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Properties

Label 23.1.86500494174...8983.1
Degree $23$
Signature $[1, 11]$
Discriminant $-\,23^{33}$
Root discriminant $89.90$
Ramified prime $23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{23}$ (as 23T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1437601, -386078, 19385596, -75769360, 67771087, -24771506, 58090042, -57748676, 24074675, -6694426, 3237457, -1033942, 165048, -105455, 48346, 3381, -7567, 3680, -1334, 184, -23, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 23*x^20 + 184*x^19 - 1334*x^18 + 3680*x^17 - 7567*x^16 + 3381*x^15 + 48346*x^14 - 105455*x^13 + 165048*x^12 - 1033942*x^11 + 3237457*x^10 - 6694426*x^9 + 24074675*x^8 - 57748676*x^7 + 58090042*x^6 - 24771506*x^5 + 67771087*x^4 - 75769360*x^3 + 19385596*x^2 - 386078*x + 1437601)
 
gp: K = bnfinit(x^23 - 23*x^20 + 184*x^19 - 1334*x^18 + 3680*x^17 - 7567*x^16 + 3381*x^15 + 48346*x^14 - 105455*x^13 + 165048*x^12 - 1033942*x^11 + 3237457*x^10 - 6694426*x^9 + 24074675*x^8 - 57748676*x^7 + 58090042*x^6 - 24771506*x^5 + 67771087*x^4 - 75769360*x^3 + 19385596*x^2 - 386078*x + 1437601, 1)
 

Normalized defining polynomial

\( x^{23} - 23 x^{20} + 184 x^{19} - 1334 x^{18} + 3680 x^{17} - 7567 x^{16} + 3381 x^{15} + 48346 x^{14} - 105455 x^{13} + 165048 x^{12} - 1033942 x^{11} + 3237457 x^{10} - 6694426 x^{9} + 24074675 x^{8} - 57748676 x^{7} + 58090042 x^{6} - 24771506 x^{5} + 67771087 x^{4} - 75769360 x^{3} + 19385596 x^{2} - 386078 x + 1437601 \)

