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Properties

Label 21.1.77689469353...9872.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{18}\cdot 7^{17}\cdot 17^{19}\cdot 127^{7}$
Root discriminant $571.06$
Ramified primes $2, 7, 17, 127$
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![-3290030542336, 4588789081088, -2226072324992, 287175699584, 861500572256, -363434677664, 120179963920, 9691332318, -10492301162, 6308769536, -1802769584, 490225342, -95264182, 21564242, -3262297, 673467, -67195, 16265, -1039, 217, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 217*x^19 - 1039*x^18 + 16265*x^17 - 67195*x^16 + 673467*x^15 - 3262297*x^14 + 21564242*x^13 - 95264182*x^12 + 490225342*x^11 - 1802769584*x^10 + 6308769536*x^9 - 10492301162*x^8 + 9691332318*x^7 + 120179963920*x^6 - 363434677664*x^5 + 861500572256*x^4 + 287175699584*x^3 - 2226072324992*x^2 + 4588789081088*x - 3290030542336)
 
gp: K = bnfinit(x^21 - 7*x^20 + 217*x^19 - 1039*x^18 + 16265*x^17 - 67195*x^16 + 673467*x^15 - 3262297*x^14 + 21564242*x^13 - 95264182*x^12 + 490225342*x^11 - 1802769584*x^10 + 6308769536*x^9 - 10492301162*x^8 + 9691332318*x^7 + 120179963920*x^6 - 363434677664*x^5 + 861500572256*x^4 + 287175699584*x^3 - 2226072324992*x^2 + 4588789081088*x - 3290030542336, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 217 x^{19} - 1039 x^{18} + 16265 x^{17} - 67195 x^{16} + 673467 x^{15} - 3262297 x^{14} + 21564242 x^{13} - 95264182 x^{12} + 490225342 x^{11} - 1802769584 x^{10} + 6308769536 x^{9} - 10492301162 x^{8} + 9691332318 x^{7} + 120179963920 x^{6} - 363434677664 x^{5} + 861500572256 x^{4} + 287175699584 x^{3} - 2226072324992 x^{2} + 4588789081088 x - 3290030542336 \)

