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Properties

Label 21.1.12020903363...0000.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{33}\cdot 3^{19}\cdot 5^{19}\cdot 7^{17}\cdot 83^{7}$
Root discriminant $726.02$
Ramified primes $2, 3, 5, 7, 83$
Class number Not computed
Class group Not computed
Galois group $S_3\times F_7$ (as 21T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![48239717281463331, 204438880713856581, 4304090964183750, 22310247806481510, 1226053598985135, 1519304409798669, 11992532164164, 52060322671200, 705666957930, 1332478142730, -4048247664, 12604874796, 1036911690, 59486130, 22352700, -1157136, 145539, 7665, 810, 30, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 9*x^20 + 30*x^19 + 810*x^18 + 7665*x^17 + 145539*x^16 - 1157136*x^15 + 22352700*x^14 + 59486130*x^13 + 1036911690*x^12 + 12604874796*x^11 - 4048247664*x^10 + 1332478142730*x^9 + 705666957930*x^8 + 52060322671200*x^7 + 11992532164164*x^6 + 1519304409798669*x^5 + 1226053598985135*x^4 + 22310247806481510*x^3 + 4304090964183750*x^2 + 204438880713856581*x + 48239717281463331)
 
gp: K = bnfinit(x^21 - 9*x^20 + 30*x^19 + 810*x^18 + 7665*x^17 + 145539*x^16 - 1157136*x^15 + 22352700*x^14 + 59486130*x^13 + 1036911690*x^12 + 12604874796*x^11 - 4048247664*x^10 + 1332478142730*x^9 + 705666957930*x^8 + 52060322671200*x^7 + 11992532164164*x^6 + 1519304409798669*x^5 + 1226053598985135*x^4 + 22310247806481510*x^3 + 4304090964183750*x^2 + 204438880713856581*x + 48239717281463331, 1)
 

