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Properties

Label 22.14.3050063118...0208.1
Degree $22$
Signature $[14, 4]$
Discriminant $2^{32}\cdot 3^{29}\cdot 7^{4}\cdot 23^{4}\cdot 137^{16}$
Root discriminant $1052.00$
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![-3859488, -33869664, 2781648, 127652544, -19547856, -169280496, 44592120, 112918896, -34917642, -46019430, 14763681, 12206088, -3792132, -2137974, 605475, 239076, -58926, -15678, 3315, 516, -96, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 6*x^21 - 96*x^20 + 516*x^19 + 3315*x^18 - 15678*x^17 - 58926*x^16 + 239076*x^15 + 605475*x^14 - 2137974*x^13 - 3792132*x^12 + 12206088*x^11 + 14763681*x^10 - 46019430*x^9 - 34917642*x^8 + 112918896*x^7 + 44592120*x^6 - 169280496*x^5 - 19547856*x^4 + 127652544*x^3 + 2781648*x^2 - 33869664*x - 3859488)
 
gp: K = bnfinit(x^22 - 6*x^21 - 96*x^20 + 516*x^19 + 3315*x^18 - 15678*x^17 - 58926*x^16 + 239076*x^15 + 605475*x^14 - 2137974*x^13 - 3792132*x^12 + 12206088*x^11 + 14763681*x^10 - 46019430*x^9 - 34917642*x^8 + 112918896*x^7 + 44592120*x^6 - 169280496*x^5 - 19547856*x^4 + 127652544*x^3 + 2781648*x^2 - 33869664*x - 3859488, 1)
 

