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Properties

Label 21.1.67074988669...0000.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{33}\cdot 3^{19}\cdot 5^{19}\cdot 7^{21}\cdot 349^{7}$
Root discriminant $1697.60$
Ramified primes $2, 3, 5, 7, 349$
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![-142292871632, -190888547920, -140692650448, -106034214832, -19136576368, 21977505536, 7646828112, 699630280, -193560080, -258483120, -44208724, 15049580, 6005034, 372946, -275051, -59199, 2849, 2681, 119, -49, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 49*x^19 + 119*x^18 + 2681*x^17 + 2849*x^16 - 59199*x^15 - 275051*x^14 + 372946*x^13 + 6005034*x^12 + 15049580*x^11 - 44208724*x^10 - 258483120*x^9 - 193560080*x^8 + 699630280*x^7 + 7646828112*x^6 + 21977505536*x^5 - 19136576368*x^4 - 106034214832*x^3 - 140692650448*x^2 - 190888547920*x - 142292871632)
 
gp: K = bnfinit(x^21 - 7*x^20 - 49*x^19 + 119*x^18 + 2681*x^17 + 2849*x^16 - 59199*x^15 - 275051*x^14 + 372946*x^13 + 6005034*x^12 + 15049580*x^11 - 44208724*x^10 - 258483120*x^9 - 193560080*x^8 + 699630280*x^7 + 7646828112*x^6 + 21977505536*x^5 - 19136576368*x^4 - 106034214832*x^3 - 140692650448*x^2 - 190888547920*x - 142292871632, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 49 x^{19} + 119 x^{18} + 2681 x^{17} + 2849 x^{16} - 59199 x^{15} - 275051 x^{14} + 372946 x^{13} + 6005034 x^{12} + 15049580 x^{11} - 44208724 x^{10} - 258483120 x^{9} - 193560080 x^{8} + 699630280 x^{7} + 7646828112 x^{6} + 21977505536 x^{5} - 19136576368 x^{4} - 106034214832 x^{3} - 140692650448 x^{2} - 190888547920 x - 142292871632 \)