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

Invariants

Degree:  $23$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 11]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-865004941741938633917747707002884268046728983=-\,23^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $89.90$
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:  $23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{15} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{16} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{35} a^{17} - \frac{2}{35} a^{16} - \frac{3}{35} a^{15} + \frac{2}{35} a^{14} + \frac{1}{35} a^{12} + \frac{2}{35} a^{11} - \frac{16}{35} a^{10} - \frac{11}{35} a^{9} + \frac{1}{35} a^{8} + \frac{2}{5} a^{7} + \frac{12}{35} a^{6} + \frac{8}{35} a^{5} + \frac{13}{35} a^{4} + \frac{1}{7} a^{2} + \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{12425} a^{18} + \frac{46}{12425} a^{17} + \frac{874}{12425} a^{16} + \frac{187}{12425} a^{15} - \frac{1143}{12425} a^{14} + \frac{1121}{12425} a^{13} - \frac{874}{12425} a^{12} - \frac{2979}{12425} a^{11} + \frac{313}{12425} a^{10} - \frac{3761}{12425} a^{9} - \frac{2654}{12425} a^{8} - \frac{158}{497} a^{7} - \frac{726}{2485} a^{6} + \frac{3596}{12425} a^{5} - \frac{713}{12425} a^{4} - \frac{2221}{12425} a^{3} - \frac{4078}{12425} a^{2} + \frac{6043}{12425} a - \frac{5414}{12425}$, $\frac{1}{12425} a^{19} - \frac{177}{12425} a^{17} + \frac{14}{1775} a^{16} - \frac{103}{2485} a^{15} + \frac{1159}{12425} a^{14} - \frac{51}{2485} a^{13} + \frac{29}{355} a^{12} - \frac{4653}{12425} a^{11} - \frac{5379}{12425} a^{10} + \frac{2082}{12425} a^{9} + \frac{4889}{12425} a^{8} - \frac{34}{497} a^{7} - \frac{3019}{12425} a^{6} - \frac{6024}{12425} a^{5} - \frac{754}{1775} a^{4} + \frac{6143}{12425} a^{3} - \frac{4814}{12425} a^{2} - \frac{4007}{12425} a - \frac{6201}{12425}$, $\frac{1}{12425} a^{20} + \frac{3}{497} a^{17} - \frac{136}{1775} a^{16} - \frac{887}{12425} a^{15} - \frac{216}{12425} a^{14} + \frac{632}{12425} a^{13} - \frac{1021}{12425} a^{12} + \frac{2678}{12425} a^{11} + \frac{1748}{12425} a^{10} + \frac{3042}{12425} a^{9} + \frac{3317}{12425} a^{8} - \frac{1399}{12425} a^{7} + \frac{3956}{12425} a^{6} - \frac{98}{1775} a^{5} + \frac{4902}{12425} a^{4} - \frac{2816}{12425} a^{3} + \frac{6197}{12425} a^{2} - \frac{673}{2485} a - \frac{3683}{12425}$, $\frac{1}{11617375} a^{21} - \frac{347}{11617375} a^{20} + \frac{333}{11617375} a^{19} + \frac{333}{11617375} a^{18} - \frac{18218}{2323475} a^{17} - \frac{80526}{1659625} a^{16} - \frac{20648}{683375} a^{15} - \frac{9193}{163625} a^{14} + \frac{46103}{11617375} a^{13} - \frac{73632}{1056125} a^{12} + \frac{3874686}{11617375} a^{11} - \frac{5251947}{11617375} a^{10} + \frac{2327211}{11617375} a^{9} + \frac{5104192}{11617375} a^{8} + \frac{3434124}{11617375} a^{7} - \frac{812796}{2323475} a^{6} + \frac{90952}{331925} a^{5} + \frac{1652327}{11617375} a^{4} - \frac{27029}{331925} a^{3} + \frac{343198}{2323475} a^{2} - \frac{656542}{2323475} a - \frac{196844}{1056125}$, $\frac{1}{2461648796411609512235684626517239192903941282592887871768946015296800625} a^{22} + \frac{717919717207772954988950749474727730815186122890201966118887836}{22583933912033114791153069968048065989944415436631998823568312066943125} a^{21} + \frac{853554757478822634488790126039858661514328991478402954761188980014}{22583933912033114791153069968048065989944415436631998823568312066943125} a^{20} - \frac{21659351521636686841759737754774999308426623841454111258533688654149}{2461648796411609512235684626517239192903941282592887871768946015296800625} a^{19} - \frac{31025228949168194547353229797284243677282143941091080400558162971342}{2461648796411609512235684626517239192903941282592887871768946015296800625} a^{18} + \frac{19458250793229746064590924326802502140028037448958744828489063677306658}{2461648796411609512235684626517239192903941282592887871768946015296800625} a^{17} + \frac{1350854969754432918183499194819003464957251626625234196635266794359553}{50237730539012439025218053602392636589876352705977303505488694189730625} a^{16} - \frac{789990115237838555727067932769994205366463309804787944912363039631361}{50237730539012439025218053602392636589876352705977303505488694189730625} a^{15} - \frac{82550543627437307150626988424691399185722778238081827024721646447533}{1152997094337990403857463525300814610259457275219151227994822489600375} a^{14} + \frac{5020162742340958947724243044696471770875370521387616329696965392239641}{223786254219237228385062238774294472082176480235717079251722365026981875} a^{13} - \frac{221496429840752319624995560486727397486860271648460945143263071652577706}{2461648796411609512235684626517239192903941282592887871768946015296800625} a^{12} + \frac{481043938227295801090895268234547747222471197986910505412459145780623204}{2461648796411609512235684626517239192903941282592887871768946015296800625} a^{11} - \frac{1057395824741148196312791109670645778340477993007085208774772054739610071}{2461648796411609512235684626517239192903941282592887871768946015296800625} a^{10} - \frac{1142865039647050597542513852252333691515585386091541211346826137124211072}{2461648796411609512235684626517239192903941282592887871768946015296800625} a^{9} - \frac{7234959067927394525729048047615218940948939728333564463261454283819662}{144802870377153500719746154501014070170820075446640463045232118546870625} a^{8} - \frac{197453557348580125146182799218501290114066859468316150927248062482102796}{2461648796411609512235684626517239192903941282592887871768946015296800625} a^{7} + \frac{4256993490973321206507137368828069634717221985543559247857166710382963}{11449529285635393080165975007056926478622982709734362194274167513008375} a^{6} + \frac{572081992844176198450487433756948805110721718279504469529501346073320197}{2461648796411609512235684626517239192903941282592887871768946015296800625} a^{5} + \frac{391343694795237921818551024156960265011120317338820468188236910714538172}{2461648796411609512235684626517239192903941282592887871768946015296800625} a^{4} + \frac{187373614973337101652003910089878544580474374929414398042355687068156928}{492329759282321902447136925303447838580788256518577574353789203059360125} a^{3} - \frac{2071862662224077993015623006335801231035233272448245512269815240129951}{98465951856464380489427385060689567716157651303715514870757840611872025} a^{2} + \frac{90862712593420715254229004591919598190798217284881282931708493870905311}{223786254219237228385062238774294472082176480235717079251722365026981875} a + \frac{335266723755056727675337719404681206405112223745556740184290258893074}{2053084901093919526468460906186187817267674130602908983960755642449375}$

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:  $11$
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:  \( 35038470243900 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{23}$ (as 23T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 46
The 13 conjugacy class representatives for $D_{23}$
Character table for $D_{23}$

Intermediate fields

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

Sibling fields

Galois closure: 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 $23$ $23$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $23$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ R $23$ $23$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $23$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $23$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $23$

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
23Data not computed