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Properties

Label 21.3.15986407370...2767.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 211^{18}$
Root discriminant $474.65$
Ramified primes $7, 211$
Class number $343$ (GRH)
Class group $[7, 7, 7]$ (GRH)
Galois group $F_7$ (as 21T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![745947224512, -858652083072, -214646951456, -43889160896, 22083422172, 83230171864, 5267658761, -12471465894, -350840512, 73289981, -5679616, 24379691, -5891416, -856933, 28738, -36395, 12232, 29, -34, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 29*x^17 + 12232*x^16 - 36395*x^15 + 28738*x^14 - 856933*x^13 - 5891416*x^12 + 24379691*x^11 - 5679616*x^10 + 73289981*x^9 - 350840512*x^8 - 12471465894*x^7 + 5267658761*x^6 + 83230171864*x^5 + 22083422172*x^4 - 43889160896*x^3 - 214646951456*x^2 - 858652083072*x + 745947224512)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 29*x^17 + 12232*x^16 - 36395*x^15 + 28738*x^14 - 856933*x^13 - 5891416*x^12 + 24379691*x^11 - 5679616*x^10 + 73289981*x^9 - 350840512*x^8 - 12471465894*x^7 + 5267658761*x^6 + 83230171864*x^5 + 22083422172*x^4 - 43889160896*x^3 - 214646951456*x^2 - 858652083072*x + 745947224512, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} + 12232 x^{16} - 36395 x^{15} + 28738 x^{14} - 856933 x^{13} - 5891416 x^{12} + 24379691 x^{11} - 5679616 x^{10} + 73289981 x^{9} - 350840512 x^{8} - 12471465894 x^{7} + 5267658761 x^{6} + 83230171864 x^{5} + 22083422172 x^{4} - 43889160896 x^{3} - 214646951456 x^{2} - 858652083072 x + 745947224512 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-159864073705644817826866721926705417106647399937729252767=-\,7^{17}\cdot 211^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $474.65$
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:  $7, 211$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{3}$, $\frac{1}{16880} a^{18} + \frac{183}{16880} a^{17} - \frac{593}{16880} a^{16} - \frac{43}{1055} a^{15} - \frac{411}{3376} a^{14} + \frac{291}{1688} a^{13} - \frac{1059}{16880} a^{12} + \frac{143}{1055} a^{11} - \frac{2257}{16880} a^{10} - \frac{27}{1688} a^{9} - \frac{4117}{16880} a^{8} + \frac{549}{8440} a^{7} - \frac{6171}{16880} a^{6} - \frac{73}{1688} a^{5} + \frac{1957}{8440} a^{4} + \frac{6533}{16880} a^{3} + \frac{2689}{8440} a^{2} + \frac{9}{2110} a - \frac{443}{2110}$, $\frac{1}{33760} a^{19} - \frac{1}{33760} a^{18} - \frac{101}{6752} a^{17} - \frac{81}{2110} a^{16} + \frac{2157}{33760} a^{15} + \frac{123}{3376} a^{14} - \frac{559}{33760} a^{13} + \frac{189}{2110} a^{12} - \frac{5469}{33760} a^{11} + \frac{729}{16880} a^{10} + \frac{7583}{33760} a^{9} - \frac{487}{16880} a^{8} - \frac{1423}{33760} a^{7} - \frac{6553}{16880} a^{6} + \frac{3707}{16880} a^{5} - \frac{4683}{33760} a^{4} + \frac{3003}{16880} a^{3} + \frac{93}{211} a^{2} - \frac{2099}{4220} a + \frac{333}{1055}$, $\frac{1}{519178361982612598698809985764009041895160468766130052116579676495171291589439880274390384929640691272415074240} a^{20} + \frac{173386406719594675464324399597255858194598769757657755927711100723347806686095733790357818052132055885931}{519178361982612598698809985764009041895160468766130052116579676495171291589439880274390384929640691272415074240} a^{19} - \frac{4801711339327169117714778450819281163604902357535214657393144566397301720855263472755959241120785693079717}{519178361982612598698809985764009041895160468766130052116579676495171291589439880274390384929640691272415074240} a^{18} - \frac{15133871793895874861851135946095516663172008542858154217572369390034553326603062899430933534169764097254246849}{129794590495653149674702496441002260473790117191532513029144919123792822897359970068597596232410172818103768560} a^{17} + \frac{12320254366636428698814421930071162759059567671419701856945714901309594705661476158100433320459372224475503081}{103835672396522519739761997152801808379032093753226010423315935299034258317887976054878076985928138254483014848} a^{16} - \frac{22764566912828421446934461319662738084328174971060421709828883972747692055928152384931447953649434952138964563}{259589180991306299349404992882004520947580234383065026058289838247585645794719940137195192464820345636207537120} a^{15} + \frac{39272165740624768638474451056165510093411721047