Properties

Label 23.23.126...801.1
Degree $23$
Signature $[23, 0]$
Discriminant $1.269\times 10^{61}$
Root discriminant $453.60$
Ramified prime $599$
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 - 286*x^21 + 61*x^20 + 32489*x^19 + 14967*x^18 - 1928206*x^17 - 2239956*x^16 + 65439786*x^15 + 128285990*x^14 - 1279328514*x^13 - 3758213062*x^12 + 12892455511*x^11 + 58719404143*x^10 - 31105194350*x^9 - 448954503771*x^8 - 491325793721*x^7 + 1084631540509*x^6 + 3053149451318*x^5 + 2305895458302*x^4 - 281144721703*x^3 - 1007901237378*x^2 - 154197339893*x + 118359867167)
 
gp: K = bnfinit(x^23 - x^22 - 286*x^21 + 61*x^20 + 32489*x^19 + 14967*x^18 - 1928206*x^17 - 2239956*x^16 + 65439786*x^15 + 128285990*x^14 - 1279328514*x^13 - 3758213062*x^12 + 12892455511*x^11 + 58719404143*x^10 - 31105194350*x^9 - 448954503771*x^8 - 491325793721*x^7 + 1084631540509*x^6 + 3053149451318*x^5 + 2305895458302*x^4 - 281144721703*x^3 - 1007901237378*x^2 - 154197339893*x + 118359867167, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![118359867167, -154197339893, -1007901237378, -281144721703, 2305895458302, 3053149451318, 1084631540509, -491325793721, -448954503771, -31105194350, 58719404143, 12892455511, -3758213062, -1279328514, 128285990, 65439786, -2239956, -1928206, 14967, 32489, 61, -286, -1, 1]);
 

\(x^{23} - x^{22} - 286 x^{21} + 61 x^{20} + 32489 x^{19} + 14967 x^{18} - 1928206 x^{17} - 2239956 x^{16} + 65439786 x^{15} + 128285990 x^{14} - 1279328514 x^{13} - 3758213062 x^{12} + 12892455511 x^{11} + 58719404143 x^{10} - 31105194350 x^{9} - 448954503771 x^{8} - 491325793721 x^{7} + 1084631540509 x^{6} + 3053149451318 x^{5} + 2305895458302 x^{4} - 281144721703 x^{3} - 1007901237378 x^{2} - 154197339893 x + 118359867167\)  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:  \(126\!\cdots\!801\)\(\medspace = 599^{22}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $453.60$
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:  $599$
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:  \(599\)
Dirichlet character group:    $\lbrace$$\chi_{599}(384,·)$, $\chi_{599}(1,·)$, $\chi_{599}(322,·)$, $\chi_{599}(323,·)$, $\chi_{599}(324,·)$, $\chi_{599}(441,·)$, $\chi_{599}(578,·)$, $\chi_{599}(18,·)$, $\chi_{599}(405,·)$, $\chi_{599}(151,·)$, $\chi_{599}(39,·)$, $\chi_{599}(103,·)$, $\chi_{599}(221,·)$, $\chi_{599}(480,·)$, $\chi_{599}(102,·)$, $\chi_{599}(423,·)$, $\chi_{599}(233,·)$, $\chi_{599}(426,·)$, $\chi_{599}(427,·)$, $\chi_{599}(498,·)$, $\chi_{599}(57,·)$, $\chi_{599}(379,·)$, $\chi_{599}(254,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{19} a^{13} - \frac{6}{19} a^{12} + \frac{3}{19} a^{11} - \frac{7}{19} a^{9} + \frac{4}{19} a^{8} - \frac{8}{19} a^{7} - \frac{2}{19} a^{6} + \frac{5}{19} a^{5} - \frac{5}{19} a^{4} + \frac{1}{19} a^{3} + \frac{9}{19} a^{2} + \frac{5}{19} a$, $\frac{1}{19} a^{14} + \frac{5}{19} a^{12} - \frac{1}{19} a^{11} - \frac{7}{19} a^{10} - \frac{3}{19} a^{8} + \frac{7}{19} a^{7} - \frac{7}{19} a^{6} + \frac{6}{19} a^{5} + \frac{9}{19} a^{4} - \frac{4}{19} a^{3} + \frac{2}{19} a^{2} - \frac{8}{19} a$, $\frac{1}{19} a^{15} - \frac{9}{19} a^{12} - \frac{3}{19} a^{11} - \frac{6}{19} a^{9} + \frac{6}{19} a^{8} - \frac{5}{19} a^{7} - \frac{3}{19} a^{6} + \frac{3}{19} a^{5} + \frac{2}{19} a^{4} - \frac{3}{19} a^{3} + \frac{4}{19} a^{2} - \frac{6}{19} a$, $\frac{1}{19} a^{16} + \frac{8}{19} a^{11} - \frac{6}{19} a^{10} - \frac{7}{19} a^{8} + \frac{1}{19} a^{7} + \frac{4}{19} a^{6} + \frac{9}{19} a^{5} + \frac{9}{19} a^{4} - \frac{6}{19} a^{3} - \frac{1}{19} a^{2} + \frac{7}{19} a$, $\frac{1}{19} a^{17} + \frac{8}{19} a^{12} - \frac{6}{19} a^{11} - \frac{7}{19} a^{9} + \frac{1}{19} a^{8} + \frac{4}{19} a^{7} + \frac{9}{19} a^{6} + \frac{9}{19} a^{5} - \frac{6}{19} a^{4} - \frac{1}{19} a^{3} + \frac{7}{19} a^{2}$, $\frac{1}{19} a^{18} + \frac{4}{19} a^{12} - \frac{5}{19} a^{11} - \frac{7}{19} a^{10} - \frac{9}{19} a^{8} - \frac{3}{19} a^{7} + \frac{6}{19} a^{6} - \frac{8}{19} a^{5} + \frac{1}{19} a^{4} - \frac{1}{19} a^{3} + \frac{4}{19} a^{2} - \frac{2}{19} a$, $\frac{1}{19} a^{19} - \frac{1}{19} a$, $\frac{1}{361} a^{20} + \frac{1}{361} a^{19} + \frac{6}{361} a^{18} + \frac{4}{361} a^{17} + \frac{1}{361} a^{16} - \frac{9}{361} a^{15} + \frac{4}{361} a^{14} - \frac{5}{361} a^{13} - \frac{98}{361} a^{12} + \frac{4}{19} a^{11} - \frac{3}{19} a^{10} + \frac{23}{361} a^{9} - \frac{162}{361} a^{8} + \frac{17}{361} a^{7} + \frac{47}{361} a^{6} + \frac{7}{361} a^{5} - \frac{4}{361} a^{4} + \frac{47}{361} a^{3} + \frac{91}{361} a^{2} - \frac{161}{361} a + \frac{9}{19}$, $\frac{1}{482235713} a^{21} + \frac{506172}{482235713} a^{20} - \frac{10573825}{482235713} a^{19} - \frac{8592775}{482235713} a^{18} - \frac{8479940}{482235713} a^{17} - \frac{9427684}{482235713} a^{16} - \frac{31020}{1335833} a^{15} - \frac{1568658}{482235713} a^{14} + \frac{3778377}{482235713} a^{13} - \frac{70926387}{482235713} a^{12} + \frac{10331432}{25380827} a^{11} - \frac{17909073}{482235713} a^{10} + \frac{169923646}{482235713} a^{9} - \frac{226873184}{482235713} a^{8} + \frac{204588377}{482235713} a^{7} + \frac{215883882}{482235713} a^{6} - \frac{108521581}{482235713} a^{5} + \frac{89534121}{482235713} a^{4} + \frac{5550649}{25380827} a^{3} - \frac{94702912}{482235713} a^{2} - \frac{43551595}{482235713} a - \frac{29816}{60287}$, $\frac{1}{933498861856968310907750652351989036348155790263090067093932176032391941587410163935012580461501049} a^{22} - \frac{645540470883649225755699286611344990029145075275387604069915257002690300045225093385168597}{933498861856968310907750652351989036348155790263090067093932176032391941587410163935012580461501049} a^{21} - \frac{453416331601488771356434435636142379737906164300295963729625222861734394462325249224959282969275}{933498861856968310907750652351989036348155790263090067093932176032391941587410163935012580461501049} a^{20} + \frac{9484607249406711162526085599376602552503391633568862687234606083985640023448981896428080318536790}{933498861856968310907750652351989036348155790263090067093932176032391941587410163935012580461501049} a^{19} - \frac{1284825625308078524485579150154104198948115098112241464992606335833652634131083092826620628396244}{933498861856968310907750652351989036348155790263090067093932176032391941587410163935012580461501049} a^{18} + \frac{5022273146666083055074117986719255568962390071806826757701787028820040415852371957981595619275110}{933498861856968310907750652351989036348155790263090067093932176032391941587410163935012580461501049} a^{17} + \frac{22699571891122412229823456878932534547334928538763625378029262883150292458649007011116885952746784}{933498861856968310907750652351989036348155790263090067093932176032391941587410163935012580461501049} a^{16} + \frac{19442725421927418241902851469215305388851506909423313076066687763280744577696130589602439436669651}{933498861856968310907750652351989036348155790263090067093932176032391941587410163935012580461501049} a^{15} - \frac{15548692191885904643874635743713477998228857621855252656167121589690369030650501271182661594171807}{933498861856968310907750652351989036348155790263090067093932176032391941587410163935012580461501049} a^{14