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Properties

Label 22.2.63974568766...0073.1
Degree $22$
Signature $[2, 10]$
Discriminant $3^{21}\cdot 11^{19}$
Root discriminant $22.64$
Ramified primes $3, 11$
Class number $1$ (GRH)
Class group Trivial (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![-8, 98, -429, 1016, -1166, -27, 1485, -915, -891, 1273, -121, -780, 473, 205, -297, 27, 99, -42, -11, 14, 0, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 14*x^19 - 11*x^18 - 42*x^17 + 99*x^16 + 27*x^15 - 297*x^14 + 205*x^13 + 473*x^12 - 780*x^11 - 121*x^10 + 1273*x^9 - 891*x^8 - 915*x^7 + 1485*x^6 - 27*x^5 - 1166*x^4 + 1016*x^3 - 429*x^2 + 98*x - 8)
 
gp: K = bnfinit(x^22 - x^21 + 14*x^19 - 11*x^18 - 42*x^17 + 99*x^16 + 27*x^15 - 297*x^14 + 205*x^13 + 473*x^12 - 780*x^11 - 121*x^10 + 1273*x^9 - 891*x^8 - 915*x^7 + 1485*x^6 - 27*x^5 - 1166*x^4 + 1016*x^3 - 429*x^2 + 98*x - 8, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 14 x^{19} - 11 x^{18} - 42 x^{17} + 99 x^{16} + 27 x^{15} - 297 x^{14} + 205 x^{13} + 473 x^{12} - 780 x^{11} - 121 x^{10} + 1273 x^{9} - 891 x^{8} - 915 x^{7} + 1485 x^{6} - 27 x^{5} - 1166 x^{4} + 1016 x^{3} - 429 x^{2} + 98 x - 8 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(639745687664639805566853620073=3^{21}\cdot 11^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.64$
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:  $3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{41123519104772983538978115802} a^{21} - \frac{15095270186859282682956046029}{41123519104772983538978115802} a^{20} - \frac{254516396540282589795355310}{20561759552386491769489057901} a^{19} - \frac{5505213952698693958564830767}{20561759552386491769489057901} a^{18} - \frac{9634372344752377025218457493}{41123519104772983538978115802} a^{17} + \frac{4451557926769372913025604561}{20561759552386491769489057901} a^{16} - \frac{11302927999547228950932312487}{41123519104772983538978115802} a^{15} - \frac{17833311120406287385644221573}{41123519104772983538978115802} a^{14} + \frac{15616542471080159316722229485}{41123519104772983538978115802} a^{13} - \frac{17749829187916097747204042879}{41123519104772983538978115802} a^{12} + \frac{6421209351823217310672425395}{41123519104772983538978115802} a^{11} + \frac{6859531605379419086560661882}{20561759552386491769489057901} a^{10} + \frac{8247905435865559879366814131}{41123519104772983538978115802} a^{9} - \frac{1101428697990366197498550233}{41123519104772983538978115802} a^{8} + \frac{537635490066710496999387495}{41123519104772983538978115802} a^{7} - \frac{11072676989530835607936095839}{41123519104772983538978115802} a^{6} + \frac{6285306585654715625455125595}{41123519104772983538978115802} a^{5} - \frac{19319712491047423425877191635}{41123519104772983538978115802} a^{4} + \frac{8310760584463466725038277306}{20561759552386491769489057901} a^{3} + \frac{5950519614004267877954969117}{20561759552386491769489057901} a^{2} - \frac{19365438812901176130425661729}{41123519104772983538978115802} a - \frac{5394087174870999487236789891}{20561759552386491769489057901}$

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:  $11$
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:  \( 5215069.85605 \) (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

\(\Q(\sqrt{33}) \), 11.1.139234453205859.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ $22$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
3Data not computed
11Data not computed