↑

Properties

Label 22.0.83110763880...8768.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{10}\cdot 3^{10}\cdot 11^{18}\cdot 47^{11}$
Root discriminant $110.10$
Ramified primes $2, 3, 11, 47$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group 22T25

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![23837, -44286, 20691, -44770, 132495, -164197, 138424, -80344, 32791, 13915, -31603, 32934, -18469, 6919, 935, -2816, 2684, -1529, 704, -220, 54, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 6*x^21 + 54*x^20 - 220*x^19 + 704*x^18 - 1529*x^17 + 2684*x^16 - 2816*x^15 + 935*x^14 + 6919*x^13 - 18469*x^12 + 32934*x^11 - 31603*x^10 + 13915*x^9 + 32791*x^8 - 80344*x^7 + 138424*x^6 - 164197*x^5 + 132495*x^4 - 44770*x^3 + 20691*x^2 - 44286*x + 23837)
 
gp: K = bnfinit(x^22 - 6*x^21 + 54*x^20 - 220*x^19 + 704*x^18 - 1529*x^17 + 2684*x^16 - 2816*x^15 + 935*x^14 + 6919*x^13 - 18469*x^12 + 32934*x^11 - 31603*x^10 + 13915*x^9 + 32791*x^8 - 80344*x^7 + 138424*x^6 - 164197*x^5 + 132495*x^4 - 44770*x^3 + 20691*x^2 - 44286*x + 23837, 1)
 

Normalized defining polynomial

\( x^{22} - 6 x^{21} + 54 x^{20} - 220 x^{19} + 704 x^{18} - 1529 x^{17} + 2684 x^{16} - 2816 x^{15} + 935 x^{14} + 6919 x^{13} - 18469 x^{12} + 32934 x^{11} - 31603 x^{10} + 13915 x^{9} + 32791 x^{8} - 80344 x^{7} + 138424 x^{6} - 164197 x^{5} + 132495 x^{4} - 44770 x^{3} + 20691 x^{2} - 44286 x + 23837 \)

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:  \(-831107638802333745804311925084476212698528768=-\,2^{10}\cdot 3^{10}\cdot 11^{18}\cdot 47^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $110.10$
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:  $2, 3, 11, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} + \frac{5}{11} a^{11} - \frac{1}{11} a^{10}$, $\frac{1}{22} a^{13} - \frac{1}{22} a^{12} - \frac{9}{22} a^{11} - \frac{5}{22} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{22} a^{14} - \frac{4}{11} a^{11} - \frac{2}{11} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{22} a^{15} - \frac{4}{11} a^{11} + \frac{3}{22} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{44} a^{16} - \frac{1}{44} a^{14} - \frac{1}{22} a^{12} - \frac{3}{44} a^{11} - \frac{1}{22} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{132} a^{17} + \frac{1}{132} a^{16} - \frac{1}{44} a^{15} - \frac{1}{44} a^{14} + \frac{1}{66} a^{13} + \frac{1}{44} a^{12} - \frac{37}{132} a^{11} + \frac{1}{132} a^{10} + \frac{1}{4} a^{9} - \frac{5}{12} a^{8} + \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{5}{12} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{1}{12} a - \frac{5}{12}$, $\frac{1}{132} a^{18} - \frac{1}{132} a^{16} + \frac{1}{66} a^{14} + \frac{1}{132} a^{13} + \frac{1}{66} a^{12} + \frac{5}{132} a^{11} - \frac{1}{6} a^{10} + \frac{1}{12} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{3} + \frac{1}{12} a^{2} - \frac{1}{3}$, $\frac{1}{264} a^{19} - \frac{1}{264} a^{18} + \frac{1}{132} a^{16} + \frac{5}{264} a^{15} + \frac{1}{132} a^{14} + \frac{1}{88} a^{13} - \frac{1}{44} a^{12} - \frac{1}{3} a^{11} - \frac{23}{66} a^{10} - \frac{1}{24} a^{9} + \frac{1}{24} a^{8} - \frac{11}{24} a^{7} - \frac{1}{6} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{5}{24} a^{3} - \frac{7}{24} a^{2} - \frac{3}{8} a + \frac{5}{24}$, $\frac{1}{264} a^{20} - \frac{1}{264} a^{18} - \frac{1}{264} a^{16} + \frac{1}{264} a^{15} + \frac{5}{264} a^{14} + \frac{5}{264} a^{13} - \frac{1}{66} a^{12} - \frac{13}{66} a^{11} + \frac{9}{88} a^{10} - \frac{1}{2} a^{8} - \frac{5}{24} a^{7} - \frac{3}{8} a^{6} - \frac{5}{24} a^{5} + \frac{7}{24} a^{4} + \frac{1}{4} a^{3} - \frac{1}{6} a^{2} + \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{13520404791967437935385843170251623988848} a^{21} + \frac{13795040303673563702142072898959849007}{13520404791967437935385843170251623988848} a^{20} + \frac{6468671035496743218837192828128363935}{13520404791967437935385843170251623988848} a^{19} + \frac{830213552963026494129252615963771807}{409709236120225391981389186977321939056} a^{18} + \frac{1537867203477529220160356742101171605}{409709236120225391981389186977321939056} a^{17} - \frac{153118532740477350664780030927121777}{307281927090169043986041890232991454292} a^{16} - \frac{2008536412494016703508010558636961297}{614563854180338087972083780465982908584} a^{15} + \frac{148907055887421803787885750201169939}{307281927090169043986041890232991454292} a^{14} - \frac{7592764527059078406576306004657771247}{409709236120225391981389186977321939056} a^{13} + \frac{6225080772607049491963747898263179967}{614563854180338087972083780465982908584} a^{12} + \frac{43898135692727028866254848278401251715}{1229127708360676175944167560931965817168} a^{11} + \frac{197452904973205511350933858626583257627}{409709236120225391981389186977321939056} a^{10} - \frac{250822209463013462737139889303985845661}{614563854180338087972083780465982908584} a^{9} - \frac{47718195681279853853476515947084069611}{111738882578243288722197050993815074288} a^{8} + \frac{7472831212980680664867744751820195513}{27934720644560822180549262748453768572} a^{7} + \frac{5210524619954889103714586319920718657}{18623147096373881453699508498969179048} a^{6} + \frac{750398304787555729019704949194033198}{2327893387046735181712438562371147381} a^{5} + \frac{9332533766090439161586928998416001755}{37246294192747762907399016997938358096} a^{4} + \frac{2167853260717908813705405542182665113}{55869441289121644361098525496907537144} a^{3} + \frac{1902581507339679666484042424558858465}{9311573548186940726849754249484589524} a^{2} + \frac{901370381622073810543798663792520591}{111738882578243288722197050993815074288} a + \frac{8692599621395452625317191142299960229}{111738882578243288722197050993815074288}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{5}$, which has order $5$ (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:  \( 49111075804800 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T25:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12100
The 49 conjugacy class representatives for t22n25
Character table for t22n25 is not computed

Intermediate fields

\(\Q(\sqrt{-47}) \)

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

Sibling fields

Degree 44 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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ $22$ R ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$0$Trivial$[\ ]$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
2.11.10.1$x^{11} - 2$$11$$1$$10$$F_{11}$$[\ ]_{11}^{10}$
$3$$\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$$[\ ]_{11}^{5}$
11Data not computed
$47$47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.10.5.2$x^{10} - 4879681 x^{2} + 688035021$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
47.10.5.2$x^{10} - 4879681 x^{2} + 688035021$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$