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Properties

Label 23.23.9313472568...0329.1
Degree $23$
Signature $[23, 0]$
Discriminant $23^{20}\cdot 277^{22}$
Root discriminant $3314.33$
Ramified primes $23, 277$
Class number Not computed
Class group Not computed
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![4639460945705837181, -88548356899300876754, -1226738150814548351153, -791389848161205290501, 862348894552066379264, 1018760816151118188540, 328159724854144381281, 14340621876424475050, -10639739683422990020, -1307367517981252668, 127640804228029831, 22707143025850299, -765546003229229, -189575719371646, 2526286588076, 888193853662, -4712063343, -2419508022, 4709554, 3753073, -1939, -3047, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 3047*x^21 - 1939*x^20 + 3753073*x^19 + 4709554*x^18 - 2419508022*x^17 - 4712063343*x^16 + 888193853662*x^15 + 2526286588076*x^14 - 189575719371646*x^13 - 765546003229229*x^12 + 22707143025850299*x^11 + 127640804228029831*x^10 - 1307367517981252668*x^9 - 10639739683422990020*x^8 + 14340621876424475050*x^7 + 328159724854144381281*x^6 + 1018760816151118188540*x^5 + 862348894552066379264*x^4 - 791389848161205290501*x^3 - 1226738150814548351153*x^2 - 88548356899300876754*x + 4639460945705837181)
 
gp: K = bnfinit(x^23 - 3047*x^21 - 1939*x^20 + 3753073*x^19 + 4709554*x^18 - 2419508022*x^17 - 4712063343*x^16 + 888193853662*x^15 + 2526286588076*x^14 - 189575719371646*x^13 - 765546003229229*x^12 + 22707143025850299*x^11 + 127640804228029831*x^10 - 1307367517981252668*x^9 - 10639739683422990020*x^8 + 14340621876424475050*x^7 + 328159724854144381281*x^6 + 1018760816151118188540*x^5 + 862348894552066379264*x^4 - 791389848161205290501*x^3 - 1226738150814548351153*x^2 - 88548356899300876754*x + 4639460945705837181, 1)
 

Normalized defining polynomial

\( x^{23} - 3047 x^{21} - 1939 x^{20} + 3753073 x^{19} + 4709554 x^{18} - 2419508022 x^{17} - 4712063343 x^{16} + 888193853662 x^{15} + 2526286588076 x^{14} - 189575719371646 x^{13} - 765546003229229 x^{12} + 22707143025850299 x^{11} + 127640804228029831 x^{10} - 1307367517981252668 x^{9} - 10639739683422990020 x^{8} + 14340621876424475050 x^{7} + 328159724854144381281 x^{6} + 1018760816151118188540 x^{5} + 862348894552066379264 x^{4} - 791389848161205290501 x^{3} - 1226738150814548351153 x^{2} - 88548356899300876754 x + 4639460945705837181 \)

