↑

Properties

Label 23.23.4344981734...2241.1
Degree $23$
Signature $[23, 0]$
Discriminant $23^{30}\cdot 47^{22}$
Root discriminant $2374.46$
Ramified primes $23, 47$
Class number $23$ (GRH)
Class group $[23]$ (GRH)
Galois group $C_{23}:C_{11}$ (as 23T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12862216057817467245, -78929282232647458634, 199912058984322799237, -269173314235796280477, 204445625295748936871, -81864894718917833350, 9167317157190582864, 5949260040976823570, -2826995282155771181, 474243077814357335, -7798660667836453, -8796561210816172, 1102283075184770, 6885232378569, -10294412120640, 512000507352, 29686323629, -2724444300, -23233933, 5382299, -6486, -4324, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 4324*x^21 - 6486*x^20 + 5382299*x^19 - 23233933*x^18 - 2724444300*x^17 + 29686323629*x^16 + 512000507352*x^15 - 10294412120640*x^14 + 6885232378569*x^13 + 1102283075184770*x^12 - 8796561210816172*x^11 - 7798660667836453*x^10 + 474243077814357335*x^9 - 2826995282155771181*x^8 + 5949260040976823570*x^7 + 9167317157190582864*x^6 - 81864894718917833350*x^5 + 204445625295748936871*x^4 - 269173314235796280477*x^3 + 199912058984322799237*x^2 - 78929282232647458634*x + 12862216057817467245)
 
gp: K = bnfinit(x^23 - 4324*x^21 - 6486*x^20 + 5382299*x^19 - 23233933*x^18 - 2724444300*x^17 + 29686323629*x^16 + 512000507352*x^15 - 10294412120640*x^14 + 6885232378569*x^13 + 1102283075184770*x^12 - 8796561210816172*x^11 - 7798660667836453*x^10 + 474243077814357335*x^9 - 2826995282155771181*x^8 + 5949260040976823570*x^7 + 9167317157190582864*x^6 - 81864894718917833350*x^5 + 204445625295748936871*x^4 - 269173314235796280477*x^3 + 199912058984322799237*x^2 - 78929282232647458634*x + 12862216057817467245, 1)
 

Normalized defining polynomial

\( x^{23} - 4324 x^{21} - 6486 x^{20} + 5382299 x^{19} - 23233933 x^{18} - 2724444300 x^{17} + 29686323629 x^{16} + 512000507352 x^{15} - 10294412120640 x^{14} + 6885232378569 x^{13} + 1102283075184770 x^{12} - 8796561210816172 x^{11} - 7798660667836453 x^{10} + 474243077814357335 x^{9} - 2826995282155771181 x^{8} + 5949260040976823570 x^{7} + 9167317157190582864 x^{6} - 81864894718917833350 x^{5} + 204445625295748936871 x^{4} - 269173314235796280477 x^{3} + 199912058984322799237 x^{2} - 78929282232647458634 x + 12862216057817467245 \)

