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Properties

Label 22.0.26522354285...4144.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{20}\cdot 7^{10}\cdot 11^{23}$
Root discriminant $55.79$
Ramified primes $2, 7, 11$
Class number $22$ (GRH)
Class group $[22]$ (GRH)
Galois group $C_2\times F_{11}$ (as 22T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![45, -913, 4334, 99, 47520, -11385, 237369, -284306, 373593, 27313, -78386, -61939, 14498, 13915, 3619, -5698, 1331, 495, -264, -33, 44, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 44*x^20 - 33*x^19 - 264*x^18 + 495*x^17 + 1331*x^16 - 5698*x^15 + 3619*x^14 + 13915*x^13 + 14498*x^12 - 61939*x^11 - 78386*x^10 + 27313*x^9 + 373593*x^8 - 284306*x^7 + 237369*x^6 - 11385*x^5 + 47520*x^4 + 99*x^3 + 4334*x^2 - 913*x + 45)
 
gp: K = bnfinit(x^22 - 11*x^21 + 44*x^20 - 33*x^19 - 264*x^18 + 495*x^17 + 1331*x^16 - 5698*x^15 + 3619*x^14 + 13915*x^13 + 14498*x^12 - 61939*x^11 - 78386*x^10 + 27313*x^9 + 373593*x^8 - 284306*x^7 + 237369*x^6 - 11385*x^5 + 47520*x^4 + 99*x^3 + 4334*x^2 - 913*x + 45, 1)
 

Normalized defining polynomial

\( x^{22} - 11 x^{21} + 44 x^{20} - 33 x^{19} - 264 x^{18} + 495 x^{17} + 1331 x^{16} - 5698 x^{15} + 3619 x^{14} + 13915 x^{13} + 14498 x^{12} - 61939 x^{11} - 78386 x^{10} + 27313 x^{9} + 373593 x^{8} - 284306 x^{7} + 237369 x^{6} - 11385 x^{5} + 47520 x^{4} + 99 x^{3} + 4334 x^{2} - 913 x + 45 \)

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:  \(-265223542853498303707290831275917574144=-\,2^{20}\cdot 7^{10}\cdot 11^{23}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.79$
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:  $2, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{61454767620954569963998460374080173688640465174323620733093403523} a^{21} - \frac{26828495142163967224776274660494436802295296319087880101320946765}{61454767620954569963998460374080173688640465174323620733093403523} a^{20} - \frac{8141116069407380959133738028701237820672340286543991585710717727}{61454767620954569963998460374080173688640465174323620733093403523} a^{19} + \frac{18557522830484888259097523840338504236334608819472913813306975358}{61454767620954569963998460374080173688640465174323620733093403523} a^{18} - \frac{17715641863109256604183664428255352161287197104351857403293361205}{61454767620954569963998460374080173688640465174323620733093403523} a^{17} + \frac{22142554860698286199844944885345582462302522269241712623332050388}{61454767620954569963998460374080173688640465174323620733093403523} a^{16} - \frac{13495113900801863289248310168253029237652499875605922967921487191}{61454767620954569963998460374080173688640465174323620733093403523} a^{15} + \frac{26957422420586421220605337082216983701480455489749360814189939897}{61454767620954569963998460374080173688640465174323620733093403523} a^{14} + \frac{27801421540514943370466030750536198036468870893613998864677686151}{61454767620954569963998460374080173688640465174323620733093403523} a^{13} - \frac{15176183903613996858531857992360673593243266711862333669290494920}{61454767620954569963998460374080173688640465174323620733093403523} a^{12} + \frac{16959745858999355678048474998455184492872348555940301014663453352}{61454767620954569963998460374080173688640465174323620733093403523} a^{11} - \frac{11824969141359108384992380226580868716614096216217973966886455331}{61454767620954569963998460374080173688640465174323620733093403523} a^{10} + \frac{13520328733276757261561171382783395034770243997494435394360348665}{61454767620954569963998460374080173688640465174323620733093403523} a^{9} - \frac{18307423334580410127192865480111906965489274124997243466223925368}{61454767620954569963998460374080173688640465174323620733093403523} a^{8} + \frac{21033198194507986074041154464966843413805523168779072315641586594}{61454767620954569963998460374080173688640465174323620733093403523} a^{7} - \frac{2006435805433758434899734439201699683742486853286459164386292214}{61454767620954569963998460374080173688640465174323620733093403523} a^{6} - \frac{25472505824510971207852352778771782900679095379786143227223263090}{61454767620954569963998460374080173688640465174323620733093403523} a^{5} + \frac{29766746146586079967827931657609854590878790473548393882577863613}{61454767620954569963998460374080173688640465174323620733093403523} a^{4} + \frac{4717065032492216389345231015533983112681204569058140294702857330}{61454767620954569963998460374080173688640465174323620733093403523} a^{3} + \frac{10420170593699373039967146761818926061968514700432428024224785640}{61454767620954569963998460374080173688640465174323620733093403523} a^{2} + \frac{14081428084827835716921196259822788014721778098778989412443032389}{61454767620954569963998460374080173688640465174323620733093403523} a + \frac{12588351202653201477346459692850515670192400253465887370507042027}{61454767620954569963998460374080173688640465174323620733093403523}$

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

Class group and class number

$C_{22}$, which has order $22$ (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:  \( 2013788572.23 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_{11}$ (as 22T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 220
The 22 conjugacy class representatives for $C_2\times F_{11}$
Character table for $C_2\times F_{11}$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), 11.11.4910318845910094848.1

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

Sibling fields

Degree 22 sibling: data not computed
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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
7Data not computed
11Data not computed