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Properties

Label 21.1.27796743871...0000.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{33}\cdot 3^{19}\cdot 5^{19}\cdot 7^{17}\cdot 13^{7}$
Root discriminant $391.36$
Ramified primes $2, 3, 5, 7, 13$
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![-82620315048384, -23356753720704, -30425924350656, -4534363813248, -2909582113104, -209353969512, -139865038686, -28494481803, -10903896024, 334966533, -224961030, -53377557, -25576020, -2020725, -180030, -19449, 6876, -4161, -438, 21, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 + 21*x^19 - 438*x^18 - 4161*x^17 + 6876*x^16 - 19449*x^15 - 180030*x^14 - 2020725*x^13 - 25576020*x^12 - 53377557*x^11 - 224961030*x^10 + 334966533*x^9 - 10903896024*x^8 - 28494481803*x^7 - 139865038686*x^6 - 209353969512*x^5 - 2909582113104*x^4 - 4534363813248*x^3 - 30425924350656*x^2 - 23356753720704*x - 82620315048384)
 
gp: K = bnfinit(x^21 + 21*x^19 - 438*x^18 - 4161*x^17 + 6876*x^16 - 19449*x^15 - 180030*x^14 - 2020725*x^13 - 25576020*x^12 - 53377557*x^11 - 224961030*x^10 + 334966533*x^9 - 10903896024*x^8 - 28494481803*x^7 - 139865038686*x^6 - 209353969512*x^5 - 2909582113104*x^4 - 4534363813248*x^3 - 30425924350656*x^2 - 23356753720704*x - 82620315048384, 1)
 

Normalized defining polynomial

\( x^{21} + 21 x^{19} - 438 x^{18} - 4161 x^{17} + 6876 x^{16} - 19449 x^{15} - 180030 x^{14} - 2020725 x^{13} - 25576020 x^{12} - 53377557 x^{11} - 224961030 x^{10} + 334966533 x^{9} - 10903896024 x^{8} - 28494481803 x^{7} - 139865038686 x^{6} - 209353969512 x^{5} - 2909582113104 x^{4} - 4534363813248 x^{3} - 30425924350656 x^{2} - 23356753720704 x - 82620315048384 \)

