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Properties

Label 22.22.3075626510...6437.1
Degree $22$
Signature $[22, 0]$
Discriminant $13^{11}\cdot 23^{20}$
Root discriminant $62.36$
Ramified primes $13, 23$
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![277, -6581, 37580, -39833, -255689, 767825, -284987, -1379831, 1491233, 606757, -1389694, 109894, 550877, -145468, -107702, 41692, 10055, -5542, -275, 357, -18, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 9*x^21 - 18*x^20 + 357*x^19 - 275*x^18 - 5542*x^17 + 10055*x^16 + 41692*x^15 - 107702*x^14 - 145468*x^13 + 550877*x^12 + 109894*x^11 - 1389694*x^10 + 606757*x^9 + 1491233*x^8 - 1379831*x^7 - 284987*x^6 + 767825*x^5 - 255689*x^4 - 39833*x^3 + 37580*x^2 - 6581*x + 277)
 
gp: K = bnfinit(x^22 - 9*x^21 - 18*x^20 + 357*x^19 - 275*x^18 - 5542*x^17 + 10055*x^16 + 41692*x^15 - 107702*x^14 - 145468*x^13 + 550877*x^12 + 109894*x^11 - 1389694*x^10 + 606757*x^9 + 1491233*x^8 - 1379831*x^7 - 284987*x^6 + 767825*x^5 - 255689*x^4 - 39833*x^3 + 37580*x^2 - 6581*x + 277, 1)
 

Normalized defining polynomial

\( x^{22} - 9 x^{21} - 18 x^{20} + 357 x^{19} - 275 x^{18} - 5542 x^{17} + 10055 x^{16} + 41692 x^{15} - 107702 x^{14} - 145468 x^{13} + 550877 x^{12} + 109894 x^{11} - 1389694 x^{10} + 606757 x^{9} + 1491233 x^{8} - 1379831 x^{7} - 284987 x^{6} + 767825 x^{5} - 255689 x^{4} - 39833 x^{3} + 37580 x^{2} - 6581 x + 277 \)

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:  \(3075626510913487571920886830127053316437=13^{11}\cdot 23^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.36$
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:  $13, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(299=13\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{299}(64,·)$, $\chi_{299}(1,·)$, $\chi_{299}(131,·)$, $\chi_{299}(196,·)$, $\chi_{299}(261,·)$, $\chi_{299}(12,·)$, $\chi_{299}(77,·)$, $\chi_{299}(142,·)$, $\chi_{299}(144,·)$, $\chi_{299}(209,·)$, $\chi_{299}(259,·)$, $\chi_{299}(25,·)$, $\chi_{299}(27,·)$, $\chi_{299}(220,·)$, $\chi_{299}(285,·)$, $\chi_{299}(246,·)$, $\chi_{299}(105,·)$, $\chi_{299}(170,·)$, $\chi_{299}(116,·)$, $\chi_{299}(118,·)$, $\chi_{299}(233,·)$, $\chi_{299}(248,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{367} a^{20} + \frac{175}{367} a^{19} - \frac{42}{367} a^{18} + \frac{91}{367} a^{17} - \frac{134}{367} a^{16} - \frac{158}{367} a^{15} - \frac{39}{367} a^{14} + \frac{19}{367} a^{13} + \frac{150}{367} a^{12} - \frac{161}{367} a^{11} - \frac{97}{367} a^{10} + \frac{81}{367} a^{9} - \frac{19}{367} a^{8} - \frac{127}{367} a^{7} - \frac{35}{367} a^{6} - \frac{81}{367} a^{5} - \frac{3}{367} a^{4} - \frac{84}{367} a^{3} - \frac{148}{367} a^{2} - \frac{80}{367} a + \frac{93}{367}$, $\frac{1}{641904678234023036736096420278574686835663577} a^{21} + \frac{576753779023090166251566255688927781647123}{641904678234023036736096420278574686835663577} a^{20} - \frac{52594101256677690818011092901589342578780060}{641904678234023036736096420278574686835663577} a^{19} + \frac{262228908332976170868509486300654696050349032}{641904678234023036736096420278574686835663577} a^{18} + \frac{116111569484736643506078294558533099086464985}{641904678234023036736096420278574686835663577} a^{17} - \frac{212978128000834684080272955997451350998778446}{641904678234023036736096420278574686835663577} a^{16} - \frac{166719380572895283989909371134984575891629332}{641904678234023036736096420278574686835663577} a^{15} + \frac{66521441526614842608852807445114938370116410}{641904678234023036736096420278574686835663577} a^{14} - \frac{209228099944680722340144775520732376491491026}{641904678234023036736096420278574686835663577} a^{13} - \frac{24677958370940791917121027477851438285516529}{641904678234023036736096420278574686835663577} a^{12} + \frac{226830827458245311510736365601447569337477304}{641904678234023036736096420278574686835663577} a^{11} + \frac{241874532594201628094945373596786321086880062}{641904678234023036736096420278574686835663577} a^{10} + \frac{198399459215204109848959543469548681705414552}{641904678234023036736096420278574686835663577} a^{9} + \frac{153040747775229326037187509821691197213569437}{641904678234023036736096420278574686835663577} a^{8} + \frac{70436921781925456779565106378279921920292623}{641904678234023036736096420278574686835663577} a^{7} + \frac{159728231117724302262358778193501056741938141}{641904678234023036736096420278574686835663577} a^{6} - \frac{278252748879797446274603356471464143987264656}{641904678234023036736096420278574686835663577} a^{5} - \frac{317682895240662054778867224805993505119176056}{641904678234023036736096420278574686835663577} a^{4} - \frac{75346719285738061517884691726902467644726247}{641904678234023036736096420278574686835663577} a^{3} - \frac{38832794858665735522307268033030271620046043}{641904678234023036736096420278574686835663577} a^{2} - \frac{157966794850664077907819914719368685432914147}{641904678234023036736096420278574686835663577} a + \frac{256507705218927077868966620368974298346826378}{641904678234023036736096420278574686835663577}$

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:  \( 3411497799150 \) (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

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{23})^+\)

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 $22$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ $22$ $22$ $22$ R ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ $22$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ $22$ $22$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ $22$

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
13Data not computed
$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}$