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Properties

Label 21.1.25471539222...0000.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{33}\cdot 3^{18}\cdot 5^{19}\cdot 7^{17}\cdot 29^{7}$
Root discriminant $485.29$
Ramified primes $2, 3, 5, 7, 29$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
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![2927357027960148, -3793231674684900, 1982371881158172, -384001652308932, -83034819969948, 60981098550876, -10017806215788, -1456671621720, 812925249780, -79222959900, -18571849464, 5222680680, -128880486, -113584254, 14121359, 789901, -277431, 8981, 2359, -189, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 189*x^19 + 2359*x^18 + 8981*x^17 - 277431*x^16 + 789901*x^15 + 14121359*x^14 - 113584254*x^13 - 128880486*x^12 + 5222680680*x^11 - 18571849464*x^10 - 79222959900*x^9 + 812925249780*x^8 - 1456671621720*x^7 - 10017806215788*x^6 + 60981098550876*x^5 - 83034819969948*x^4 - 384001652308932*x^3 + 1982371881158172*x^2 - 3793231674684900*x + 2927357027960148)
 
gp: K = bnfinit(x^21 - 7*x^20 - 189*x^19 + 2359*x^18 + 8981*x^17 - 277431*x^16 + 789901*x^15 + 14121359*x^14 - 113584254*x^13 - 128880486*x^12 + 5222680680*x^11 - 18571849464*x^10 - 79222959900*x^9 + 812925249780*x^8 - 1456671621720*x^7 - 10017806215788*x^6 + 60981098550876*x^5 - 83034819969948*x^4 - 384001652308932*x^3 + 1982371881158172*x^2 - 3793231674684900*x + 2927357027960148, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 189 x^{19} + 2359 x^{18} + 8981 x^{17} - 277431 x^{16} + 789901 x^{15} + 14121359 x^{14} - 113584254 x^{13} - 128880486 x^{12} + 5222680680 x^{11} - 18571849464 x^{10} - 79222959900 x^{9} + 812925249780 x^{8} - 1456671621720 x^{7} - 10017806215788 x^{6} + 60981098550876 x^{5} - 83034819969948 x^{4} - 384001652308932 x^{3} + 1982371881158172 x^{2} - 3793231674684900 x + 2927357027960148 \)