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:  \(67074988669695514157661184969648385566174950359040000000000000000000=2^{33}\cdot 3^{19}\cdot 5^{19}\cdot 7^{21}\cdot 349^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1697.60$
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:  $2, 3, 5, 7, 349$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} + \frac{1}{3} a^{6} - \frac{1}{2} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{9} + \frac{1}{3} a^{7} - \frac{1}{2} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{6} a^{12} - \frac{1}{3} a^{6} - \frac{1}{2} a^{4} - \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{3} a^{7} - \frac{1}{2} a^{5} - \frac{1}{3} a$, $\frac{1}{180} a^{14} - \frac{1}{20} a^{13} - \frac{7}{180} a^{12} - \frac{1}{60} a^{11} + \frac{1}{20} a^{10} + \frac{1}{60} a^{9} + \frac{13}{180} a^{8} - \frac{9}{20} a^{7} + \frac{13}{90} a^{6} - \frac{13}{30} a^{5} - \frac{1}{10} a^{4} - \frac{4}{15} a^{3} + \frac{19}{45} a^{2} - \frac{1}{5} a + \frac{2}{45}$, $\frac{1}{180} a^{15} + \frac{1}{90} a^{13} - \frac{1}{30} a^{12} + \frac{1}{15} a^{11} - \frac{1}{30} a^{10} + \frac{1}{18} a^{9} + \frac{1}{30} a^{8} - \frac{73}{180} a^{7} + \frac{11}{30} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{14}{45} a^{3} + \frac{4}{15} a^{2} + \frac{11}{45} a + \frac{1}{15}$, $\frac{1}{180} a^{16} + \frac{1}{15} a^{13} - \frac{1}{45} a^{12} - \frac{2}{45} a^{10} - \frac{1}{20} a^{8} + \frac{4}{15} a^{7} + \frac{17}{45} a^{6} + \frac{1}{5} a^{5} - \frac{1}{9} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{7}{15} a + \frac{11}{45}$, $\frac{1}{1080} a^{17} - \frac{1}{360} a^{16} - \frac{1}{1080} a^{15} - \frac{1}{1080} a^{14} - \frac{1}{72} a^{13} + \frac{49}{1080} a^{12} - \frac{71}{1080} a^{11} - \frac{29}{360} a^{10} - \frac{29}{540} a^{9} - \frac{1}{108} a^{8} + \frac{2}{9} a^{7} - \frac{34}{135} a^{6} - \frac{131}{270} a^{5} - \frac{2}{5} a^{4} + \frac{25}{54} a^{3} - \frac{56}{135} a^{2} + \frac{1}{5} a - \frac{61}{135}$, $\frac{1}{14040} a^{18} - \frac{1}{4680} a^{17} - \frac{19}{14040} a^{16} - \frac{19}{14040} a^{15} + \frac{1}{4680} a^{14} + \frac{107}{2808} a^{13} - \frac{29}{1080} a^{12} + \frac{301}{4680} a^{11} - \frac{23}{1755} a^{10} + \frac{281}{3510} a^{9} + \frac{7}{585} a^{8} + \frac{17}{702} a^{7} + \frac{23}{135} a^{6} - \frac{59}{390} a^{5} - \frac{68}{1755} a^{4} + \frac{763}{1755} a^{3} - \frac{31}{65} a^{2} + \frac{67}{351} a - \frac{82}{195}$, $\frac{1}{232993800} a^{19} - \frac{878}{29124225} a^{18} + \frac{1282}{9708075} a^{17} + \frac{1753}{995700} a^{16} + \frac{19411}{9708075} a^{15} - \frac{8339}{9708075} a^{14} - \frac{741454}{9708075} a^{13} - \frac{771527}{9708075} a^{12} + \frac{261371}{77664600} a^{11} + \frac{212813}{3236025} a^{10} + \frac{97307}{1941615} a^{9} - \frac{102553}{7766460} a^{8} + \frac{701677}{2588820} a^{7} + \frac{24682}{215735} a^{6} + \frac{138773}{1941615} a^{5} - \frac{1436659}{3236025} a^{4} - \frac{4667282}{9708075} a^{3} - \frac{1980896}{9708075} a^{2} + \frac{7375153}{29124225} a + \frac{1290611}{29124225}$, $\frac{1}{979789661597092237727731871332859577058813078279858620256163518208570721596725800} a^{20} + \frac{179226496918406596379455997380203681306427048929718782514429868223158087}{97978966159709223772773187133285957705881307827985862025616351820857072159672580} a^{19} - \frac{228832557206588895570097823136048485593305202726514554101914629768134376531}{75368435507468633671363990102527659773754852175373740019704886016043901661286600} a^{18} - \frac{108027602330101232841808559128985599055661261794681726354965687110540883051579}{244947415399273059431932967833214894264703269569964655064040879552142680399181450} a^{17} + \frac{128857866108666638518088883279533017669216310110242087825350816109866786590723}{244947415399273059431932967833214894264703269569964655064040879552142680399181450} a^{16} - \frac{201306794917146475966620388320390632856697973033195923125449492119600452839941}{97978966159709223772773187133285957705881307827985862025616351820857072159672580} a^{15} + \frac{72279309002111696510513045955486947959006465408451529529980792884436036166477}{97978966159709223772773187133285957705881307827985862025616351820857072159672580} a^{14} - \frac{9859300597268919214506406303455142337336122490809251244403279373885841105679353}{244947415399273059431932967833214894264703269569964655064040879552142680399181450} a^{13} + \frac{42850650715502796181818529740569613442983105444871252578559014756141103170476101}{979789661597092237727731871332859577058813078279858620256163518208570721596725800} a^{12} - \frac{17254685192202722302860954655707802796283089338044985223133805190511238315445981}{489894830798546118863865935666429788529406539139929310128081759104285360798362900} a^{11} - \frac{3949197322069197956601338728224092901537908067388109853687290634114034157608547}{75368435507468633671363990102527659773754852175373740019704886016043901661286600} a^{10} - \frac{1954607838857949778480478760899457432423319818435922479587582503460547101240827}{24494741539927305943193296783321489426470326956996465506404087955214268039918145} a^{9} + \frac{269365229330173145445399353217119347721617719889766552685118750014007773918061}{7536843550746863367136399010252765977375485217537374001970488601604390166128660} a^{8} - \frac{1261811853888360184181781242100441712950103982197927965504999496447227854075801}{19595793231941844754554637426657191541176261565597172405123270364171414431934516} a^{7} - \frac{3666498996915556585644253984278431519821615719888428621144551098099351048894956}{24494741539927305943193296783321489426470326956996465506404087955214268039918145} a^{6} - \frac{70097756463756505905951798075821499864113690011806596479477861543142464369669327}{244947415399273059431932967833214894264703269569964655064040879552142680399181450} a^{5} + \frac{10193863717988704041586175616816120134351781470514271609324572726028557911157388}{24494741539927305943193296783321489426470326956996465506404087955214268039918145} a^{4} + \frac{100084475626911416079082093696127918041334879330445802189819420204865254430433041}{244947415399273059431932967833214894264703269569964655064040879552142680399181450} a^{3} + \frac{2549960909105970181359622135111470855605934362125726001146161056403555838640717}{40824569233212176571988827972202482377450544928327442510673479925357113399863575} a^{2} + \frac{1213297313953518176329628811085805885781377012217050868596208781910952137785429}{4536063248134686285776536441355831375272282769814160278963719991706345933318175} a + \frac{1456669916146332553833833609845636754550805401351057513116072605381155192251299}{122473707699636529715966483916607447132351634784982327532020439776071340199590725}$

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

3.1.41880.1, 7.1.600362847000000.33

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ $21$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{7}^{3}$
2.14.27.13$x^{14} - 4 x^{12} + 8 x^{11} - 6 x^{10} - 4 x^{9} - 6 x^{8} + 4 x^{7} + 6 x^{6} - 4 x^{5} + 4 x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 6$$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$$[\ ]_{7}^{6}$
5.14.13.1$x^{14} - 5$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
7Data not computed
349Data not computed