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:  \(7768946935337343279213431536688153817536377176369532239872=2^{18}\cdot 7^{17}\cdot 17^{19}\cdot 127^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $571.06$
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:  $2, 7, 17, 127$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{136} a^{16} + \frac{9}{136} a^{15} - \frac{27}{136} a^{14} + \frac{5}{136} a^{13} + \frac{45}{136} a^{12} + \frac{1}{8} a^{11} + \frac{47}{136} a^{10} + \frac{27}{136} a^{9} + \frac{3}{68} a^{8} - \frac{9}{68} a^{7} - \frac{1}{4} a^{6} + \frac{3}{17} a^{5} - \frac{6}{17} a^{4} + \frac{3}{68} a^{3} - \frac{33}{68} a^{2} + \frac{8}{17} a - \frac{4}{17}$, $\frac{1}{272} a^{17} - \frac{1}{272} a^{16} + \frac{19}{272} a^{15} + \frac{3}{272} a^{14} - \frac{5}{272} a^{13} - \frac{25}{272} a^{12} - \frac{123}{272} a^{11} + \frac{101}{272} a^{10} - \frac{8}{17} a^{9} + \frac{29}{136} a^{8} + \frac{5}{136} a^{7} + \frac{23}{68} a^{6} + \frac{15}{34} a^{5} - \frac{29}{136} a^{4} - \frac{63}{136} a^{3} + \frac{11}{68} a^{2} + \frac{1}{34} a + \frac{3}{17}$, $\frac{1}{120224} a^{18} - \frac{45}{120224} a^{17} + \frac{267}{120224} a^{16} + \frac{371}{7072} a^{15} - \frac{15845}{120224} a^{14} + \frac{37799}{120224} a^{13} + \frac{33685}{120224} a^{12} + \frac{8573}{120224} a^{11} + \frac{47}{578} a^{10} + \frac{18451}{60112} a^{9} - \frac{28063}{60112} a^{8} + \frac{7903}{30056} a^{7} + \frac{3469}{7514} a^{6} + \frac{87}{272} a^{5} - \frac{1455}{4624} a^{4} + \frac{12379}{30056} a^{3} + \frac{925}{7514} a^{2} + \frac{492}{3757} a - \frac{1120}{3757}$, $\frac{1}{240448} a^{19} - \frac{1}{240448} a^{18} + \frac{55}{240448} a^{17} + \frac{375}{240448} a^{16} - \frac{28289}{240448} a^{15} - \frac{44117}{240448} a^{14} - \frac{14583}{240448} a^{13} - \frac{2999}{14144} a^{12} - \frac{601}{3536} a^{11} - \frac{21069}{120224} a^{10} + \frac{4977}{120224} a^{9} - \frac{21623}{60112} a^{8} + \frac{81}{289} a^{7} - \frac{30877}{120224} a^{6} - \frac{3019}{9248} a^{5} - \frac{28051}{60112} a^{4} + \frac{2247}{7514} a^{3} - \frac{31}{884} a^{2} - \frac{1670}{3757} a - \frac{330}{3757}$, $\frac{1}{2346086780760625824484709679937400866864083564982653510877127777075002985985019585055452102738996867571401251968} a^{20} - \frac{1058303430682720214323160241822429454659117730194202705933522845610029714810442967622252669097431510474645}{2346086780760625824484709679937400866864083564982653510877127777075002985985019585055452102738996867571401251968} a^{19} - \frac{8893444783014182730675175987131612355270560326043912875792072713060090052169366152171100055177136138892321}{2346086780760625824484709679937400866864083564982653510877127777075002985985019585055452102738996867571401251968} a^{18} + \frac{3108707320976784794780389065955182529646050706997345797423072038227599804517839720983380166245947859654661231}{2346086780760625824484709679937400866864083564982653510877127777075002985985019585055452102738996867571401251968} a^{17} + \frac{3591006073883438236320444024354268389817793577442986810802084150794885281723900172168639353887044757976260743}{2346086780760625824484709679937400866864083564982653510877127777075002985985019585055452102738996867571401251968} a^{16} + \frac{3312613483879893648515957154622996235531809557326398182976314748396559548268589313610163811780676195002172595}{2346086780760625824484709679937400866864083564982653510877127777075002985985019585055452102738996867571401251968} a^{15} - \frac{199552630962072993098674744857422537532024079149355752044339270900903999797564261564865914327901127035349938815}{2346086780760625824484709679937400866864083564982653510877127777075002985985019585055452102738996867571401251968} a^{14} - \frac{218867817238567761052602855200922039559871965850506158606968761209931730324550048173463740387582998682645416151}{2346086780760625824484709679937400866864083564982653510877127777075002985985019585055452102738996867571401251968} a^{13} - \frac{107228713832973900353423233375849201813547627093960453054665657044980372380934894942181556087229054163226459}{618041828440628510138226996822286845854605786349487226258463587216807952050848152016715517054530260161064608} a^{12} - \frac{403608062537961698154066786954553337415533905233158579147963857575969283028263826578300550710866047513760675871}{1173043390380312912242354839968700433432041782491326755438563888537501492992509792527726051369498433785700625984} a^{11} - \frac{576879405456849998122471501833743948553465423872218009068252296311433506104699634358085458776480710079678193063}{1173043390380312912242354839968700433432041782491326755438563888537501492992509792527726051369498433785700625984} a^{10} + \frac{4805725057760234285378529126150527186438622970777500305038324175284378565325322203118147785543025180989810073}{45117053476165881240090570768026939747386222403512567516867841866826980499711915097220232744980708991757716384} a^{9} - \frac{105939243981885363217991675777619050258287230035889610513252894850133133139470627626274254142891095378848114911}{293260847595078228060588709992175108358010445622831688859640972134375373248127448131931512842374608446425156496} a^{8} - \frac{103078652944042910957488265645334485997583906500349748736794767548534217351036946270452437945088758923450634205}{1173043390380312912242354839968700433432041782491326755438563888537501492992509792527726051369498433785700625984} a^{7} + \frac{490644091592125319356635965102365761983743916318380105500479403090113597217132520191397908606324905326106567397}{1173043390380312912242354839968700433432041782491326755438563888537501492992509792527726051369498433785700625984} a^{6} - \frac{33126751105789651987399937763767562957369906381574935826843093135473211667349891676828572997004516168494476111}{586521695190156456121177419984350216716020891245663377719281944268750746496254896263863025684749216892850312992} a^{5} + \frac{119330170258940565152983017591560913034207492124980831371521662974889359033055351672206089976913180849994896953}{293260847595078228060588709992175108358010445622831688859640972134375373248127448131931512842374608446425156496} a^{4} + \frac{16443125351206053245224229268202232049387984167149385135895204090106563722260858368877451625056966089489633761}{146630423797539114030294354996087554179005222811415844429820486067187686624063724065965756421187304223212578248} a^{3} + \frac{15962830439849606458596560719213314911065238895048374812172565655127847770253945367084487747908626544543117651}{73315211898769557015147177498043777089502611405707922214910243033593843312031862032982878210593652111606289124} a^{2} - \frac{6676987725888467050855520460195039738521177722617263121423340221816154699529944679370985763577466859984467143}{36657605949384778507573588749021888544751305702853961107455121516796921656015931016491439105296826055803144562} a + \frac{1148395479174733326363451889418386652407546550122267210942810264724517649271587015742829004497697106645932765}{18328802974692389253786794374510944272375652851426980553727560758398460828007965508245719552648413027901572281}$

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

3.1.105791.2, 7.1.25963527819712.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ $21$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
7Data not computed
$17$17.7.6.1$x^{7} - 17$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
17.14.13.1$x^{14} - 17$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
$127$$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$