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Properties

Label 22.2.18498813244...6129.1
Degree $22$
Signature $[2, 10]$
Discriminant $23^{20}\cdot 47^{6}$
Root discriminant $49.43$
Ramified primes $23, 47$
Class number $32$ (GRH)
Class group $[2, 2, 2, 4]$ (GRH)
Galois group 22T23

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-247433, 973620, -1835434, 1534908, -1129686, 202039, -958963, 162572, -752931, -120476, -552925, -129306, -179279, -8523, -26905, 1764, -2477, 482, -55, 12, 16, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + 16*x^20 + 12*x^19 - 55*x^18 + 482*x^17 - 2477*x^16 + 1764*x^15 - 26905*x^14 - 8523*x^13 - 179279*x^12 - 129306*x^11 - 552925*x^10 - 120476*x^9 - 752931*x^8 + 162572*x^7 - 958963*x^6 + 202039*x^5 - 1129686*x^4 + 1534908*x^3 - 1835434*x^2 + 973620*x - 247433)
 
gp: K = bnfinit(x^22 - 2*x^21 + 16*x^20 + 12*x^19 - 55*x^18 + 482*x^17 - 2477*x^16 + 1764*x^15 - 26905*x^14 - 8523*x^13 - 179279*x^12 - 129306*x^11 - 552925*x^10 - 120476*x^9 - 752931*x^8 + 162572*x^7 - 958963*x^6 + 202039*x^5 - 1129686*x^4 + 1534908*x^3 - 1835434*x^2 + 973620*x - 247433, 1)
 

Normalized defining polynomial

\( x^{22} - 2 x^{21} + 16 x^{20} + 12 x^{19} - 55 x^{18} + 482 x^{17} - 2477 x^{16} + 1764 x^{15} - 26905 x^{14} - 8523 x^{13} - 179279 x^{12} - 129306 x^{11} - 552925 x^{10} - 120476 x^{9} - 752931 x^{8} + 162572 x^{7} - 958963 x^{6} + 202039 x^{5} - 1129686 x^{4} + 1534908 x^{3} - 1835434 x^{2} + 973620 x - 247433 \)

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:  \(18498813244074511635519301831008736129=23^{20}\cdot 47^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.43$
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:  $23, 47$
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}{321652142673446306620491292525233806696156067043839679423314607962500509601689} a^{21} + \frac{114881474888945617292163198801039950553751116468716932636684126762749474354121}{321652142673446306620491292525233806696156067043839679423314607962500509601689} a^{20} + \frac{120202115518455865232070576156993643142033493118722162824802361126936080727788}{321652142673446306620491292525233806696156067043839679423314607962500509601689} a^{19} - \frac{39935331817423353984497367525644778363733588132561457255962058652256226785467}{321652142673446306620491292525233806696156067043839679423314607962500509601689} a^{18} + \frac{49965874534678041205629678770404838344256528816915368004808336893179612285549}{321652142673446306620491292525233806696156067043839679423314607962500509601689} a^{17} + \frac{126812197253061594246668142889787365536447216278469757318965512653457700660202}{321652142673446306620491292525233806696156067043839679423314607962500509601689} a^{16} - \frac{92588265354863346879778133554169935218227244533112017706427032376877545943846}{321652142673446306620491292525233806696156067043839679423314607962500509601689} a^{15} + \frac{58764267234696177711041347053006275521080806188744641533541130749545666496186}{321652142673446306620491292525233806696156067043839679423314607962500509601689} a^{14} + \frac{55195292335116604110108063643961062666626823871041871625201741737321888052105}{321652142673446306620491292525233806696156067043839679423314607962500509601689} a^{13} + \frac{101793557327691316620708997957874665191634870803918383480601431848840877341108}{321652142673446306620491292525233806696156067043839679423314607962500509601689} a^{12} + \frac{2507964495266438626876061746987756661794165608633179571608810838610100741440}{6843662610073325672776410479260293759492682277528503817517332084308521480887} a^{11} + \frac{151811306790579599569737235994830419969219565935527009807190860915492079887949}{321652142673446306620491292525233806696156067043839679423314607962500509601689} a^{10} + \frac{48027589876152968457976909231260480375571148072796922562041965504231329569888}{321652142673446306620491292525233806696156067043839679423314607962500509601689} a^{9} + \frac{83031783874116421641163453568192591157857205043513072097060726959831806189417}{321652142673446306620491292525233806696156067043839679423314607962500509601689} a^{8} - \frac{46002444551446989873051175906746162963328552934113764378410955538137468441351}{321652142673446306620491292525233806696156067043839679423314607962500509601689} a^{7} - \frac{32203565652812576800859546577676335378432040547452948146089429273633707718811}{321652142673446306620491292525233806696156067043839679423314607962500509601689} a^{6} - \frac{45796933806153844925700135924900371488952724480047996940412935651906088839022}{321652142673446306620491292525233806696156067043839679423314607962500509601689} a^{5} - \frac{48812769328820757231558926656741912499833192832307614018328214883550514692923}{321652142673446306620491292525233806696156067043839679423314607962500509601689} a^{4} - \frac{130037422488317493959017566917167065087161938343950040521924538094243975570527}{321652142673446306620491292525233806696156067043839679423314607962500509601689} a^{3} + \frac{95909077737065002830335604805004067006725940730535415041548830014843473234583}{321652142673446306620491292525233806696156067043839679423314607962500509601689} a^{2} + \frac{25597895948535681346082039519187523671436870523259075576618861697339864957256}{321652142673446306620491292525233806696156067043839679423314607962500509601689} a - \frac{11962714409869806732367617367432720739799561859476340679562050267819524050187}{321652142673446306620491292525233806696156067043839679423314607962500509601689}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}$, which has order $32$ (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:  \( 113968730.636 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T23:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 11264
The 104 conjugacy class representatives for t22n23 are not computed
Character table for t22n23 is not computed

Intermediate fields

\(\Q(\zeta_{23})^+\)

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

Sibling fields

Degree 22 siblings: data not computed
Degree 44 siblings: 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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ R ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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}$
47Data not computed