Normalized defining polynomial

\( x^{21} - 9 x^{20} + 30 x^{19} + 810 x^{18} + 7665 x^{17} + 145539 x^{16} - 1157136 x^{15} + 22352700 x^{14} + 59486130 x^{13} + 1036911690 x^{12} + 12604874796 x^{11} - 4048247664 x^{10} + 1332478142730 x^{9} + 705666957930 x^{8} + 52060322671200 x^{7} + 11992532164164 x^{6} + 1519304409798669 x^{5} + 1226053598985135 x^{4} + 22310247806481510 x^{3} + 4304090964183750 x^{2} + 204438880713856581 x + 48239717281463331 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1202090336315934431360526607958288705617920000000000000000000=2^{33}\cdot 3^{19}\cdot 5^{19}\cdot 7^{17}\cdot 83^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $726.02$
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:  $2, 3, 5, 7, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{12} a^{8} + \frac{1}{4}$, $\frac{1}{12} a^{9} + \frac{1}{4} a$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a + \frac{1}{8}$, $\frac{1}{72} a^{11} - \frac{1}{24} a^{9} - \frac{1}{4} a^{6} - \frac{1}{6} a^{5} + \frac{1}{12} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} + \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{144} a^{12} - \frac{1}{48} a^{8} + \frac{1}{6} a^{6} - \frac{1}{12} a^{5} - \frac{1}{16} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{3}{16}$, $\frac{1}{144} a^{13} - \frac{1}{48} a^{9} - \frac{1}{12} a^{6} - \frac{1}{16} a^{5} - \frac{1}{4} a^{2} + \frac{3}{16} a$, $\frac{1}{1440} a^{14} + \frac{1}{360} a^{13} + \frac{1}{1440} a^{12} + \frac{1}{360} a^{11} + \frac{7}{480} a^{10} + \frac{1}{120} a^{9} + \frac{11}{480} a^{8} + \frac{1}{120} a^{7} - \frac{59}{480} a^{6} + \frac{2}{15} a^{5} + \frac{77}{480} a^{4} + \frac{1}{10} a^{3} - \frac{61}{160} a^{2} + \frac{1}{10} a - \frac{1}{160}$, $\frac{1}{1440} a^{15} - \frac{1}{288} a^{13} + \frac{1}{288} a^{11} - \frac{1}{120} a^{10} - \frac{1}{32} a^{9} - \frac{1}{24} a^{8} + \frac{1}{96} a^{7} - \frac{5}{24} a^{6} + \frac{31}{480} a^{5} + \frac{5}{24} a^{4} + \frac{7}{32} a^{3} + \frac{1}{4} a^{2} - \frac{7}{32} a + \frac{2}{5}$, $\frac{1}{2880} a^{16} - \frac{1}{240} a^{11} - \frac{1}{48} a^{10} - \frac{1}{96} a^{8} - \frac{1}{24} a^{7} + \frac{9}{40} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{5}{16} a^{3} - \frac{7}{16} a^{2} + \frac{9}{20} a + \frac{9}{64}$, $\frac{1}{2880} a^{17} + \frac{1}{360} a^{12} - \frac{1}{144} a^{11} + \frac{1}{32} a^{9} + \frac{1}{48} a^{8} - \frac{1}{40} a^{7} - \frac{1}{12} a^{6} + \frac{1}{8} a^{5} + \frac{1}{12} a^{4} + \frac{3}{16} a^{3} + \frac{9}{20} a^{2} + \frac{17}{64} a + \frac{7}{16}$, $\frac{1}{708480} a^{18} + \frac{1}{39360} a^{17} - \frac{7}{78720} a^{16} + \frac{1}{7380} a^{15} + \frac{1}{3690} a^{14} + \frac{61}{19680} a^{13} + \frac{11}{19680} a^{12} - \frac{163}{59040} a^{11} - \frac{1}{192} a^{10} - \frac{761}{19680} a^{9} + \frac{1573}{39360} a^{8} + \frac{349}{9840} a^{7} + \frac{983}{4920} a^{6} + \frac{377}{19680} a^{5} + \frac{1339}{6560} a^{4} + \frac{133}{6560} a^{3} - \frac{449}{5248} a^{2} + \frac{1131}{2624} a + \frac{8559}{26240}$, $\frac{1}{97085139840} a^{19} + \frac{3119}{7468087680} a^{18} + \frac{204037}{6472342656} a^{17} + \frac{2237699}{32361713280} a^{16} - \frac{225293}{809042832} a^{15} + \frac{2748667}{8090428320} a^{14} + \frac{99095}{269680944} a^{13} - \frac{1544737}{1348404720} a^{12} + \frac{439237}{82978752} a^{11} + \frac{31890697}{5393618880} a^{10} - \frac{169623}{359574592} a^{9} + \frac{73386771}{1797872960} a^{8} + \frac{41941}{224734120} a^{7} + \frac{240038729}{2696809440} a^{6} + \frac{35424913}{674202360} a^{5} - \frac{142244957}{674202360} a^{4} + \frac{1067788983}{3595745920} a^{3} - \frac{1235563127}{3595745920} a^{2} + \frac{1778443293}{3595745920} a - \frac{128894829}{276595840}$, $\frac{1}{1697368158088373909864944974256326411310429358503435771487280121497050780643070038731730622655893729448346090212074681906880886264960} a^{20} - \frac{172490385559206168861809414531580754588912177665767734574681775315416795452220478318822189775790075776604110577398964391}{62865487336606441106109073120604681900386272537164287832862226722112991875669260693767800839107175164753558896743506737291884676480} a^{19} + \frac{129256197223566416583813717998324912585144222851762882707060440442222740338566022549721270756514356015868930988622352935948671}{282894693014728984977490829042721068551738226417239295247880020249508463440511673121955103775982288241391015035345780317813481044160} a^{18} - \frac{18261685345067619866262092853821030182970185160402807404349665591301945022519253135051368369159954029253172999364574558734300077}{113157877205891593990996331617088427420695290566895718099152008099803385376204669248782041510392915296556406014138312127125392417664} a^{17} + \frac{5912553206721700788422264817710141599950954937541072785671556772497264129114858091722145911222977187628260280315906735482595461}{565789386029457969954981658085442137103476452834478590495760040499016926881023346243910207551964576482782030070691560635626962088320} a^{16} - \frac{647056706194040487142654637943478639958017825100519218316086791659677365524569776656936219743463053446625407372082597522449776}{4420229578355140390273294203792516696120909787769363988248125316398569741257994892530548496499723253771734609927277817465835641315} a^{15} - \frac{1570226543044371036412551908726474630