} - \frac{12968655457262858918956201988376090086095570236625408067309640273355883401377272994143246806338609}{933498861856968310907750652351989036348155790263090067093932176032391941587410163935012580461501049} a^{13} - \frac{105233662005660974585968970678003162827502596179725763975575789792643513055248609331012808611095530}{933498861856968310907750652351989036348155790263090067093932176032391941587410163935012580461501049} a^{12} - \frac{27220126405573734983179334235685983039174080429920411084384701820625727035296853937224900212644970}{933498861856968310907750652351989036348155790263090067093932176032391941587410163935012580461501049} a^{11} + \frac{228389891027783625467723808394391015140738587051053479374231889468607220897494081287940195100860744}{933498861856968310907750652351989036348155790263090067093932176032391941587410163935012580461501049} a^{10} + \frac{160995188934897405333940838364198892134204212771308943664830312530165052994406824076148908101298829}{933498861856968310907750652351989036348155790263090067093932176032391941587410163935012580461501049} a^{9} + \frac{296767417712595315946268754510512318397479848601285991640874937535279589380030542763416736403077998}{933498861856968310907750652351989036348155790263090067093932176032391941587410163935012580461501049} a^{8} - \frac{440855670652562317100168973622746121057499906991314836493572642205574930499035787455193089272741400}{933498861856968310907750652351989036348155790263090067093932176032391941587410163935012580461501049} a^{7} - \frac{285756184143543804289073713419484177163062847005822651710287946800425964214087057205546857578947355}{933498861856968310907750652351989036348155790263090067093932176032391941587410163935012580461501049} a^{6} + \frac{219320458098110871353314655509122283423470280126976267204661098215165193631922313546130684699151377}{933498861856968310907750652351989036348155790263090067093932176032391941587410163935012580461501049} a^{5} + \frac{141565598392590450369397329718334359846415583820227393499816092408191484294924719770782854530230217}{933498861856968310907750652351989036348155790263090067093932176032391941587410163935012580461501049} a^{4} + \frac{154921642654569578355164704448379339393176374617173249031961333339467570436311717129459285076710342}{933498861856968310907750652351989036348155790263090067093932176032391941587410163935012580461501049} a^{3} - \frac{1863260604566893844738716998774831653983243629373413040365240676003889693390806431578989440215508}{5589813544053702460525453008095742732623687366844850701161270515164023602319821340928219044679647} a^{2} - \frac{314382443460021697655510741384639329254991847931774571085200047537003496208944953580263287466765809}{933498861856968310907750652351989036348155790263090067093932176032391941587410163935012580461501049} a - \frac{40699349156907659244388946450717077206015843610355399339019061431943336127964096356511365298937}{116701945475305452044974453350667462976391522723226661719456454060806593522616597566572394106951}$  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:  \( 342364856486088370000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{23}\cdot(2\pi)^{0}\cdot 342364856486088370000000 \cdot 1}{2\sqrt{12687910093209570367523581223700258577242701778943851443146801}}\approx 0.403138278184045$ (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 $23$ $23$ $23$ $23$ $23$ $23$ $23$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{23}$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$

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
599Data not computed