Properties

Label 23.23.294...281.1
Degree $23$
Signature $[23, 0]$
Discriminant $2.941\times 10^{62}$
Root discriminant $520.02$
Ramified prime $691$
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 - 330*x^21 + 491*x^20 + 46010*x^19 - 87998*x^18 - 3534734*x^17 + 8097381*x^16 + 163274386*x^15 - 432276476*x^14 - 4617725871*x^13 + 13958868541*x^12 + 76934101895*x^11 - 269759119969*x^10 - 654647210801*x^9 + 2901393808025*x^8 + 1396549312920*x^7 - 14112769866979*x^6 + 10842020004862*x^5 + 12214116095559*x^4 - 17293268074281*x^3 + 3381350907752*x^2 + 1669609496979*x - 220900980203)
 
gp: K = bnfinit(x^23 - x^22 - 330*x^21 + 491*x^20 + 46010*x^19 - 87998*x^18 - 3534734*x^17 + 8097381*x^16 + 163274386*x^15 - 432276476*x^14 - 4617725871*x^13 + 13958868541*x^12 + 76934101895*x^11 - 269759119969*x^10 - 654647210801*x^9 + 2901393808025*x^8 + 1396549312920*x^7 - 14112769866979*x^6 + 10842020004862*x^5 + 12214116095559*x^4 - 17293268074281*x^3 + 3381350907752*x^2 + 1669609496979*x - 220900980203, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-220900980203, 1669609496979, 3381350907752, -17293268074281, 12214116095559, 10842020004862, -14112769866979, 1396549312920, 2901393808025, -654647210801, -269759119969, 76934101895, 13958868541, -4617725871, -432276476, 163274386, 8097381, -3534734, -87998, 46010, 491, -330, -1, 1]);
 