705858105786037532303585702624237474527641617900712198279071001}{519178361982612598698809985764009041895160468766130052116579676495171291589439880274390384929640691272415074240} a^{14} + \frac{7449336486411238148511807105494733316457925795269990768204647333719197349754487725393941125596729262071987199}{129794590495653149674702496441002260473790117191532513029144919123792822897359970068597596232410172818103768560} a^{13} - \frac{5157835429384953699492118085217915085317125003865651227127255332111251144360963885105028064986177373258061697}{39936797075585584515293075828000695530396959135856157855121513576551637814572298482645414225356976251724236480} a^{12} - \frac{5863396581530933120625453644123198364609950584181103634248254770243057615199643942583349608289167541256787889}{51917836198261259869880998576400904189516046876613005211657967649517129158943988027439038492964069127241507424} a^{11} - \frac{53073451744369729356994157078241460093859321384586939700001682202830667245630509997832673917403599306802055481}{519178361982612598698809985764009041895160468766130052116579676495171291589439880274390384929640691272415074240} a^{10} + \frac{1341449834593677377174369693943323838702145806976469418175485222638826406703781095278180414270078560486278451}{259589180991306299349404992882004520947580234383065026058289838247585645794719940137195192464820345636207537120} a^{9} + \frac{58718935763396186993430188820641416679104827659157541794062433768045081928774177991959153223703865053124413449}{519178361982612598698809985764009041895160468766130052116579676495171291589439880274390384929640691272415074240} a^{8} + \frac{47639129130012113233756979024447935211608438626231690644139705037399784096810352659109817286840181570217047669}{259589180991306299349404992882004520947580234383065026058289838247585645794719940137195192464820345636207537120} a^{7} + \frac{51208516460618038858395090464975921242624278227285508492143512099654970227858056762077530681941940110597727111}{259589180991306299349404992882004520947580234383065026058289838247585645794719940137195192464820345636207537120} a^{6} - \frac{51718158847892533940698637552263172321769260706590239276650963795636432997128659520976929658448139229109126695}{103835672396522519739761997152801808379032093753226010423315935299034258317887976054878076985928138254483014848} a^{5} + \frac{1113508783642850844299832766790607479665815030045767314796725433238388065454315575169086257114009109920787329}{3993679707558558451529307582800069553039695913585615785512151357655163781457229848264541422535697625172423648} a^{4} - \frac{12290061369846909503877315438753005293016774320479298567756889234571775361119350513730630002487625446315656613}{32448647623913287418675624110250565118447529297883128257286229780948205724339992517149399058102543204525942140} a^{3} - \frac{3040995232476619024053791363041560445542436436081193318797953269937148696115233113965310735996512709239002929}{64897295247826574837351248220501130236895058595766256514572459561896411448679985034298798116205086409051884280} a^{2} + \frac{62743774683228957805407391593765782841233594972264791165348213427495554694309044045817562795360393248932001}{624012454306024758051454309812510867662452486497752466486273649633619340852692163791334597271202753933191195} a + \frac{3080815728534444939877080789540606035344298464219459106134515036055517203929832501582652563046021892475846778}{8112161905978321854668906027562641279611882324470782064321557445237051431084998129287349764525635801131485535}$

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

Class group and class number

$C_{7}\times C_{7}\times C_{7}$, which has order $343$ (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:  $11$
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:  \( 482362851034983940 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_7$ (as 21T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 7 conjugacy class representatives for $F_7$
Character table for $F_7$

Intermediate fields

\(\Q(\zeta_{7})^+\), Deg 7

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

Sibling fields

Galois closure: data not computed
Degree 7 sibling: data not computed
Degree 14 sibling: 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
211Data not computed