Properties

Label 23.23.477...489.1
Degree $23$
Signature $[23, 0]$
Discriminant $4.779\times 10^{65}$
Root discriminant $717.18$
Ramified prime $967$
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 - 462*x^21 + 519*x^20 + 78046*x^19 - 96292*x^18 - 6163501*x^17 + 7373917*x^16 + 250455980*x^15 - 290053856*x^14 - 5501340108*x^13 + 6875865393*x^12 + 66926124686*x^11 - 96263348909*x^10 - 438046106473*x^9 + 740824871406*x^8 + 1383038185176*x^7 - 2863260767644*x^6 - 1570567685623*x^5 + 5193710248505*x^4 - 862947136357*x^3 - 3368621429671*x^2 + 2154713129200*x - 341763031283)
 
gp: K = bnfinit(x^23 - x^22 - 462*x^21 + 519*x^20 + 78046*x^19 - 96292*x^18 - 6163501*x^17 + 7373917*x^16 + 250455980*x^15 - 290053856*x^14 - 5501340108*x^13 + 6875865393*x^12 + 66926124686*x^11 - 96263348909*x^10 - 438046106473*x^9 + 740824871406*x^8 + 1383038185176*x^7 - 2863260767644*x^6 - 1570567685623*x^5 + 5193710248505*x^4 - 862947136357*x^3 - 3368621429671*x^2 + 2154713129200*x - 341763031283, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-341763031283, 2154713129200, -3368621429671, -862947136357, 5193710248505, -1570567685623, -2863260767644, 1383038185176, 740824871406, -438046106473, -96263348909, 66926124686, 6875865393, -5501340108, -290053856, 250455980, 7373917, -6163501, -96292, 78046, 519, -462, -1, 1]);
 