Normalized defining polynomial

\( x^{22} - 6 x^{21} - 96 x^{20} + 516 x^{19} + 3315 x^{18} - 15678 x^{17} - 58926 x^{16} + 239076 x^{15} + 605475 x^{14} - 2137974 x^{13} - 3792132 x^{12} + 12206088 x^{11} + 14763681 x^{10} - 46019430 x^{9} - 34917642 x^{8} + 112918896 x^{7} + 44592120 x^{6} - 169280496 x^{5} - 19547856 x^{4} + 127652544 x^{3} + 2781648 x^{2} - 33869664 x - 3859488 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3050063118130674940421242764072714852846127200026846650975238750208=2^{32}\cdot 3^{29}\cdot 7^{4}\cdot 23^{4}\cdot 137^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1052.00$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{18} a^{11} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2}$, $\frac{1}{36} a^{12} - \frac{1}{36} a^{11} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{4} a^{4} - \frac{1}{12} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{36} a^{13} - \frac{1}{36} a^{11} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{12} a^{5} + \frac{1}{6} a^{4} - \frac{1}{4} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{36} a^{14} - \frac{1}{36} a^{11} + \frac{1}{6} a^{8} + \frac{1}{3} a^{7} + \frac{1}{4} a^{6} + \frac{1}{3} a^{5} - \frac{1}{4} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{36} a^{15} - \frac{1}{36} a^{11} - \frac{1}{6} a^{8} - \frac{5}{12} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{72} a^{16} - \frac{1}{72} a^{12} - \frac{1}{36} a^{11} - \frac{1}{8} a^{8} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{1}{8} a^{4} + \frac{5}{12} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{72} a^{17} - \frac{1}{72} a^{13} - \frac{1}{36} a^{11} + \frac{1}{24} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{2} a^{6} + \frac{7}{24} a^{5} + \frac{1}{6} a^{4} + \frac{1}{4} a^{3} - \frac{1}{6} a^{2}$, $\frac{1}{576} a^{18} - \frac{1}{288} a^{17} - \frac{1}{144} a^{16} - \frac{1}{144} a^{15} - \frac{1}{576} a^{14} + \frac{1}{96} a^{13} + \frac{1}{96} a^{12} - \frac{1}{48} a^{11} - \frac{11}{192} a^{10} + \frac{5}{96} a^{9} + \frac{5}{24} a^{8} - \frac{1}{6} a^{7} + \frac{7}{192} a^{6} + \frac{17}{96} a^{5} - \frac{15}{32} a^{4} + \frac{7}{16} a^{2} - \frac{3}{8} a + \frac{3}{8}$, $\frac{1}{1728} a^{19} - \frac{1}{144} a^{16} - \frac{1}{192} a^{15} - \frac{1}{144} a^{14} - \frac{1}{288} a^{13} - \frac{1}{192} a^{11} + \frac{5}{144} a^{10} + \frac{1}{16} a^{9} - \frac{1}{12} a^{8} - \frac{17}{64} a^{7} + \frac{1}{6} a^{6} + \frac{35}{96} a^{5} - \frac{5}{16} a^{4} + \frac{1}{16} a^{3} - \frac{1}{2} a^{2} - \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{7776000} a^{20} - \frac{47}{1296000} a^{19} - \frac{181}{259200} a^{18} + \frac{323}{81000} a^{17} + \frac{6829}{2592000} a^{16} - \frac{767}{144000} a^{15} + \frac{31}{12960} a^{14} - \frac{613}{108000} a^{13} + \frac{4379}{864000} a^{12} - \frac{15139}{1296000} a^{11} + \frac{8719}{144000} a^{10} + \frac{5321}{216000} a^{9} + \frac{39967}{288000} a^{8} - \frac{13909}{86400} a^{7} + \frac{1939}{36000} a^{6} + \frac{32137}{216000} a^{5} - \frac{14731}{36000} a^{4} + \frac{16843}{36000} a^{3} - \frac{3241}{27000} a^{2} - \frac{3883}{18000} a - \frac{1849}{18000}$, $\frac{1}{35287823480245076801529067342928309341026604128000} a^{21} - \frac{1289363610364960746264320606363750462213879}{35287823480245076801529067342928309341026604128000} a^{20} - \frac{598416461844891157451454062460556143679668923}{2940651956687089733460755611910692445085550344000} a^{19} - \frac{483543772008412645257666758627608478784508783}{653478212597131051880167913757931654463455632000} a^{18} - \frac{17202016155123600342390456601482006640936737763}{11762607826748358933843022447642769780342201376000} a^{17} + \frac{1689614230279161091949336224247507772009645389}{405607166439598583925621463711819647598006944000} a^{16} - \frac{67121409487923961550192653806319806128991558809}{5881303913374179466921511223821384890171100688000} a^{15} - \frac{10444182294768055971169258334029301100038758139}{1470325978343544866730377805955346222542775172000} a^{14} - \frac{1007020642383278675620770188207309632290258437}{435652141731420701253445275838621102975637088000} a^{13} - \frac{1242854615776753146593388384511573274728372067}{11762607826748358933843022447642769780342201376000} a^{12} + \frac{5466341170757563300738207624231652878635138727}{2940651956687089733460755611910692445085550344000} a^{11} - \frac{11078301700679385711259162327956751665163893587}{1960434637791393155640503741273794963390366896000} a^{10} + \frac{211919503056373869125259844536481203827776863353}{3920869275582786311281007482547589926780733792000} a^{9} - \frac{147494104865636349381164234783467641341977180987}{3920869275582786311281007482547589926780733792000} a^{8} - \frac{652218102567733728376798347954437870397552385367}{1960434637791393155640503741273794963390366896000} a^{7} + \frac{404653099377583459166724469210256136379192781389}{980217318895696577820251870636897481695183448000} a^{6} - \frac{8722648207297773847346606891698996373957159}{35872545979714421878142794899794967308149440} a^{5} - \frac{7152685384466348035374154334231108642518753}{54456517716427587656680659479827637871954636} a^{4} - \frac{211521332752508235084119051436854933303584555777}{490108659447848288910125935318448740847591724000} a^{3} - \frac{4477740170731469569389923931612714446982271729}{49010865944784828891012593531844874084759172400} a^{2} + \frac{18433621238158927394765944166595105820345737401}{40842388287320690742510494609870728403965977000} a - \frac{10093308908313610675647845737562674824339086647}{81684776574641381485020989219741456807931954000}$

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:  $17$
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:  \( 852811169305000000000000000 \) (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 ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ $22$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ $22$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.12.18.79$x^{12} + 2 x^{9} + 2 x^{7} + 2 x^{2} - 2$$12$$1$$18$$C_2 \times S_4$$[4/3, 4/3, 2]_{3}^{2}$
3Data not computed
7Data not computed
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.7.0.1$x^{7} - x + 8$$1$$7$$0$$C_7$$[\ ]^{7}$
23.7.0.1$x^{7} - x + 8$$1$$7$$0$$C_7$$[\ ]^{7}$
$137$137.2.0.1$x^{2} - x + 6$$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}$