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:  \(931347256889446325436632107655346061164193665348344821578377438399536607931200329=23^{20}\cdot 277^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3314.33$
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:  $23, 277$
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}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691} a^{22} - \frac{1921286305066212061442258274684416666382379230848714634278968122929844380944688792373209467920715625283206733664081059704595648205874082086767447825697119892543673434514490847482019547295406509620789842297427372148131945475668850019735266}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691} a^{21} + \frac{12745174637823468417543888160233698122341610697495640354876180250873340019920566095474635345069085572582550544740858154151169973719352639566560457254169788979505745745120367353693069840950338365578999974796622389602300027695910258953231595}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691} a^{20} + \frac{19311181484760840683186624427299804433576215345597044057121208944418997754069008429848367536908669489498534550198803011348973531434927444762771409513089303981984792579644662119984342783551549923337824274567312997291089196163875361329805904}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691} a^{19} + \frac{2404271545139624945190479384390069872908987120644172847447367135267714423082332693396600255472786030841352962806265635850761345091204243029358153245728511387514859914734748757667146054905467614178689818949657873800673896335523368978488919}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691} a^{18} + \frac{12582006627762368362697152864695820139850688071836847204126056381672078814648146222148129480248517055356258507970120788248311205916698202883059292736094866326661245340031879397680836650266541033856716413580759192751908240739837696930405215}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691} a^{17} + \frac{9648213732528462310086768520254627848791499526733429482204159148699699749210025846032725935984199174979581386080567656870181792562593957737718402316596741317543457789445192743303344842932384362351912946830029183417018945030705430214183890}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691} a^{16} + \frac{14460994833325903426057670661518512586395181226805863729627165157011763285809929644589977340207036245770780669101792621488086618464221118428429782713742920264896092489605681558536796031664774429649901946502552482266043652600825318467017301}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691} a^{15} - \frac{6310994652882749384375845772253521118914651985366446830730733570433842469756178866119015706462271774541241806771640660921767467086206748277954011963578427319766607868802117564002680862616499532750464458721871492758148783649958798847762526}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691} a^{14} + \frac{8152350831459176069437299761085957564133104862638630960260171440659197936378731742688357548442729155554603498062572001460504605696262711956140442950980660082998739489702463328469329950010422182267932364693824095044955632997016788896513986}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691} a^{13} + \frac{2913597038139980293841324701178410521150882165457424126554687470759427373485994972929826136923946558594412234023019462808516262835315177005778289093208932876696757080364797199025567985662584185432578479916113524893305226426637048120103133}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691} a^{12} + \frac{19104832911704838312151651217722664408059193018702623457455381004348096086097445340572832337394853355518017230674001140834261556433618586763113065555445170223636795656141901396523954869673659630613258906041031037483050647875761557896760935}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691} a^{11} + \frac{3360335167713155834769713301834635870491071585304575180163187984689850001493462303579737772569381729781349643989747284298607142171730398554383683925363578365408425886695480985583151448025515932944920974011536237634486545319029350522824534}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691} a^{10} - \frac{19311003291610460152835481197550574792816854707720310789098244029438402817378086382280590002239152605274511168356828536905614290006425761031712697983864716964053283498660608584311787669989277119915726042747404061879876789099153567623164972}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691} a^{9} - \frac{18828722039765015587016463394729231359414164562816521970189948774360252459252206958578512403151391622191274360152442951970873410476348801695333478803095185274277011517825690503152807513769920872493521382465127776549440405762303466666743386}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691} a^{8} - \frac{7332546387980318145552913732219097296083270807711211451068405328561718033352749453648801923869365451817559857412977330710350883211346764045219394768616295583184355100009673331756108403895367210447374966667985476476397138881026921589288930}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691} a^{7} - \frac{246347135358352541700265459218065145490753294452444630751580937037000050507441016247785977082190092400891728175561774130136127524262606417348605274072200983790221431915896029505406117641617575691686449606456959189230568923349060307214108}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691} a^{6} - \frac{10856942552345946046545269458040547288169655893136330514492488333291063146733649794686868882246605909153648417625948821161549353773533513422087704366125866702710820813577969297254359181426615649354498484721036727671535623096615412870277014}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691} a^{5} + \frac{12663909020147377849315414497365823266386361411066980394036190773037062037855313405287963739541460186908060696316771109977835180246715940438484565704339345614305385240785007892514522652982818447415868804915424580276233854190867763768845255}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691} a^{4} + \frac{5377519235092961862029132894476365843463345535884605436557864478456734437831737569397070703950475253930131691145518362612722284432169887988146955522670895847629567415493424950103839357050181116539911516869781525847589323502116908932366084}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691} a^{3} + \frac{3783597477158144343000340226774175262185232018502368813280270393722951903654764376244686729604620837619862914900399217356616415166496001309428227016854375432042959882983485357476502621812671134062999565316012519958256452043634956484490728}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691} a^{2} - \frac{9036081243557360916513636536555285884964301261441795140551102565057174741398817310774404156978371189843039453630022413542858229869372199593776814648389485986415968051446626467053816596252981670851237759050328685686925587968541584565797119}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691} a + \frac{5060269380252948910059607903340788500270780639633458623825879662511952064075126421744336528858475149265035346705797093575747082803240108989315705255808290465614622449832416559616571103028793130412918176604844577996598890427778074050620445}{39260337486431234973290428768524892954837782776302968548580025786973013542039504278364124950904116321159068949337131733880602090497731045195266361125782850522380091293230712865673427918040924303876368295433206936947124616967461344291987691}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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 ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $23$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.11.10.10$x^{11} - 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$
23.11.10.10$x^{11} - 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$
277Data not computed