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

Invariants

Degree:  $23$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[23, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(434498173468775074445875189356350743903198920600715375079386035604631941932241=23^{30}\cdot 47^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2374.46$
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:  $23, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947} a^{22} - \frac{3416332037831599162208451014675042526308208577426917958423306124602218996019710627695761237926324701157678503882427138093361829700022558166629449133978063076536571131229969344131261491778654736068415042064469720260336157487}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947} a^{21} - \frac{3901850341066539655875885961106025620852717857741833717350593331469903494669617415175237684624282289240553106403278601268201880792372132293784209839235846223465010335260365298906403275877790504284465788113333782520386662610}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947} a^{20} + \frac{659525551791490574936746953999774310213280773136210979681923119794667328155299989178836839998887988603167024387666149735805158229774195153828779719710072180851830936177300230476590652075366449822888536937458622443838089448}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947} a^{19} - \frac{3437419229944011518784422576340676238291732334950417327538223631548068469953253317712210020329112399116129594801763049283866320056336380895299435923824131990771649842677738465963375825964192009357680159042464706657930976938}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947} a^{18} + \frac{2547126157742674920068429470478174838625873262468822260515191149408639994237461355690528360660613431512326203910174768848300653397711254327923904988320714143776050601592180561871568462250188746364752613515617768600030315717}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947} a^{17} - \frac{1062018458674504939682468545966390163038065952105588878869562306720483755733280325138141415714830581476122109811556185850754053255575725128489389235813118542457191085831379981666151892838030850605331813894811523473742667844}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947} a^{16} + \frac{1060377626710373026483573541757575287218539017484460195113039115756957161613845947877144792070016255759673042129696628915654837661726695715644488844731039973485309320830960366169058703469012940454981746162794241916308363397}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947} a^{15} - \frac{1078178419021352538142456361462298451464670945505685680206816423036283388380047788974370025828249273046350558086322414347597213698441518611470909063396422336264602475170353185421616340404660080903157076302739671473473739119}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947} a^{14} + \frac{3346177962604571795733707670370989892834127069660599416051468662265571594683515079145350911197637701973150901721099468924080299426593705411914408116963030254209501832827258126367818081108992745167545892298841682739061522687}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947} a^{13} + \frac{1925729053001290695508301187080743173247699049685263576705484832157758914941542433673984342453215275891233317175844355253921634308230000493332951362667731166988926701888426636411418328057502154589296903836600820648573170509}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947} a^{12} + \frac{725306545659727204803060850341379138968805113777960460526985987525146197214432449636599000778780537370875629885069388922795109527730335236564026495531470899322444895444088661703783613437466666611940559863067969491389297740}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947} a^{11} + \frac{2584821584393563243018028553965066738594914319185315330276121514298247887211448766419288543915795099423982521823195200898896094741311794947446881156749531790816260397945729051753841230520460813063775537795467070110552534233}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947} a^{10} - \frac{1078594555044544813365568390631199645383168598553812638980573074414831236882092281333946741590722896174605246516655519865440533218972601259711311494141960841771403064262079759154699992712995015969099012995904387069998775122}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947} a^{9} + \frac{682499432228234151524886071730504984506608921341368357399306783053676322076326313803043931426348095102193840589911248940985826462490897562993097484902126504650771028541602098923677020888881296240434062889624905768418744270}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947} a^{8} + \frac{2725133062170851796439385235414669149687157795234589440520918218860564290017074972740914129173293871125764131697648918764674584731936703576355693203756840133270390111801585896800316719714184015571579047074314051697455439496}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947} a^{7} - \frac{523500053029999258517198370160340243276054954833056021199192815785512303277057882121070834291530072154834517046297453603416606518762306290393009460797492741358757052287436362634979325662004646136774267679420148250839798001}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947} a^{6} - \frac{3413504780986209219618232517253239134212800102321252754025227720647101891330643027374210529208837864191252127958666186496984657865539112244340360782627464388501523917564259091036499669863206876338904419522843238865417169189}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947} a^{5} - \frac{3095744594655436939793599953967909863740006239120344287787011492596669579850729940046878650238191010989311451345289860282951718470554475317374343077740417596280014305605345583090482290527630268794559639607904637758313811343}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947} a^{4} - \frac{1760754338611342674926300309928046572909228806587947322547050344462556386721848546007850726333027430447123726385629016492034190261415082344253352140533516849524610513953632415703534033694779078628914023787633065894588522509}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947} a^{3} - \frac{3758501983620005997175938214491556615193522552516871455393141513943003132230365155170996271007910283419902140751408958772828613697828100894304869210284300333439360595271516627148491046365813987542009816963768282133182864360}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947} a^{2} + \frac{362560954648650230974909440985221430100575489489289708210194230191251106257724208057073043181106017613308438786077539326462610343560616969810461156757671603957073869073957055443355548411476322116879420029073060796243842234}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947} a - \frac{1642038896155454394119683652317529099628433851138512122333602302225639471280029144048751235776301399357207820049380946816177346685852804881145835569349739323657591864267356329643507054966599838155767844784452497575422155799}{8548184765437022956750565394066723848474305546405946694889821209868245611222781804074888754521622678209384366174212755401634314616016522034175022755709852005967182515936726559659753191341049595410241242950147999030977040947}$

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

Class group and class number

$C_{23}$, which has order $23$ (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:  $22$
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:  \( 6651212114850000000000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{23}:C_{11}$ (as 23T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 253
The 13 conjugacy class representatives for $C_{23}:C_{11}$
Character table for $C_{23}:C_{11}$

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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ R ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
47Data not computed