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Properties

Label 22.22.3943707157...9249.1
Degree $22$
Signature $[22, 0]$
Discriminant $3^{11}\cdot 67^{21}$
Root discriminant $95.86$
Ramified primes $3, 67$
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![-116981, -391563, 394142, 2696745, 1572041, -5207551, -7233811, 1113268, 7232279, 3053082, -2370346, -2109350, 63350, 530058, 107774, -58584, -21240, 2739, 1709, -27, -65, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - 65*x^20 - 27*x^19 + 1709*x^18 + 2739*x^17 - 21240*x^16 - 58584*x^15 + 107774*x^14 + 530058*x^13 + 63350*x^12 - 2109350*x^11 - 2370346*x^10 + 3053082*x^9 + 7232279*x^8 + 1113268*x^7 - 7233811*x^6 - 5207551*x^5 + 1572041*x^4 + 2696745*x^3 + 394142*x^2 - 391563*x - 116981)
 
gp: K = bnfinit(x^22 - x^21 - 65*x^20 - 27*x^19 + 1709*x^18 + 2739*x^17 - 21240*x^16 - 58584*x^15 + 107774*x^14 + 530058*x^13 + 63350*x^12 - 2109350*x^11 - 2370346*x^10 + 3053082*x^9 + 7232279*x^8 + 1113268*x^7 - 7233811*x^6 - 5207551*x^5 + 1572041*x^4 + 2696745*x^3 + 394142*x^2 - 391563*x - 116981, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} - 65 x^{20} - 27 x^{19} + 1709 x^{18} + 2739 x^{17} - 21240 x^{16} - 58584 x^{15} + 107774 x^{14} + 530058 x^{13} + 63350 x^{12} - 2109350 x^{11} - 2370346 x^{10} + 3053082 x^{9} + 7232279 x^{8} + 1113268 x^{7} - 7233811 x^{6} - 5207551 x^{5} + 1572041 x^{4} + 2696745 x^{3} + 394142 x^{2} - 391563 x - 116981 \)

