Properties

Label 23.23.542...529.1
Degree $23$
Signature $[23, 0]$
Discriminant $5.427\times 10^{53}$
Root discriminant $216.91$
Ramified prime $277$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{23}$ (as 23T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^23 - x^22 - 132*x^21 + 269*x^20 + 6722*x^19 - 19118*x^18 - 167488*x^17 + 601226*x^16 + 2127300*x^15 - 9664345*x^14 - 12698450*x^13 + 83446410*x^12 + 19888027*x^11 - 390244377*x^10 + 110929757*x^9 + 994369765*x^8 - 511651057*x^7 - 1370318108*x^6 + 775297206*x^5 + 972267257*x^4 - 466392148*x^3 - 312747839*x^2 + 85444683*x + 38737459)
 
gp: K = bnfinit(x^23 - x^22 - 132*x^21 + 269*x^20 + 6722*x^19 - 19118*x^18 - 167488*x^17 + 601226*x^16 + 2127300*x^15 - 9664345*x^14 - 12698450*x^13 + 83446410*x^12 + 19888027*x^11 - 390244377*x^10 + 110929757*x^9 + 994369765*x^8 - 511651057*x^7 - 1370318108*x^6 + 775297206*x^5 + 972267257*x^4 - 466392148*x^3 - 312747839*x^2 + 85444683*x + 38737459, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![38737459, 85444683, -312747839, -466392148, 972267257, 775297206, -1370318108, -511651057, 994369765, 110929757, -390244377, 19888027, 83446410, -12698450, -9664345, 2127300, 601226, -167488, -19118, 6722, 269, -132, -1, 1]);
 

\(x^{23} - x^{22} - 132 x^{21} + 269 x^{20} + 6722 x^{19} - 19118 x^{18} - 167488 x^{17} + 601226 x^{16} + 2127300 x^{15} - 9664345 x^{14} - 12698450 x^{13} + 83446410 x^{12} + 19888027 x^{11} - 390244377 x^{10} + 110929757 x^{9} + 994369765 x^{8} - 511651057 x^{7} - 1370318108 x^{6} + 775297206 x^{5} + 972267257 x^{4} - 466392148 x^{3} - 312747839 x^{2} + 85444683 x + 38737459\)  Toggle raw display

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

Invariants

Degree:  $23$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[23, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(542693874230042671882983450092579839839306394717839529\)\(\medspace = 277^{22}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $216.91$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $277$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $23$
This field is Galois and abelian over $\Q$.
Conductor:  \(277\)
Dirichlet character group:    $\lbrace$$\chi_{277}(256,·)$, $\chi_{277}(1,·)$, $\chi_{277}(131,·)$, $\chi_{277}(69,·)$, $\chi_{277}(264,·)$, $\chi_{277}(201,·)$, $\chi_{277}(203,·)$, $\chi_{277}(16,·)$, $\chi_{277}(273,·)$, $\chi_{277}(19,·)$, $\chi_{277}(84,·)$, $\chi_{277}(213,·)$, $\chi_{277}(218,·)$, $\chi_{277}(155,·)$, $\chi_{277}(157,·)$, $\chi_{277}(30,·)$, $\chi_{277}(27,·)$, $\chi_{277}(164,·)$, $\chi_{277}(169,·)$, $\chi_{277}(236,·)$, $\chi_{277}(175,·)$, $\chi_{277}(211,·)$, $\chi_{277}(52,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{1013} a^{21} - \frac{327}{1013} a^{20} + \frac{367}{1013} a^{19} - \frac{421}{1013} a^{18} + \frac{41}{1013} a^{17} + \frac{47}{1013} a^{16} + \frac{142}{1013} a^{15} - \frac{32}{1013} a^{14} + \frac{25}{1013} a^{13} + \frac{362}{1013} a^{12} + \frac{486}{1013} a^{11} - \frac{251}{1013} a^{10} + \frac{278}{1013} a^{9} + \frac{316}{1013} a^{8} + \frac{389}{1013} a^{7} + \frac{398}{1013} a^{6} - \frac{431}{1013} a^{5} + \frac{57}{1013} a^{4} - \frac{132}{1013} a^{3} + \frac{212}{1013} a^{2} - \frac{227}{1013} a - \frac{415}{1013}$, $\frac{1}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739} a^{22} - \frac{10515748538181309881151437316398131922250721372503825228361195713622046382414669758}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739} a^{21} - \frac{9262750842696334233313874233575406859264193138557553305157961269602227306979758913672}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739} a^{20} - \frac{2336234889593665803225998160382056290551003062029618818017279768752170193219081044896}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739} a^{19} - \frac{7448545303333453971189988367342905680122491869960027794022414472572639717355485043877}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739} a^{18} - \frac{1228155082069522763549217853616629560868489146954445182178642261841985699196098757054}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739} a^{17} + \frac{14790932328629458415830148987896101641493210518272821124689308049917816500994696752881}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739} a^{16} - \frac{16606398511390007372508321036683947251445295941982005771615138513305110468903909791307}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739} a^{15} - \frac{18845306050345569186292132251837098460303136416863523731867159045522469094638446821047}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739} a^{14} + \frac{4983269681705881165186670980322311381434422423231840094983142264649549448608676406919}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739} a^{13} - \frac{19394897448130012602457985571336336954644100154030189009205956579267298558835600727688}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739} a^{12} - \frac{816250505079287569436087382334046265751485425825176395096199715337252280202185115532}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739} a^{11} - \frac{8351248388359053695277025552028996903778922810327174107039253807303135431022593075849}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739} a^{10} + \frac{22802137051503419020610689469308228770053063508268828493229884816176001880360143440292}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739} a^{9} - \frac{5702456238633522521414508366376970022489041416521223700383493791074572666332333435216}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739} a^{8} + \frac{12607490638192053081673515000158142878576091510145385150713798439840773815861327086577}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739} a^{7} - \frac{5941471315445100325601392838531812081600245319755132390118650744172338214709153638677}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739} a^{6} - \frac{17868160330429865506375661306506015353069482955583304241593964652650253390458471080594}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739} a^{5} + \frac{2773627444529236465994298539702556863795798570852033171406535937575674343233890952092}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739} a^{4} - \frac{13436631819666901757476805677241762573304242809068136039275063826570971997760386358834}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739} a^{3} - \frac{19513464101539587219205642765385364405611334383580628700108318206561585602608045966070}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739} a^{2} + \frac{18982205009368513739690189252135484155071092905318144602749637382633258773159122730191}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739} a - \frac{5631540621346237655237554950801365452224068336437461380650510276202838666255693124773}{46337890549234016298171853702530849247972658950954890006735473863887995788008751327739}$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $22$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 17443707994029808000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{23}\cdot(2\pi)^{0}\cdot 17443707994029808000 \cdot 1}{2\sqrt{542693874230042671882983450092579839839306394717839529}}\approx 0.0993164649374737$ (assuming GRH)

Galois group

$C_{23}$ (as 23T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 23
The 23 conjugacy class representatives for $C_{23}$
Character table for $C_{23}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
277Data not computed