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:  \(254715392223928332796590537567913082880000000000000000000=2^{33}\cdot 3^{18}\cdot 5^{19}\cdot 7^{17}\cdot 29^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $485.29$
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:  $2, 3, 5, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6}$, $\frac{1}{7} a^{7}$, $\frac{1}{7} a^{8}$, $\frac{1}{7} a^{9}$, $\frac{1}{7} a^{10}$, $\frac{1}{49} a^{11} + \frac{1}{7} a^{5}$, $\frac{1}{49} a^{12}$, $\frac{1}{49} a^{13}$, $\frac{1}{1470} a^{14} + \frac{11}{1470} a^{13} + \frac{1}{490} a^{12} + \frac{1}{210} a^{11} - \frac{13}{210} a^{10} + \frac{3}{70} a^{9} - \frac{11}{210} a^{8} - \frac{13}{210} a^{7} - \frac{2}{35} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{4410} a^{15} - \frac{1}{4410} a^{14} + \frac{1}{210} a^{13} + \frac{1}{4410} a^{12} + \frac{1}{882} a^{11} + \frac{1}{42} a^{10} + \frac{1}{630} a^{9} - \frac{31}{630} a^{8} - \frac{1}{105} a^{7} - \frac{26}{105} a^{5} + \frac{2}{15} a^{4} + \frac{1}{5} a^{3} - \frac{2}{15} a^{2} + \frac{1}{3} a + \frac{2}{5}$, $\frac{1}{30870} a^{16} - \frac{1}{4410} a^{14} + \frac{5}{882} a^{13} + \frac{1}{294} a^{12} - \frac{37}{4410} a^{11} - \frac{173}{4410} a^{10} - \frac{13}{210} a^{9} - \frac{4}{63} a^{8} - \frac{1}{42} a^{7} + \frac{4}{105} a^{6} - \frac{4}{35} a^{5} - \frac{1}{105} a^{4} - \frac{1}{3} a^{3} + \frac{2}{15} a + \frac{2}{5}$, $\frac{1}{30870} a^{17} + \frac{1}{105} a^{13} - \frac{1}{245} a^{12} + \frac{4}{735} a^{11} + \frac{1}{35} a^{10} + \frac{1}{42} a^{9} + \frac{19}{315} a^{8} - \frac{1}{21} a^{7} + \frac{2}{35} a^{6} - \frac{17}{35} a^{5} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{15} a + \frac{1}{5}$, $\frac{1}{710010} a^{18} + \frac{1}{710010} a^{17} - \frac{11}{710010} a^{16} + \frac{1}{10143} a^{15} + \frac{31}{101430} a^{14} - \frac{803}{101430} a^{13} + \frac{125}{20286} a^{12} - \frac{557}{101430} a^{11} - \frac{697}{50715} a^{10} - \frac{11}{2415} a^{9} - \frac{11}{161} a^{8} - \frac{181}{4830} a^{7} + \frac{11}{483} a^{6} + \frac{439}{2415} a^{5} - \frac{479}{2415} a^{4} + \frac{34}{345} a^{3} + \frac{11}{115} a^{2} + \frac{22}{115} a - \frac{34}{115}$, $\frac{1}{7100100} a^{19} + \frac{1}{7100100} a^{18} - \frac{19}{2366700} a^{17} + \frac{47}{7100100} a^{16} + \frac{11}{144900} a^{15} - \frac{7}{20700} a^{14} + \frac{257}{1014300} a^{13} - \frac{6859}{1014300} a^{12} - \frac{599}{72450} a^{11} + \frac{8279}{507150} a^{10} + \frac{284}{7245} a^{9} + \frac{319}{7245} a^{8} - \frac{73}{1610} a^{7} - \frac{179}{4830} a^{6} - \frac{1991}{4830} a^{5} + \frac{694}{12075} a^{4} - \frac{133}{1725} a^{3} - \frac{289}{1725} a^{2} + \frac{409}{1725} a - \frac{3}{25}$, $\frac{1}{1104120902241304409962373256034315798262076442251910685478828671519230248435648518186979745395300} a^{20} + \frac{7789793379361762816141456724325322643405188768121420730534604887905665733933257832985877}{276030225560326102490593314008578949565519110562977671369707167879807562108912129546744936348825} a^{19} + \frac{69228419925261598990165024922403745918975800837826613843679205972779816559643290969849883}{110412090224130440996237325603431579826207644225191068547882867151923024843564851818697974539530} a^{18} - \frac{1500490742145523236145531824518210070950849211855743796916575154106485158249404202946180163}{276030225560326102490593314008578949565519110562977671369707167879807562108912129546744936348825} a^{17} - \frac{8885751057174388500346255547601747153777799362459131210185754044825463298496350489216426341}{552060451120652204981186628017157899131038221125955342739414335759615124217824259093489872697650} a^{16} + \frac{1312273477256603618072551497535314438938738473552800822948477557993082427102282267434958603}{78865778731521743568740946859593985590148317303707906105630619394230732031117751299069981813950} a^{15} - \frac{1666378361033723040826181122595298474840023371723872696246171159780035900476972594684335961}{39432889365760871784370473429796992795074158651853953052815309697115366015558875649534990906975} a^{14} + \frac{14966726183164016294023218299307149877454402665495603585144864871413979413108442934007617143}{7886577873152174356874094685959398559014831730370790610563061939423073203111775129906998181395} a^{13} + \frac{214551151577806980205802521861977064867657297889067909552607266890402900240221092258707624483}{22533079637577641019640270531312567311470947801059401744465891255494494866033643228305709089700} a^{12} - \frac{232743697602782362697625310123385134182997410929275654116826604009705246187964654193376606041}{39432889365760871784370473429796992795074158651853953052815309697115366015558875649534990906975} a^{11} - \frac{4398910018605820758692332638246833225758274435211337265187839475680200484332606260923353946167}{78865778731521743568740946859593985590148317303707906105630619394230732031117751299069981813950} a^{10} + \frac{5595920179079023336206668739450017427997054531433016695946427111408691914192393695055668633}{125183775764320227886890391840625373952616376672552231913699395863858304811298017935031717165} a^{9} + \frac{68663156263770354475177928390294253854774337010271664578793082293298770567480023663642950949}{1126653981878882050982013526565628365573547390052970087223294562774724743301682161415285454485} a^{8} + \frac{29780710502433648171412015822724840231870071943912886855453092486948344763967765477641923727}{751102654585921367321342351043752243715698260035313391482196375183149828867788107610190302990} a^{7} + \frac{1218808076171275153467611372319042959338727043561326582933309695290837642663710214167756551}{75110265458592136732134235104375224371569826003531339148219637518314982886778810761019030299} a^{6} + \frac{201497183657925098847071696868981951008813246121455795232273670530027078886814611468350470249}{536501896132800976658101679316965888368355900025223851058711696559392734905562934007278787850} a^{5} + \frac{932161092490170237784629831006642388451670857287113576278750991240662884493853183039494732552}{1877756636464803418303355877609380609289245650088283478705490937957874572169470269025475757475} a^{4} - \frac{13415946975349292998171353858767749923415078416855595828911723726167529872935421098298338394}{53650189613280097665810167931696588836835590002522385105871169655939273490556293400727878785} a^{3} - \frac{37717099947335536595565904089451851324556064364028896326141675605941481206374901878599783198}{89416982688800162776350279886160981394725983337537308509785282759898789150927155667879797975} a^{2} + \frac{128247747577985826256647064069005211308955410551964014033119683743259198759453410675189557976}{268250948066400488329050839658482944184177950012611925529355848279696367452781467003639393925} a - \frac{7641664948227436733513056568521637289503308073600094599343195727319463762867480896251044398}{89416982688800162776350279886160981394725983337537308509785282759898789150927155667879797975}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $10$
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:  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:  \( 362534892447484540000 \) (assuming GRH)
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

3.1.56840.1, 7.1.12252303000000.6

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $21$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$[\ ]_{7}^{3}$
2.14.27.121$x^{14} - 4 x^{13} - 4 x^{12} + 8 x^{11} + 8 x^{10} - 6 x^{8} + 8 x^{7} + 6 x^{6} - 4 x^{5} - 2 x^{4} - 4 x^{3} + 4 x^{2} + 8 x + 6$$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$$[\ ]_{7}^{6}$
3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
$5$5.7.6.1$x^{7} - 5$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
5.14.13.2$x^{14} + 10$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
7Data not computed
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$