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Properties

Label 21.3.23744507219...4663.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 167^{18}$
Root discriminant $388.43$
Ramified primes $7, 167$
Class number $1$ (GRH)
Class group Trivial (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![-146043093056, -76802664064, 87604194608, 73337235104, -59139292492, -7267451240, 13211621361, -3295825882, -117158588, -36546963, 48390428, -15927429, 2227436, -306733, -111234, 37133, -9692, 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 - 9692*x^16 + 37133*x^15 - 111234*x^14 - 306733*x^13 + 2227436*x^12 - 15927429*x^11 + 48390428*x^10 - 36546963*x^9 - 117158588*x^8 - 3295825882*x^7 + 13211621361*x^6 - 7267451240*x^5 - 59139292492*x^4 + 73337235104*x^3 + 87604194608*x^2 - 76802664064*x - 146043093056)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 29*x^17 - 9692*x^16 + 37133*x^15 - 111234*x^14 - 306733*x^13 + 2227436*x^12 - 15927429*x^11 + 48390428*x^10 - 36546963*x^9 - 117158588*x^8 - 3295825882*x^7 + 13211621361*x^6 - 7267451240*x^5 - 59139292492*x^4 + 73337235104*x^3 + 87604194608*x^2 - 76802664064*x - 146043093056, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} - 9692 x^{16} + 37133 x^{15} - 111234 x^{14} - 306733 x^{13} + 2227436 x^{12} - 15927429 x^{11} + 48390428 x^{10} - 36546963 x^{9} - 117158588 x^{8} - 3295825882 x^{7} + 13211621361 x^{6} - 7267451240 x^{5} - 59139292492 x^{4} + 73337235104 x^{3} + 87604194608 x^{2} - 76802664064 x - 146043093056 \)

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:  \(-2374450721934441204800704599474362981402341385467744663=-\,7^{17}\cdot 167^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $388.43$
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:  $7, 167$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{14} + \frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{1}{8} a^{6} - \frac{3}{8} a^{4} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} + \frac{1}{8} a^{11} + \frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{16} - \frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{17} - \frac{1}{16} a^{16} - \frac{1}{16} a^{15} - \frac{1}{16} a^{13} + \frac{1}{8} a^{12} + \frac{3}{16} a^{11} - \frac{1}{4} a^{10} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{3}{16} a^{7} - \frac{1}{8} a^{6} - \frac{5}{16} a^{5} - \frac{3}{8} a^{4} - \frac{1}{4} a^{3} - \frac{7}{16} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{14589120} a^{18} + \frac{179267}{14589120} a^{17} + \frac{57163}{1122240} a^{16} - \frac{5729}{182364} a^{15} - \frac{35843}{2917824} a^{14} + \frac{9235}{1458912} a^{13} - \frac{432129}{4863040} a^{12} + \frac{25187}{607880} a^{11} - \frac{343907}{4863040} a^{10} - \frac{25007}{7294560} a^{9} - \frac{368967}{4863040} a^{8} + \frac{261953}{7294560} a^{7} + \frac{1560749}{14589120} a^{6} - \frac{51283}{208416} a^{5} + \frac{126589}{1042080} a^{4} - \frac{31073}{160320} a^{3} + \frac{941}{6240} a^{2} + \frac{86441}{521040} a - \frac{114203}{260520}$, $\frac{1}{26202059520} a^{19} + \frac{123}{8734019840} a^{18} - \frac{159616309}{8734019840} a^{17} - \frac{75615257}{4367009920} a^{16} + \frac{304482701}{5240411904} a^{15} - \frac{5776817}{436700992} a^{14} + \frac{663456673}{26202059520} a^{13} + \frac{403062609}{4367009920} a^{12} + \frac{232638201}{1746803968} a^{11} - \frac{533328349}{6550514880} a^{10} - \frac{3804593209}{26202059520} a^{9} + \frac{54163243}{935787840} a^{8} - \frac{705147693}{8734019840} a^{7} + \frac{1041518347}{6550514880} a^{6} - \frac{304391701}{1871575680} a^{5} + \frac{136440429}{1247717120} a^{4} - \frac{33082741}{155964640} a^{3} + \frac{174058139}{467893920} a^{2} + \frac{3801121}{58486740} a - \frac{34554733}{233946960}$, $\frac{1}{589459228899050678026859483653576470480888102663839754873754932939033882278224226979680893271040} a^{20} + \frac{506723794470252085030246082108660219521191908156774717417077440164739480211828456267}{589459228899050678026859483653576470480888102663839754873754932939033882278224226979680893271040} a^{19} + \frac{1798421750675607772518302446054996039110551270517551935236272678116759853187529029520159}{117891845779810135605371896730715294096177620532767950974750986587806776455644845395936178654208} a^{18} + \frac{2312325664984525745332457871951083471481274472445076510738120308216567849389137540920043969633}{147364807224762669506714870913394117620222025665959938718438733234758470569556056744920223317760} a^{17} - \frac{201864972485504465785419955583394627310511293315654009893013850368458184888155199912593751789}{15114339202539760975047679068040422320022771863175378330096280331770099545595492999478997263360} a^{16} + \frac{41723872104423842197356064831842706508831975415704107039288939262257779625599228598641355391}{4534301760761928292514303720412126696006831558952613499028884099531029863678647899843699179008} a^{15} + \frac{4903496327228276284887893031708775066000430663794282765683557202715337787129872710211540436239}{84208461271292954003837069093368067211555443237691393553393561848433411754032032425668699038720} a^{14} - \frac{89550318184921359821673892260063556449741609833672888714396738650906868211790690338895505557}{6140200301031777896113119621391421567509251069414997446601613884781602940398169031038342638240} a^{13} + \frac{183084215117258762581727083024286167222753371941100953072895341904730708241844915432787999629}{1085560274215562942959225568422792763316552675255690156305257703386802729794151430901806433280} a^{12} - \frac{73679399383581314708297096687163041018390967219114787872798787084305250426736680124482201142783}{294729614449525339013429741826788235240444051331919877436877466469516941139112113489840446635520} a^{11} - \frac{13455101277028658009499597039933466234038892065448785663296619076238213235635309497553291570657}{589459228899050678026859483653576470480888102663839754873754932939033882278224226979680893271040} a^{10} + \frac{16875890466608005597377606226147373269041152147081831552589741386510103682811493818553769364741}{294729614449525339013429741826788235240444051331919877436877466469516941139112113489840446635520} a^{9} + \frac{126914790668095161769213044550227235793622175130744978416984950293020306957263665733461245168033}{589459228899050678026859483653576470480888102663839754873754932939033882278224226979680893271040} a^{8} + \frac{823893221608627574162827302027447845271849263377315546146296724624090671528741130851601494557}{7557169601269880487523839534020211160011385931587689165048140165885049772797746499739498631680} a^{7} - \frac{2331587705276109930536295927222609032578544686030301851431133937125925158544307238038012990057}{19648640963301689267561982788452549016029603422127991829125164431301129409274140899322696442368} a^{6} + \frac{39089878907691863307575806750638029389886849534451544130349202942823215549178805656658988436771}{84208461271292954003837069093368067211555443237691393553393561848433411754032032425668699038720} a^{5} - \frac{2186980020647128326081412991009774687383418881768929047501829686307862632818551612554750329809}{42104230635646477001918534546684033605777721618845696776696780924216705877016016212834349519360} a^{4} + \frac{211700901919285239512650148599503586256569327212524110275274538784504632959675129832350033687}{701737177260774616698642242444733893429628693647428279611613015403611764616933603547239158656} a^{3} - \frac{160662463695172818861707657754633485281112178363147393389945095637085382990065807744136270879}{350868588630387308349321121222366946714814346823714139805806507701805882308466801773619579328} a^{2} + \frac{189333187598636419741236578937358561218126253991299503867192730653465367676637559700139243379}{5263028829455809625239816818335504200722215202355712097087097615527088234627002026604293689920} a + \frac{398197530066028017186544488234545436089343820563164193014277953635272686732246247126575654981}{877171471575968270873302803055917366787035867059285349514516269254514705771167004434048948320}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 20918292900910133000 \) (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}$
$167$167.7.6.1$x^{7} - 167$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
167.7.6.1$x^{7} - 167$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
167.7.6.1$x^{7} - 167$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$