\(x^{23} - x^{22} - 462 x^{21} + 519 x^{20} + 78046 x^{19} - 96292 x^{18} - 6163501 x^{17} + 7373917 x^{16} + 250455980 x^{15} - 290053856 x^{14} - 5501340108 x^{13} + 6875865393 x^{12} + 66926124686 x^{11} - 96263348909 x^{10} - 438046106473 x^{9} + 740824871406 x^{8} + 1383038185176 x^{7} - 2863260767644 x^{6} - 1570567685623 x^{5} + 5193710248505 x^{4} - 862947136357 x^{3} - 3368621429671 x^{2} + 2154713129200 x - 341763031283\)  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:  \(477\!\cdots\!489\)\(\medspace = 967^{22}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $717.18$
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:  $967$
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:  \(967\)
Dirichlet character group:    $\lbrace$$\chi_{967}(1,·)$, $\chi_{967}(196,·)$, $\chi_{967}(133,·)$, $\chi_{967}(641,·)$, $\chi_{967}(72,·)$, $\chi_{967}(714,·)$, $\chi_{967}(332,·)$, $\chi_{967}(916,·)$, $\chi_{967}(667,·)$, $\chi_{967}(474,·)$, $\chi_{967}(283,·)$, $\chi_{967}(157,·)$, $\chi_{967}(926,·)$, $\chi_{967}(69,·)$, $\chi_{967}(795,·)$, $\chi_{967}(873,·)$, $\chi_{967}(349,·)$, $\chi_{967}(696,·)$, $\chi_{967}(953,·)$, $\chi_{967}(187,·)$, $\chi_{967}(893,·)$, $\chi_{967}(574,·)$, $\chi_{967}(703,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{151} a^{19} - \frac{69}{151} a^{18} - \frac{19}{151} a^{17} - \frac{67}{151} a^{16} + \frac{69}{151} a^{15} - \frac{14}{151} a^{14} - \frac{60}{151} a^{13} + \frac{44}{151} a^{12} + \frac{21}{151} a^{11} - \frac{1}{151} a^{10} + \frac{49}{151} a^{9} - \frac{55}{151} a^{8} + \frac{30}{151} a^{7} + \frac{16}{151} a^{6} + \frac{37}{151} a^{5} - \frac{70}{151} a^{3} - \frac{69}{151} a^{2} + \frac{56}{151} a$, $\frac{1}{151} a^{20} + \frac{52}{151} a^{18} - \frac{19}{151} a^{17} - \frac{24}{151} a^{16} + \frac{66}{151} a^{15} + \frac{31}{151} a^{14} - \frac{19}{151} a^{13} + \frac{37}{151} a^{12} - \frac{62}{151} a^{11} - \frac{20}{151} a^{10} + \frac{4}{151} a^{9} + \frac{10}{151} a^{8} - \frac{28}{151} a^{7} - \frac{67}{151} a^{6} - \frac{14}{151} a^{5} - \frac{70}{151} a^{4} - \frac{67}{151} a^{3} - \frac{24}{151} a^{2} - \frac{62}{151} a$, $\frac{1}{151} a^{21} - \frac{55}{151} a^{18} + \frac{58}{151} a^{17} - \frac{74}{151} a^{16} + \frac{67}{151} a^{15} - \frac{46}{151} a^{14} - \frac{14}{151} a^{13} + \frac{66}{151} a^{12} - \frac{55}{151} a^{11} + \frac{56}{151} a^{10} + \frac{29}{151} a^{9} - \frac{37}{151} a^{8} + \frac{34}{151} a^{7} + \frac{60}{151} a^{6} - \frac{31}{151} a^{5} - \frac{67}{151} a^{4} - \frac{8}{151} a^{3} + \frac{53}{151} a^{2} - \frac{43}{151} a$, $\frac{1}{7154323715164870718340959796293814827822786071343889568069877412997898783312644773754880255607615432246055958198527722361331936336527} a^{22} + \frac{16628712852238660183462596690451762642774709094691090503876865245631254247153169567615188244391403457788080302046329821605490491482}{7154323715164870718340959796293814827822786071343889568069877412997898783312644773754880255607615432246055958198527722361331936336527} a^{21} - \frac{13174214121051995752605890637267926504693047638552770166635777246755529214638756009294680176177683568500897450204603918414890455758}{7154323715164870718340959796293814827822786071343889568069877412997898783312644773754880255607615432246055958198527722361331936336527} a^{20} + \frac{65379440333081034237103818080513992526880261642055610528049128773538122714728141057701903833565552777260644728380330185724252410}{47379627252747488200933508584727250515382689214197944159403161675482773399421488567913114275547122067854675219857799485836635339977} a^{19} + \frac{3488806919007788310244343147733619686105187272078891866044874899188936047114583640328235714725766496555043280885412974591417896204374}{7154323715164870718340959796293814827822786071343889568069877412997898783312644773754880255607615432246055958198527722361331936336527} a^{18} - \frac{1482487552840309506682949891290571272822564292947515520226550274136879018149730172865762027302732121570007245835668956882698525861947}{7154323715164870718340959796293814827822786071343889568069877412997898783312644773754880255607615432246055958198527722361331936336527} a^{17} - \frac{1238813310354266810703198055710951961892436353500936503352354953971681317269784567346357924778280409915810023558753681977713733494690}{7154323715164870718340959796293814827822786071343889568069877412997898783312644773754880255607615432246055958198527722361331936336527} a^{16} + \frac{3130337774632094618337163120182704094722992561532257600219772686568850508451119572870191873908943915887031447980803776510083831344895}{7154323715164870718340959796293814827822786071343889568069877412997898783312644773754880255607615432246055958198527722361331936336527} a^{15} - \frac{3163972446470427666290304452506735933341359464706600401936350030787450191448004099828577295584617650331772463783201287659598001436089}{7154323715164870718340959796293814827822786071343889568069877412997898783312644773754880255607615432246055958198527722361331936336527} a^{14} - \frac{1085761051516029188399918260650207827280414160916802943536994178817933567209819129628274045617171053215504223292125521783133342459288}{7154323715164870718340959796293814827822786071343889568069877412997898783312644773754880255607615432246055958198527722361331936336527} a^{13} + \frac{177566274023243933724134944556160928883254064910282529839406854865159927532740226280012638924539244789085914614842175558426198582799}{7154323715164870718340959796293814827822786071343889568069877412997898783312644773754880255607615432246055958198527722361331936336527} a^{12} + \frac{184299170468054584700879575112866870553709471234140481230972771964305580803632204771740018733851018763782245571237885933069287816818}{7154323715164870718340959796293814827822786071343889568069877412997898783312644773754880255607615432246055958198527722361331936336527} a^{11} - \frac{743618118452191929350977954033913433441182211696076435895215026922079543928572253648476354243531429737294631544691976480127872288374}{7154323715164870718340959796293814827822786071343889568069877412997898783312644773754880255607615432246055958198527722361331936336527} a^{10} - \frac{3242897351517269114278316713726685424256235571463192073305397685629873028668388541757507666113995512393734942121315803636681858120140}{7154323715164870718340959796293814827822786071343889568069877412997898783312644773754880255607615432246055958198527722361331936336527} a^{9} - \frac{941654260663993294844745469427422120469251489742997941291350672614876649683488534533926506434971646753749893495546347063739074222279}{7154323715164870718340959796293814827822786071343889568069877412997898783312644773754880255607615432246055958198527722361331936336527} a^{8} - \frac{1723372363286534101632840855610039607886207739413523287595367515575152607082138044283849735811183708939615379417010403927032610050323}{7154323715164870718340959796293814827822786071343889568069877412997898783312644773754880255607615432246055958198527722361331936336527} a^{7} + \frac{1521842667986888192012051223933668701304909172601270669621751651198646612105059451858196437948497131538215255215739144287428229862731}{7154323715164870718340959796293814827822786071343889568069877412997898783312644773754880255607615432246055958198527722361331936336527} a^{6} - \frac{2718725749964085389924239855257357862304182618639408578967997335542761968931808134697973276742045113833567940189098885005441928968651}{7154323715164870718340959796293814827822786071343889568069877412997898783312644773754880255607615432246055958198527722361331936336527} a^{5} - \frac{1602215066837710415196705886123759289164195502763383783426708395918116927325050991289940773819598888111132749644569686126814569567989}{7154323715164870718340959796293814827822786071343889568069877412997898783312644773754880255607615432246055958198527722361331936336527} a^{4} + \frac{2758631331660170852541202980211167858075771446261008440006362616547676955087907279722158534933806574024583150818122723483422200772915}{7154323715164870718340959796293814827822786071343889568069877412997898783312644773754880255607615432246055958198527722361331936336527} a^{3} - \frac{3392135399519242521127692946512978267712035327602154000091765585374276046427687355436954595944791617465937818938441864161584634887708}{7154323715164870718340959796293814827822786071343889568069877412997898783312644773754880255607615432246055958198527722361331936336527} a^{2} + \frac{2186642097582517307216206524792540190446646807697657697519768404658094608416828062613777861031182068662110810825267255877939645628444}{7154323715164870718340959796293814827822786071343889568069877412997898783312644773754880255607615432246055958198527722361331936336527} a + \frac{13030198824292055741350954289649859588160279691210392225454324302956280993334412631362220213078493344784404704900972231722575502}{261765896424019271828361925882470997322556294001093614140348959533053996681886677170790686605232718606931907292032041358213454917}$  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:  \( 27464009362584775000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{23}\cdot(2\pi)^{0}\cdot 27464009362584775000000000 \cdot 1}{2\sqrt{477949960252640343082669666642744217085836889062320413931701625489}}\approx 0.166622109369944$ (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
967Data not computed