16} - \frac{1}{36} a^{14} - \frac{1}{36} a^{12} - \frac{1}{36} a^{11} - \frac{1}{6} a^{9} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{108} a^{18} - \frac{1}{36} a^{16} - \frac{1}{36} a^{15} - \frac{1}{36} a^{14} - \frac{1}{36} a^{13} - \frac{1}{36} a^{11} - \frac{1}{18} a^{9} + \frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{108} a^{19} - \frac{1}{36} a^{15} - \frac{1}{36} a^{11} - \frac{1}{6} a^{9} - \frac{1}{3} a^{8} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{729000} a^{20} + \frac{539}{121500} a^{19} - \frac{1}{250} a^{18} + \frac{16}{1215} a^{17} + \frac{6589}{243000} a^{16} - \frac{146}{10125} a^{15} - \frac{71}{30375} a^{14} - \frac{233}{10125} a^{13} + \frac{2173}{81000} a^{12} + \frac{59}{60750} a^{11} - \frac{179}{13500} a^{10} - \frac{833}{6750} a^{9} + \frac{584}{3375} a^{8} - \frac{253}{40500} a^{7} + \frac{19}{3375} a^{6} - \frac{34}{2025} a^{5} - \frac{251}{675} a^{4} - \frac{478}{1125} a^{3} - \frac{2699}{20250} a^{2} - \frac{1169}{6750} a + \frac{361}{2250}$, $\frac{1}{303086670176456970495967843893000} a^{21} + \frac{62877033312676226643337859}{101028890058818990165322614631000} a^{20} - \frac{58584704071888747806777657703}{16838148343136498360887102438500} a^{19} + \frac{89208343461135934688170354027}{50514445029409495082661307315500} a^{18} - \frac{550281195146542606984208821061}{101028890058818990165322614631000} a^{17} - \frac{41437599164613159493927799053}{11225432228757665573924734959000} a^{16} - \frac{84008107975105419501195207019}{10102889005881899016532261463100} a^{15} - \frac{114201825613608330176246864243}{8419074171568249180443551219250} a^{14} - \frac{732808487347834512025166960429}{33676296686272996721774204877000} a^{13} + \frac{2148363286863957936910404800003}{101028890058818990165322614631000} a^{12} - \frac{340460108101748991341890265879}{16838148343136498360887102438500} a^{11} - \frac{34813192622930723855242840063}{5612716114378832786962367479500} a^{10} + \frac{79051423906010284517875207037}{1403179028594708196740591869875} a^{9} - \frac{3505474528221712527610380808759}{16838148343136498360887102438500} a^{8} + \frac{117308749292721446023210302637}{623635123819870309662485275500} a^{7} - \frac{738263379368294124665191202744}{4209537085784124590221775609625} a^{6} + \frac{41237231905077996920002382}{2078783746066234365541617585} a^{5} + \frac{226065368423862965294102960992}{467726342864902732246863956625} a^{4} - \frac{58946005574161323203539379567}{647621090120634552341811632250} a^{3} - \frac{551960269123146876829892197919}{1403179028594708196740591869875} a^{2} + \frac{8139182683610586415124095936}{467726342864902732246863956625} a + \frac{92310343042549322083703075941}{311817561909935154831242637750}$

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:  \( -\frac{45857543861397661588583459}{404115560235275960661290458524} a^{21} - \frac{2181086504872815458901569}{44901728915030662295698939836} a^{20} - \frac{178216142039565906970027051}{44901728915030662295698939836} a^{19} - \frac{539925034001650159653401131}{33676296686272996721774204877} a^{18} - \frac{18838203925820066566983220577}{134705186745091986887096819508} a^{17} - \frac{17489914613439699792040572251}{44901728915030662295698939836} a^{16} - \frac{125478777649349894314594768633}{134705186745091986887096819508} a^{15} - \frac{42130229910541202487660866179}{7483621485838443715949823306} a^{14} - \frac{437757791793002786648961750035}{44901728915030662295698939836} a^{13} - \frac{2585127815797723748584080269551}{134705186745091986887096819508} a^{12} - \frac{485052396037237426537672653287}{44901728915030662295698939836} a^{11} - \frac{84166308867181713814939181060}{3741810742919221857974911653} a^{10} - \frac{181554584272409390007286070047}{7483621485838443715949823306} a^{9} - \frac{192874228667814523305394214015}{11225432228757665573924734959} a^{8} - \frac{40848039178800285815335173547}{7483621485838443715949823306} a^{7} - \frac{148928184816721105621690756067}{22450864457515331147849469918} a^{6} - \frac{71083581891760346712654921431}{7483621485838443715949823306} a^{5} - \frac{655827849591219147265103117}{138585583071082291036107839} a^{4} - \frac{1433494370743172721762961097}{863494786827512736455748843} a^{3} - \frac{3014364559620487801239175585}{1247270247639740619324970551} a^{2} - \frac{1097996282465898389484983687}{415756749213246873108323517} a + \frac{30621804865368512430518822}{415756749213246873108323517} \) (order $6$)
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:  \( 1462413343710000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T46:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 39916800
The 62 conjugacy class representatives for t22n46 are not computed
Character table for t22n46 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 11.7.63019333158425674204677255696384.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ R $22$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ $22$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $18{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.12.12.28$x^{12} - x^{10} + 2 x^{8} - x^{6} - 2 x^{4} + 3 x^{2} + 1$$6$$2$$12$$S_4$$[4/3, 4/3]_{3}^{2}$
3Data not computed
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.7.0.1$x^{7} - x + 2$$1$$7$$0$$C_7$$[\ ]^{7}$
7.7.0.1$x^{7} - x + 2$$1$$7$$0$$C_7$$[\ ]^{7}$
$23$23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.14.0.1$x^{14} - x + 7$$1$$14$$0$$C_{14}$$[\ ]^{14}$
$137$137.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
137.10.8.1$x^{10} - 137 x^{5} + 112614$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
137.10.8.1$x^{10} - 137 x^{5} + 112614$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$