103441499305709599654806979051320910101722066726056607012174485098509259356049658624682873}{9429823100490966165916360968090702285057940880574643174929334008316948781350389104065170125866076274713033834511526010593782701472} a^{14} - \frac{36311132891932127980379137360535459245028948074460478127608755290840452101329529710272050347783266284307490481633772822919466159}{23574557751227415414790902420226755712644852201436607937323335020792371953375972760162925314665190686782584586278815026484456753680} a^{13} - \frac{297823917651396758618960990488736884351367124585672395359614183496139797337836380733825980493707317484301708574467351472117459683}{94298231004909661659163609680907022850579408805746431749293340083169487813503891040651701258660762747130338345115260105937827014720} a^{12} + \frac{49569441201135607408086632396712920067322656812340378770689756882497516685490704900589326523883342176067736555003025547410230183}{282894693014728984977490829042721068551738226417239295247880020249508463440511673121955103775982288241391015035345780317813481044160} a^{11} - \frac{122377119553222134643430560592730960848231997935071132331934788405534159514752321590795633408421352415300136185720556916471905177}{47149115502454830829581804840453511425289704402873215874646670041584743906751945520325850629330381373565169172557630052968913507360} a^{10} + \frac{27863545694661660617072579942616770469333963126606893693548894824757361107054590847818726758545164073196251904128422720605479129}{31432743668303220553054536560302340950193136268582143916431113361056495937834630346883900419553587582376779448371753368645942338240} a^{9} + \frac{36206800918886077352789211832969511836187084452322655839112592821266638943663746229306870604257501584689790966402260439980326043}{2417903359100247734850348966177103150014856636044780301263931797004345841371894642067992339965660583259752265259365643741995564480} a^{8} - \frac{117959089248163003803492342223183403724024470704652209247992429809461461425553943219545431628147368888221748152772485106763944863}{2946819718903426926848862802528344464080606525179575992165416877599046494171996595020365664333148835847823073284851878310557094210} a^{7} - \frac{5942388120146313868292296979612411935240111485601184472058143861319486373388139355394426647533229082969330365007263667748631774353}{47149115502454830829581804840453511425289704402873215874646670041584743906751945520325850629330381373565169172557630052968913507360} a^{6} + \frac{1285626421808368371578909550333780206341582768674629336740957172781779027850403268801422407273986083527964958936601271259652045761}{23574557751227415414790902420226755712644852201436607937323335020792371953375972760162925314665190686782584586278815026484456753680} a^{5} - \frac{40666835236711274126534307447213806556039099830885059219333730824013980133951711370988205344438851514251389858715738800984340732539}{188596462009819323318327219361814045701158817611492863498586680166338975627007782081303402517321525494260676690230520211875654029440} a^{4} - \frac{2201116174596118712388853375623793380604510955768849251071197718642261151673664278826311725340081506914814292247007429127556652683}{12573097467321288221221814624120936380077254507432857566572445344422598375133852138753560167821435032950711779348701347458376935296} a^{3} - \frac{14671374186669165199327944122201125365534418929133297987360195802991446045692019011064846737397760556149356839230077107807423369261}{31432743668303220553054536560302340950193136268582143916431113361056495937834630346883900419553587582376779448371753368645942338240} a^{2} + \frac{7726926086918988297279751666406314595066769846430256188116673019265983175016045261723721448414747880513013178057571540643149641591}{62865487336606441106109073120604681900386272537164287832862226722112991875669260693767800839107175164753558896743506737291884676480} a - \frac{783408854818341630443913970378665596113679867328914557963802408332165027257019391888679533282640728051132585960390038424420174671}{4835806718200495469700697932354206300029713272089560602527863594008691682743789284135984679931321166519504530518731287483991128960}$

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:  $10$
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

$S_3\times F_7$ (as 21T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 252
The 21 conjugacy class representatives for $S_3\times F_7$
Character table for $S_3\times F_7$ is not computed

Intermediate fields

3.1.488040.1, 7.1.12252303000000.9

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 42 siblings: 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 R R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $21$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$2$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.14.27.21$x^{14} + 4 x^{12} + 4 x^{11} - 4 x^{10} + 8 x^{9} + 2 x^{8} + 8 x^{6} + 4 x^{5} - 4 x^{4} + 6 x^{2} - 4 x - 6$$14$$1$$27$$(C_7:C_3) \times C_2$$[3]_{7}^{3}$
$3$3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
3.14.13.1$x^{14} - 3$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
$5$5.7.6.1$x^{7} - 5$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
5.14.13.2$x^{14} + 10$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
7Data not computed
$83$$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$