\(x^{23} - x^{22} - 330 x^{21} + 491 x^{20} + 46010 x^{19} - 87998 x^{18} - 3534734 x^{17} + 8097381 x^{16} + 163274386 x^{15} - 432276476 x^{14} - 4617725871 x^{13} + 13958868541 x^{12} + 76934101895 x^{11} - 269759119969 x^{10} - 654647210801 x^{9} + 2901393808025 x^{8} + 1396549312920 x^{7} - 14112769866979 x^{6} + 10842020004862 x^{5} + 12214116095559 x^{4} - 17293268074281 x^{3} + 3381350907752 x^{2} + 1669609496979 x - 220900980203\)  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:  \(294\!\cdots\!281\)\(\medspace = 691^{22}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $520.02$
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:  $691$
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:  \(691\)
Dirichlet character group:    $\lbrace$$\chi_{691}(1,·)$, $\chi_{691}(195,·)$, $\chi_{691}(583,·)$, $\chi_{691}(329,·)$, $\chi_{691}(333,·)$, $\chi_{691}(399,·)$, $\chi_{691}(528,·)$, $\chi_{691}(659,·)$, $\chi_{691}(20,·)$, $\chi_{691}(608,·)$, $\chi_{691}(271,·)$, $\chi_{691}(604,·)$, $\chi_{691}(413,·)$, $\chi_{691}(670,·)$, $\chi_{691}(672,·)$, $\chi_{691}(400,·)$, $\chi_{691}(361,·)$, $\chi_{691}(51,·)$, $\chi_{691}(310,·)$, $\chi_{691}(311,·)$, $\chi_{691}(441,·)$, $\chi_{691}(379,·)$, $\chi_{691}(445,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{197} a^{20} - \frac{39}{197} a^{19} - \frac{95}{197} a^{18} + \frac{25}{197} a^{17} - \frac{29}{197} a^{16} - \frac{59}{197} a^{15} + \frac{35}{197} a^{14} + \frac{72}{197} a^{13} + \frac{98}{197} a^{12} - \frac{7}{197} a^{11} - \frac{7}{197} a^{10} - \frac{68}{197} a^{9} + \frac{69}{197} a^{8} - \frac{64}{197} a^{7} + \frac{43}{197} a^{6} - \frac{64}{197} a^{5} - \frac{80}{197} a^{4} - \frac{33}{197} a^{3} - \frac{11}{197} a^{2} + \frac{47}{197} a - \frac{96}{197}$, $\frac{1}{1981229} a^{21} - \frac{409}{1981229} a^{20} - \frac{244326}{1981229} a^{19} - \frac{47171}{1981229} a^{18} + \frac{575023}{1981229} a^{17} - \frac{340580}{1981229} a^{16} - \frac{125097}{1981229} a^{15} + \frac{35584}{1981229} a^{14} + \frac{57380}{1981229} a^{13} + \frac{132562}{1981229} a^{12} + \frac{292370}{1981229} a^{11} + \frac{344908}{1981229} a^{10} - \frac{437130}{1981229} a^{9} - \frac{976907}{1981229} a^{8} + \frac{496720}{1981229} a^{7} - \frac{785062}{1981229} a^{6} + \frac{336436}{1981229} a^{5} + \frac{581758}{1981229} a^{4} - \frac{27201}{1981229} a^{3} - \frac{205885}{1981229} a^{2} - \frac{700879}{1981229} a + \frac{3489}{17533}$, $\frac{1}{134711639425363670040712040476114962294434666705570197252723994720033253300421571933459383136373897023226162825708548965536337} a^{22} - \frac{14961161292159348786766851157410831751834333335908102836239371552769588405881025482935790208856814093561930877001931143}{134711639425363670040712040476114962294434666705570197252723994720033253300421571933459383136373897023226162825708548965536337} a^{21} - \frac{105288195290455620373270801044644340590482114261622330017888017330643957588064098540746957325505972046568296886629163977077}{134711639425363670040712040476114962294434666705570197252723994720033253300421571933459383136373897023226162825708548965536337} a^{20} - \frac{23477267926113800326921712692420583055432731140827961434546741289304209112589212395788759056031861871377598294444404125166208}{134711639425363670040712040476114962294434666705570197252723994720033253300421571933459383136373897023226162825708548965536337} a^{19} + \frac{39175361531648646393868260865396442729148783385904560459626090794926673910672494436315752559009241493085533762623783311749658}{134711639425363670040712040476114962294434666705570197252723994720033253300421571933459383136373897023226162825708548965536337} a^{18} - \frac{2955229309743410700690901228738459572599814157715127328335504984726925405678758496142672078598998045274549835384617820110268}{134711639425363670040712040476114962294434666705570197252723994720033253300421571933459383136373897023226162825708548965536337} a^{17} - \frac{25464566318519219179178301816189472776456609014211507938681167604152897250550789202723162362720538913857948259232066462923140}{134711639425363670040712040476114962294434666705570197252723994720033253300421571933459383136373897023226162825708548965536337} a^{16} + \frac{29267846601010899169816693083858704490676069832722105807217368629690018253595221096312135497599255497741340568635886344827986}{134711639425363670040712040476114962294434666705570197252723994720033253300421571933459383136373897023226162825708548965536337} a^{15} - \frac{26162036431317159849733154656953698209092862438449435191683937269779934424557944699086035954831883559777452730218897862370666}{134711639425363670040712040476114962294434666705570197252723994720033253300421571933459383136373897023226162825708548965536337} a^{14} - \frac{3584437972352037451523171853057522608903491827536616741088147932450063416216352877673536042123368442361412086687030522223774}{134711639425363670040712040476114962294434666705570197252723994720033253300421571933459383136373897023226162825708548965536337} a^{13} + \frac{67177607701829348223708107970563519087711269257497041830559831336569130708889181166027261415954885621310972654556985874685000}{134711639425363670040712040476114962294434666705570197252723994720033253300421571933459383136373897023226162825708548965536337} a^{12} + \frac{64595430384337830853461734896685526985279185894218795752350973353947504621436102746864835865085712355378479135219812289671309}{134711639425363670040712040476114962294434666705570197252723994720033253300421571933459383136373897023226162825708548965536337} a^{11} + \frac{5989501389764505942115403337810858783202167326138826298214003839641358638859471017376014771629944773846355923151035381632982}{134711639425363670040712040476114962294434666705570197252723994720033253300421571933459383136373897023226162825708548965536337} a^{10} + \frac{4850473562570017244329641040333149734031981765777110357545189637476293506039618819424545321490137348452303018547187368967113}{134711639425363670040712040476114962294434666705570197252723994720033253300421571933459383136373897023226162825708548965536337} a^{9} - \frac{37188189428800268269593885650627276709729806847090200783745395019080956569342874182556209423716777465555493384227226993130142}{134711639425363670040712040476114962294434666705570197252723994720033253300421571933459383136373897023226162825708548965536337} a^{8} - \frac{44940465953046221049712478361985791257721162077804839399672353880555543199414103661594239201725895296948027596211950271170203}{134711639425363670040712040476114962294434666705570197252723994720033253300421571933459383136373897023226162825708548965536337} a^{7} + \frac{35772244342547210997781563354052905081003880287991476277116139343653978306003657511527334228809143871568198761012569878906204}{134711639425363670040712040476114962294434666705570197252723994720033253300421571933459383136373897023226162825708548965536337} a^{6} + \frac{27138970474980605716169891040655617455372034534758855917933638926569395674684039448976226878709407330736288627249461583362155}{134711639425363670040712040476114962294434666705570197252723994720033253300421571933459383136373897023226162825708548965536337} a^{5} - \frac{28574429797478364830929435891776104612028048886019537898097702098239061250737403691154807180849656213470268151802812708344977}{134711639425363670040712040476114962294434666705570197252723994720033253300421571933459383136373897023226162825708548965536337} a^{4} - \frac{44451388907249249678679533107179376646774146124758375275124412534808661988682062920138950083142792614616493925714535852148750}{134711639425363670040712040476114962294434666705570197252723994720033253300421571933459383136373897023226162825708548965536337} a^{3} + \frac{58247293148375491742697114437874714160802079203614190508911495862380495769443895476485949319431700147973324794734409117184995}{134711639425363670040712040476114962294434666705570197252723994720033253300421571933459383136373897023226162825708548965536337} a^{2} + \frac{45183248226388135432878561854940759462931230992457146888036269313979112441361382948246522819087585803320015180375727800864629}{134711639425363670040712040476114962294434666705570197252723994720033253300421571933459383136373897023226162825708548965536337} a - \frac{582209363778276781807673990184643889409191301991157678045318070712698336331102472203472764877022418645935028326287833425209}{1192138401994368761422230446691282852163138643412125639404637121416223480534704176402295425985609708170143033855827866951649}$  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:  \( 921008224906832800000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{23}\cdot(2\pi)^{0}\cdot 921008224906832800000000 \cdot 1}{2\sqrt{294114910567428962511991419268717959339196150815159804184604281}}\approx 0.225250047948446$ (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
691Data not computed