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Properties

Label 21.3.20993200950...6503.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 113^{18}$
Root discriminant $277.91$
Ramified primes $7, 113$
Class number $7$ (GRH)
Class group $[7]$ (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![2015987008, -5170051936, 3374286608, 979864648, -1978514300, 583377214, 477537789, -399264676, -75034142, 30173537, -6603224, -2604021, -188418, -409885, 85004, -22661, 3384, 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 + 3384*x^16 - 22661*x^15 + 85004*x^14 - 409885*x^13 - 188418*x^12 - 2604021*x^11 - 6603224*x^10 + 30173537*x^9 - 75034142*x^8 - 399264676*x^7 + 477537789*x^6 + 583377214*x^5 - 1978514300*x^4 + 979864648*x^3 + 3374286608*x^2 - 5170051936*x + 2015987008)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 29*x^17 + 3384*x^16 - 22661*x^15 + 85004*x^14 - 409885*x^13 - 188418*x^12 - 2604021*x^11 - 6603224*x^10 + 30173537*x^9 - 75034142*x^8 - 399264676*x^7 + 477537789*x^6 + 583377214*x^5 - 1978514300*x^4 + 979864648*x^3 + 3374286608*x^2 - 5170051936*x + 2015987008, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} + 3384 x^{16} - 22661 x^{15} + 85004 x^{14} - 409885 x^{13} - 188418 x^{12} - 2604021 x^{11} - 6603224 x^{10} + 30173537 x^{9} - 75034142 x^{8} - 399264676 x^{7} + 477537789 x^{6} + 583377214 x^{5} - 1978514300 x^{4} + 979864648 x^{3} + 3374286608 x^{2} - 5170051936 x + 2015987008 \)

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:  \(-2099320095095383219362534880897373365668616194446503=-\,7^{17}\cdot 113^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $277.91$
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:  $7, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{3}$, $\frac{1}{18984} a^{18} - \frac{185}{2712} a^{17} + \frac{31}{2712} a^{16} + \frac{1019}{9492} a^{15} - \frac{167}{18984} a^{14} - \frac{149}{9492} a^{13} - \frac{2347}{18984} a^{12} - \frac{158}{2373} a^{11} - \frac{155}{904} a^{10} - \frac{43}{3164} a^{9} - \frac{79}{2712} a^{8} - \frac{913}{4746} a^{7} - \frac{1937}{18984} a^{6} - \frac{395}{1582} a^{5} - \frac{389}{1582} a^{4} + \frac{6679}{18984} a^{3} + \frac{173}{452} a^{2} + \frac{109}{226} a - \frac{83}{339}$, $\frac{1}{7783440} a^{19} + \frac{137}{7783440} a^{18} - \frac{26129}{222384} a^{17} - \frac{137861}{3891720} a^{16} + \frac{744581}{7783440} a^{15} + \frac{1166}{23165} a^{14} + \frac{173659}{7783440} a^{13} - \frac{54704}{486465} a^{12} - \frac{52537}{7783440} a^{11} - \frac{8571}{259448} a^{10} - \frac{1148353}{7783440} a^{9} - \frac{114386}{486465} a^{8} + \frac{126143}{518896} a^{7} + \frac{8663}{555960} a^{6} + \frac{59209}{162155} a^{5} - \frac{3445871}{7783440} a^{4} + \frac{414731}{3891720} a^{3} + \frac{10969}{46330} a^{2} + \frac{46243}{138990} a + \frac{12101}{69495}$, $\frac{1}{7670693314300855897201706997569614336511997067242318199591685889338296987014931438394400} a^{20} - \frac{9137370047279587548374523234991392219882426834297593290066150325719475420930731}{219162666122881597062905914216274695328914201921209091416905311123951342486140898239840} a^{19} + \frac{11506030320509884919696034763334209468275346503434124389414615733509070566517659351}{7670693314300855897201706997569614336511997067242318199591685889338296987014931438394400} a^{18} + \frac{59879573054352162093643239340554054801273363287352490771987096392598575529725077151293}{479418332143803493575106687348100896031999816702644887474480368083643561688433214899650} a^{17} + \frac{121559357863913278121573291511362698089142863196395524994785026600189661493356833759033}{1534138662860171179440341399513922867302399413448463639918337177867659397402986287678880} a^{16} + \frac{101149090212174241633706168991634842182758416036969588529359847869048259338754718129669}{1278448885716809316200284499594935722751999511207053033265280981556382831169155239732400} a^{15} + \frac{116514952158764550452810674431274245329729572813527049840562787203106213250062541367949}{2556897771433618632400568999189871445503999022414106066530561963112765662338310479464800} a^{14} + \frac{746488318651106271244089091716092887991442208467209756858033550484005880100209880293}{46208995869282264440974138539575989978987934140013965057781240297218656548282719508400} a^{13} + \frac{620175521638675680706090745914095474059022088296117610906672660203496777586811961682951}{7670693314300855897201706997569614336511997067242318199591685889338296987014931438394400} a^{12} - \frac{3780827525656416672249976820348825032421858517782874580204401892634629564738773226831}{479418332143803493575106687348100896031999816702644887474480368083643561688433214899650} a^{11} - \frac{273291658122081528926725012307467119198434078744086340646881237030373838949483602780219}{1095813330614407985314529571081373476644571009606045457084526555619756712430704491199200} a^{10} - \frac{22129544792747780261918412066897112537081102232907927932276769663013881882988881245243}{153413866286017117944034139951392286730239941344846363991833717786765939740298628767888} a^{9} + \frac{245044730491458531579694043367674673036355998303998774484290662277510057324074277856479}{2556897771433618632400568999189871445503999022414106066530561963112765662338310479464800} a^{8} - \frac{49827676901644347091021456751242685924060530864519305166506568367038703968033268740604}{239709166071901746787553343674050448015999908351322443737240184041821780844216607449825} a^{7} - \frac{59269878653848636552380061596675628210054331639373754802540086888009669461629482450439}{273953332653601996328632392770343369161142752401511364271131638904939178107676122799800} a^{6} + \frac{280113615375855381211494181509783562491437854825280526182398383709576647101680799553353}{1534138662860171179440341399513922867302399413448463639918337177867659397402986287678880} a^{5} + \frac{232061430968712290358534321871809755893386117309886513342453751563613654188254073602987}{639224442858404658100142249797467861375999755603526516632640490778191415584577619866200} a^{4} - \frac{608862896152513306243244126941732546046214305243136654754887677008591221218183610780683}{1917673328575213974300426749392403584127999266810579549897921472334574246753732859598600} a^{3} - \frac{48722370685173857679139504575933932154654891064092760102245473515015997874859449340451}{136976666326800998164316196385171684580571376200755682135565819452469589053838061399900} a^{2} - \frac{1444194607654148402866991771194429888002956841909923644276093856920198594943394595717}{11414722193900083180359683032097640381714281350062973511297151621039132421153171783325} a - \frac{16051001636797517684191676087961666441710118352750745026332244359608202923870913087086}{34244166581700249541079049096292921145142844050188920533891454863117397263459515349975}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( 104794948949991400 \) (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}$
$113$113.7.6.2$x^{7} + 339$$7$$1$$6$$C_7$$[\ ]_{7}$
113.7.6.2$x^{7} + 339$$7$$1$$6$$C_7$$[\ ]_{7}$
113.7.6.2$x^{7} + 339$$7$$1$$6$$C_7$$[\ ]_{7}$