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:  \(39437071573367006679286233687044038294749249=3^{11}\cdot 67^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.86$
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:  $3, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(201=3\cdot 67\)
Dirichlet character group:    $\lbrace$$\chi_{201}(64,·)$, $\chi_{201}(1,·)$, $\chi_{201}(196,·)$, $\chi_{201}(5,·)$, $\chi_{201}(193,·)$, $\chi_{201}(8,·)$, $\chi_{201}(137,·)$, $\chi_{201}(76,·)$, $\chi_{201}(82,·)$, $\chi_{201}(148,·)$, $\chi_{201}(22,·)$, $\chi_{201}(25,·)$, $\chi_{201}(91,·)$, $\chi_{201}(161,·)$, $\chi_{201}(40,·)$, $\chi_{201}(110,·)$, $\chi_{201}(176,·)$, $\chi_{201}(200,·)$, $\chi_{201}(179,·)$, $\chi_{201}(53,·)$, $\chi_{201}(119,·)$, $\chi_{201}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{37} a^{17} - \frac{2}{37} a^{16} + \frac{12}{37} a^{15} - \frac{11}{37} a^{14} + \frac{8}{37} a^{13} - \frac{5}{37} a^{12} + \frac{13}{37} a^{11} + \frac{17}{37} a^{10} + \frac{16}{37} a^{9} - \frac{10}{37} a^{8} - \frac{14}{37} a^{7} - \frac{5}{37} a^{6} + \frac{6}{37} a^{5} - \frac{14}{37} a^{4} + \frac{13}{37} a^{3} - \frac{5}{37} a^{2} - \frac{18}{37} a - \frac{2}{37}$, $\frac{1}{37} a^{18} + \frac{8}{37} a^{16} + \frac{13}{37} a^{15} - \frac{14}{37} a^{14} + \frac{11}{37} a^{13} + \frac{3}{37} a^{12} + \frac{6}{37} a^{11} + \frac{13}{37} a^{10} - \frac{15}{37} a^{9} + \frac{3}{37} a^{8} + \frac{4}{37} a^{7} - \frac{4}{37} a^{6} - \frac{2}{37} a^{5} - \frac{15}{37} a^{4} - \frac{16}{37} a^{3} + \frac{9}{37} a^{2} - \frac{1}{37} a - \frac{4}{37}$, $\frac{1}{37} a^{19} - \frac{8}{37} a^{16} + \frac{1}{37} a^{15} - \frac{12}{37} a^{14} + \frac{13}{37} a^{13} + \frac{9}{37} a^{12} - \frac{17}{37} a^{11} - \frac{3}{37} a^{10} - \frac{14}{37} a^{9} + \frac{10}{37} a^{8} - \frac{3}{37} a^{7} + \frac{1}{37} a^{6} + \frac{11}{37} a^{5} - \frac{15}{37} a^{4} + \frac{16}{37} a^{3} + \frac{2}{37} a^{2} - \frac{8}{37} a + \frac{16}{37}$, $\frac{1}{6031} a^{20} - \frac{25}{6031} a^{19} + \frac{13}{6031} a^{18} - \frac{3}{6031} a^{17} + \frac{2330}{6031} a^{16} + \frac{2338}{6031} a^{15} + \frac{1260}{6031} a^{14} - \frac{207}{6031} a^{13} - \frac{3003}{6031} a^{12} + \frac{1786}{6031} a^{11} + \frac{1573}{6031} a^{10} - \frac{976}{6031} a^{9} - \frac{449}{6031} a^{8} - \frac{608}{6031} a^{7} - \frac{2570}{6031} a^{6} + \frac{2341}{6031} a^{5} - \frac{96}{6031} a^{4} + \frac{2123}{6031} a^{3} - \frac{2075}{6031} a^{2} - \frac{1441}{6031} a - \frac{1905}{6031}$, $\frac{1}{6578144611528739693887127719} a^{21} - \frac{382464820073392314200654}{6578144611528739693887127719} a^{20} - \frac{23510102032099785253892299}{6578144611528739693887127719} a^{19} - \frac{106677589738394555161828}{177787692203479451186138587} a^{18} + \frac{61540947379205291513208526}{6578144611528739693887127719} a^{17} + \frac{374654346418917356822041774}{6578144611528739693887127719} a^{16} + \frac{1076810937232620527017239261}{6578144611528739693887127719} a^{15} + \frac{1651940490907362375636697711}{6578144611528739693887127719} a^{14} + \frac{559696310635033358491416649}{6578144611528739693887127719} a^{13} - \frac{1949312120661058960318799799}{6578144611528739693887127719} a^{12} - \frac{996899494515906535977296941}{6578144611528739693887127719} a^{11} + \frac{2554956687314247119315847765}{6578144611528739693887127719} a^{10} + \frac{1400211490380659939087718264}{6578144611528739693887127719} a^{9} - \frac{2961060105422468517224379375}{6578144611528739693887127719} a^{8} + \frac{1944135829577892440105742801}{6578144611528739693887127719} a^{7} + \frac{260758239107092886301584831}{6578144611528739693887127719} a^{6} + \frac{2189432081467543572764416612}{6578144611528739693887127719} a^{5} + \frac{1521605334036519759406615766}{6578144611528739693887127719} a^{4} + \frac{291160431269314099075792200}{6578144611528739693887127719} a^{3} + \frac{539688109879897562114936547}{6578144611528739693887127719} a^{2} - \frac{509697101061755813330951808}{6578144611528739693887127719} a + \frac{115483758791211472428235291}{6578144611528739693887127719}$

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:  \( 649525406371000 \) (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

\(\Q(\sqrt{201}) \), 11.11.1822837804551761449.1

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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ R ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{11}$ $22$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{22}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ $22$

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
67Data not computed