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:  \(2779674387135021652740220623564472320000000000000000000=2^{33}\cdot 3^{19}\cdot 5^{19}\cdot 7^{17}\cdot 13^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $391.36$
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:  $2, 3, 5, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{6} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{6} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{12} a^{16} - \frac{1}{12} a^{15} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{17} - \frac{1}{12} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{120} a^{18} - \frac{1}{60} a^{17} - \frac{1}{40} a^{16} - \frac{3}{40} a^{14} + \frac{1}{20} a^{13} - \frac{3}{40} a^{12} - \frac{1}{5} a^{11} - \frac{3}{40} a^{10} + \frac{1}{20} a^{9} - \frac{7}{40} a^{8} - \frac{1}{2} a^{7} - \frac{1}{40} a^{6} + \frac{1}{20} a^{5} - \frac{17}{40} a^{4} - \frac{1}{2} a^{3} + \frac{1}{10} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{240} a^{19} - \frac{7}{240} a^{17} - \frac{1}{40} a^{16} + \frac{11}{240} a^{15} - \frac{1}{20} a^{14} + \frac{1}{80} a^{13} + \frac{3}{40} a^{12} - \frac{19}{80} a^{11} - \frac{1}{20} a^{10} - \frac{3}{80} a^{9} - \frac{17}{40} a^{8} + \frac{19}{80} a^{7} - \frac{13}{80} a^{5} + \frac{3}{40} a^{4} + \frac{1}{20} a^{3} - \frac{1}{2} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{11274498829092983756722306485471929812853602904398980126398301358190361663055219254570001952162596849849410152183229201027087677280} a^{20} + \frac{210838207399258915119227005362819062836211431280837826397753897552800642146506438897603223580209668795918740598389476499837719}{234885392272770494931714718447331871101116727174978752633297944962299201313650401136875040670054101038529378170483941688064326610} a^{19} + \frac{5013049287449596530338712075377188896424344618949825181690775893949146971250941527843991706993666661670289578856275343077949447}{3758166276364327918907435495157309937617867634799660042132767119396787221018406418190000650720865616616470050727743067009029225760} a^{18} - \frac{2866587200705453085397164252379565513854576998269488900557773542412498884825194199123358064439313439665977907203442850036318861}{1879083138182163959453717747578654968808933817399830021066383559698393610509203209095000325360432808308235025363871533504514612880} a^{17} + \frac{62996572565006024598245327559732132065981427976661856061686621602455594716103738360974942829020405052448168644070876479180175747}{2254899765818596751344461297094385962570720580879796025279660271638072332611043850914000390432519369969882030436645840205417535456} a^{16} + \frac{14464208802252871193955276704311075127993400173085480155740541779426113126645215745310919454140149212247498182326481144701979923}{187908313818216395945371774757865496880893381739983002106638355969839361050920320909500032536043280830823502536387153350451461288} a^{15} - \frac{198320820234203607610620429126673356893624948053029355650096273405221721359783871810981606820545630122708518160307174944351190473}{11274498829092983756722306485471929812853602904398980126398301358190361663055219254570001952162596849849410152183229201027087677280} a^{14} - \frac{388680685571626769625623062505805735013740109997459480384921065281794610179832759876463763406422898521683111315201654350823517513}{1879083138182163959453717747578654968808933817399830021066383559698393610509203209095000325360432808308235025363871533504514612880} a^{13} + \frac{929875477803700113420789272050797731279363550427381090573309047282372293829542559452981169262396225199474486021940939370768534489}{3758166276364327918907435495157309937617867634799660042132767119396787221018406418190000650720865616616470050727743067009029225760} a^{12} - \frac{141070569390498873251273179950689209826166973409704088936210078243228556698168118772857076965406194526577481402215397854798683749}{939541569091081979726858873789327484404466908699915010533191779849196805254601604547500162680216404154117512681935766752257306440} a^{11} + \frac{142819674371352011679259535292049336916667734133477249005656790105505329724307738505251022633338863878225869713773479440346988501}{751633255272865583781487099031461987523573526959932008426553423879357444203681283638000130144173123323294010145548613401805845152} a^{10} - \frac{30697935233111626347899941446660219047526082945332993180668615340106133039809992460563322086447558666913788933332453996761173757}{1879083138182163959453717747578654968808933817399830021066383559698393610509203209095000325360432808308235025363871533504514612880} a^{9} + \frac{174886291127516449446376413792161310232611983422611051531000916724908838784566430764134682637762816982991304870830910128994143547}{751633255272865583781487099031461987523573526959932008426553423879357444203681283638000130144173123323294010145548613401805845152} a^{8} + \frac{109678213147627125211476432882170127454685157138301609464012225115218858625076757466237017649389659065559296619519800553719681653}{234885392272770494931714718447331871101116727174978752633297944962299201313650401136875040670054101038529378170483941688064326610} a^{7} - \frac{1824670181402272837695316282146159242110065662887816830948863187664956971711593746297225369829772622262937151620665955157772887881}{3758166276364327918907435495157309937617867634799660042132767119396787221018406418190000650720865616616470050727743067009029225760} a^{6} - \frac{720108054581214428717363126297209259105848349519142033323921651560505000195342078998871807023138175716145458641116420772221012497}{1879083138182163959453717747578654968808933817399830021066383559698393610509203209095000325360432808308235025363871533504514612880} a^{5} + \frac{16841276409139583340889383487823252589665513830066463667478125642338355392408170300449209621417429701152209982188389792058525689}{93954156909108197972685887378932748440446690869991501053319177984919680525460160454750016268021640415411751268193576675225730644} a^{4} + \frac{217665267854987761364088992216983968380382248263010919255689006278560520171768066377997311278646453243931969582037682885326995491}{469770784545540989863429436894663742202233454349957505266595889924598402627300802273750081340108202077058756340967883376128653220} a^{3} + \frac{32182237763142758729564719852420427621851799920422187086726325617289256740072846466249424358756707843309345302818267217963515449}{234885392272770494931714718447331871101116727174978752633297944962299201313650401136875040670054101038529378170483941688064326610} a^{2} + \frac{24236563119217844161355865102001636769423644548556179377837394774841719829108215038939918981210648321002342443312440182219831358}{117442696136385247465857359223665935550558363587489376316648972481149600656825200568437520335027050519264689085241970844032163305} a + \frac{19729311348823843771368151302667469283281582615242986114335336634816164796290793034514971396023482855444125586239092313395478173}{117442696136385247465857359223665935550558363587489376316648972481149600656825200568437520335027050519264689085241970844032163305}$

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

3.1.76440.1, 7.1.12252303000000.8

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{7}^{3}$
2.14.27.13$x^{14} - 4 x^{12} + 8 x^{11} - 6 x^{10} - 4 x^{9} - 6 x^{8} + 4 x^{7} + 6 x^{6} - 4 x^{5} + 4 x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 6$$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$$[\ ]_{7}^{6}$
3.14.13.2$x^{14} + 3$$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
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$