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Properties

Label 22.22.2093797597...6629.1
Degree $22$
Signature $[22, 0]$
Discriminant $23^{20}\cdot 29^{11}$
Root discriminant $93.14$
Ramified primes $23, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{22}$ (as 22T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-24468919, -138936109, -104865816, 386471695, 334596715, -485055359, -323131047, 335982569, 145940517, -135174899, -34619686, 32637502, 4332757, -4834612, -232734, 440652, -4969, -24022, 1305, 717, -62, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 9*x^21 - 62*x^20 + 717*x^19 + 1305*x^18 - 24022*x^17 - 4969*x^16 + 440652*x^15 - 232734*x^14 - 4834612*x^13 + 4332757*x^12 + 32637502*x^11 - 34619686*x^10 - 135174899*x^9 + 145940517*x^8 + 335982569*x^7 - 323131047*x^6 - 485055359*x^5 + 334596715*x^4 + 386471695*x^3 - 104865816*x^2 - 138936109*x - 24468919)
 
gp: K = bnfinit(x^22 - 9*x^21 - 62*x^20 + 717*x^19 + 1305*x^18 - 24022*x^17 - 4969*x^16 + 440652*x^15 - 232734*x^14 - 4834612*x^13 + 4332757*x^12 + 32637502*x^11 - 34619686*x^10 - 135174899*x^9 + 145940517*x^8 + 335982569*x^7 - 323131047*x^6 - 485055359*x^5 + 334596715*x^4 + 386471695*x^3 - 104865816*x^2 - 138936109*x - 24468919, 1)
 

Normalized defining polynomial

\( x^{22} - 9 x^{21} - 62 x^{20} + 717 x^{19} + 1305 x^{18} - 24022 x^{17} - 4969 x^{16} + 440652 x^{15} - 232734 x^{14} - 4834612 x^{13} + 4332757 x^{12} + 32637502 x^{11} - 34619686 x^{10} - 135174899 x^{9} + 145940517 x^{8} + 335982569 x^{7} - 323131047 x^{6} - 485055359 x^{5} + 334596715 x^{4} + 386471695 x^{3} - 104865816 x^{2} - 138936109 x - 24468919 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[22, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(20937975979670626213353681795476767790826629=23^{20}\cdot 29^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.14$
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:  $23, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(667=23\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{667}(1,·)$, $\chi_{667}(579,·)$, $\chi_{667}(581,·)$, $\chi_{667}(262,·)$, $\chi_{667}(202,·)$, $\chi_{667}(463,·)$, $\chi_{667}(144,·)$, $\chi_{667}(146,·)$, $\chi_{667}(407,·)$, $\chi_{667}(347,·)$, $\chi_{667}(349,·)$, $\chi_{667}(289,·)$, $\chi_{667}(610,·)$, $\chi_{667}(231,·)$, $\chi_{667}(233,·)$, $\chi_{667}(492,·)$, $\chi_{667}(173,·)$, $\chi_{667}(117,·)$, $\chi_{667}(376,·)$, $\chi_{667}(59,·)$, $\chi_{667}(637,·)$, $\chi_{667}(639,·)$$\rbrace$
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}{19554833962123412723531710556561503870813703089072316470128119654668166674969} a^{21} + \frac{7112366563154723195978169752290914381069754445694476087919489590710944689648}{19554833962123412723531710556561503870813703089072316470128119654668166674969} a^{20} + \frac{204259300134415462763007457160745134798551312083379897681643278951961563115}{19554833962123412723531710556561503870813703089072316470128119654668166674969} a^{19} + \frac{737619366978728519501461132214044292985644690318596151128655797058347125066}{19554833962123412723531710556561503870813703089072316470128119654668166674969} a^{18} + \frac{886061393878244162194733211580713018820968968905227126011316068344007078695}{19554833962123412723531710556561503870813703089072316470128119654668166674969} a^{17} + \frac{8870373647130594402122571838980644773212226318400655250511724316752542781295}{19554833962123412723531710556561503870813703089072316470128119654668166674969} a^{16} - \frac{5995528259352226549433399203384936469889991292689631186317814587864419565046}{19554833962123412723531710556561503870813703089072316470128119654668166674969} a^{15} - \frac{603184417501697196039611951833400788818882542098938491958881667317678934700}{19554833962123412723531710556561503870813703089072316470128119654668166674969} a^{14} + \frac{484823949265496498922207628940770641710420626908648783775352465931312954022}{19554833962123412723531710556561503870813703089072316470128119654668166674969} a^{13} + \frac{6687490224239811568615822038785564301678052295780009963163891677158483213203}{19554833962123412723531710556561503870813703089072316470128119654668166674969} a^{12} + \frac{9683035871693100488474702012333296771879320549430861919836480862251341028962}{19554833962123412723531710556561503870813703089072316470128119654668166674969} a^{11} - \frac{5213318577693920811443642654644194419594775130945062315810805371304316896569}{19554833962123412723531710556561503870813703089072316470128119654668166674969} a^{10} + \frac{6763525634057941773126409145624227095900523306527263366447396150353182304297}{19554833962123412723531710556561503870813703089072316470128119654668166674969} a^{9} + \frac{5496868698860391614972499492658063311347187109664443360473822460778404908127}{19554833962123412723531710556561503870813703089072316470128119654668166674969} a^{8} + \frac{5385094452199195428761955391989957354124667075335207299961549023883902485411}{19554833962123412723531710556561503870813703089072316470128119654668166674969} a^{7} - \frac{3086684992948294316302942224309078226821307023049478228086809404761349490694}{19554833962123412723531710556561503870813703089072316470128119654668166674969} a^{6} + \frac{6011200265493710097510288778125008931932307381466943217853172819545251834026}{19554833962123412723531710556561503870813703089072316470128119654668166674969} a^{5} + \frac{6237463101609159017076324401584919793124543941235417393984525652586439717276}{19554833962123412723531710556561503870813703089072316470128119654668166674969} a^{4} - \frac{1269700412027222241157006099281598211649893065297929355391926375221863566017}{19554833962123412723531710556561503870813703089072316470128119654668166674969} a^{3} - \frac{8511210082422022700219511606832227895482589428651200194332292105844461552972}{19554833962123412723531710556561503870813703089072316470128119654668166674969} a^{2} + \frac{6072282346941746251604349088534889736288996240204422001962589697040024042823}{19554833962123412723531710556561503870813703089072316470128119654668166674969} a + \frac{8054386175900244869571185246061614376196888591344794746055007839414129157355}{19554833962123412723531710556561503870813703089072316470128119654668166674969}$

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:  $21$
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:  \( 269532187377000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{22}$ (as 22T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 22
The 22 conjugacy class representatives for $C_{22}$
Character table for $C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\zeta_{23})^+\)

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 $22$ $22$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ $22$ $22$ R R $22$ $22$ $22$ $22$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
$23$23.11.10.10$x^{11} - 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$
23.11.10.10$x^{11} - 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$
29Data not computed