↑

Properties

Label 21.21.1425682560...9889.1
Degree $21$
Signature $[21, 0]$
Discriminant $7^{14}\cdot 71^{18}$
Root discriminant $141.32$
Ramified primes $7, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{21}$ (as 21T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![26494201, -25641148, -152780306, 234723315, 144428798, -422345627, 105332901, 208156183, -111354014, -39386623, 35371490, 1655193, -5452225, 423688, 451646, -65058, -20240, 3776, 456, -100, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^20 - 100*x^19 + 456*x^18 + 3776*x^17 - 20240*x^16 - 65058*x^15 + 451646*x^14 + 423688*x^13 - 5452225*x^12 + 1655193*x^11 + 35371490*x^10 - 39386623*x^9 - 111354014*x^8 + 208156183*x^7 + 105332901*x^6 - 422345627*x^5 + 144428798*x^4 + 234723315*x^3 - 152780306*x^2 - 25641148*x + 26494201)
 
gp: K = bnfinit(x^21 - 4*x^20 - 100*x^19 + 456*x^18 + 3776*x^17 - 20240*x^16 - 65058*x^15 + 451646*x^14 + 423688*x^13 - 5452225*x^12 + 1655193*x^11 + 35371490*x^10 - 39386623*x^9 - 111354014*x^8 + 208156183*x^7 + 105332901*x^6 - 422345627*x^5 + 144428798*x^4 + 234723315*x^3 - 152780306*x^2 - 25641148*x + 26494201, 1)
 

Normalized defining polynomial

\( x^{21} - 4 x^{20} - 100 x^{19} + 456 x^{18} + 3776 x^{17} - 20240 x^{16} - 65058 x^{15} + 451646 x^{14} + 423688 x^{13} - 5452225 x^{12} + 1655193 x^{11} + 35371490 x^{10} - 39386623 x^{9} - 111354014 x^{8} + 208156183 x^{7} + 105332901 x^{6} - 422345627 x^{5} + 144428798 x^{4} + 234723315 x^{3} - 152780306 x^{2} - 25641148 x + 26494201 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1425682560385717640661443870033677687676469889=7^{14}\cdot 71^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $141.32$
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:  $7, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(497=7\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{497}(1,·)$, $\chi_{497}(387,·)$, $\chi_{497}(261,·)$, $\chi_{497}(72,·)$, $\chi_{497}(463,·)$, $\chi_{497}(400,·)$, $\chi_{497}(403,·)$, $\chi_{497}(214,·)$, $\chi_{497}(471,·)$, $\chi_{497}(30,·)$, $\chi_{497}(32,·)$, $\chi_{497}(162,·)$, $\chi_{497}(37,·)$, $\chi_{497}(233,·)$, $\chi_{497}(172,·)$, $\chi_{497}(456,·)$, $\chi_{497}(179,·)$, $\chi_{497}(116,·)$, $\chi_{497}(375,·)$, $\chi_{497}(316,·)$, $\chi_{497}(190,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{11} - \frac{2}{5} a^{7} - \frac{2}{5} a^{5} - \frac{1}{5} a^{3} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{16} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{17} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{425} a^{18} + \frac{1}{17} a^{17} + \frac{7}{425} a^{16} - \frac{2}{425} a^{15} - \frac{38}{425} a^{14} + \frac{32}{425} a^{13} - \frac{14}{425} a^{12} + \frac{42}{425} a^{11} + \frac{196}{425} a^{10} - \frac{183}{425} a^{9} + \frac{161}{425} a^{8} + \frac{197}{425} a^{7} - \frac{4}{85} a^{6} + \frac{149}{425} a^{5} - \frac{4}{25} a^{4} + \frac{27}{425} a^{3} - \frac{189}{425} a^{2} - \frac{56}{425} a - \frac{44}{425}$, $\frac{1}{425} a^{19} - \frac{23}{425} a^{17} - \frac{7}{425} a^{16} + \frac{12}{425} a^{15} - \frac{38}{425} a^{14} + \frac{36}{425} a^{13} - \frac{33}{425} a^{12} - \frac{4}{425} a^{11} + \frac{11}{25} a^{10} - \frac{109}{425} a^{9} - \frac{3}{425} a^{8} - \frac{4}{17} a^{7} - \frac{31}{425} a^{6} + \frac{117}{425} a^{5} - \frac{143}{425} a^{4} - \frac{184}{425} a^{3} - \frac{176}{425} a^{2} - \frac{4}{425} a + \frac{16}{85}$, $\frac{1}{3532348843617354033785016708068067561719742197419411515475} a^{20} + \frac{1179859478307529327121870776183397240382950194332722193}{3532348843617354033785016708068067561719742197419411515475} a^{19} - \frac{1705977274483583093491083822118212087425170668419919703}{3532348843617354033785016708068067561719742197419411515475} a^{18} - \frac{286597568495014696032175007863608530325402605746670187566}{3532348843617354033785016708068067561719742197419411515475} a^{17} + \frac{233198143887530132030571022524916406614761711212683061731}{3532348843617354033785016708068067561719742197419411515475} a^{16} + \frac{312532044530795775198507070612083308135173369477929168778}{3532348843617354033785016708068067561719742197419411515475} a^{15} - \frac{216347727581218140703399030883588814997360154470797540023}{3532348843617354033785016708068067561719742197419411515475} a^{14} + \frac{53714423692762660403017934261506653274564634891070312066}{706469768723470806757003341613613512343948439483882303095} a^{13} - \frac{282530370625916768340715327482547771846261005208708584043}{3532348843617354033785016708068067561719742197419411515475} a^{12} + \frac{138488177139040877938287138215113622425494471707739622461}{706469768723470806757003341613613512343948439483882303095} a^{11} - \frac{22603145137021554160113504561683454897237089372596441573}{3532348843617354033785016708068067561719742197419411515475} a^{10} + \frac{272791200029472847143898494376908450230725570136886970208}{706469768723470806757003341613613512343948439483882303095} a^{9} - \frac{3851836111787036852733771666752459300187913161344230152}{207785226095138472575589218121651033042337776318788912675} a^{8} - \frac{1339135118874386040805752810639271900341430235504608630771}{3532348843617354033785016708068067561719742197419411515475} a^{7} + \frac{1070546458464488701905563316103739059529045843382739122609}{3532348843617354033785016708068067561719742197419411515475} a^{6} - \frac{1130391097356404673245923234814747561065710018796242774982}{3532348843617354033785016708068067561719742197419411515475} a^{5} - \frac{1334948618063218142001240811096052347490779389502687190403}{3532348843617354033785016708068067561719742197419411515475} a^{4} + \frac{1178214087787939714369012976256351139904120683237015842992}{3532348843617354033785016708068067561719742197419411515475} a^{3} - \frac{342005625775923086361583121011219256153067407972294867547}{3532348843617354033785016708068067561719742197419411515475} a^{2} + \frac{551456118648812161739235617920827293348689086546870747828}{3532348843617354033785016708068067561719742197419411515475} a - \frac{100655764083920782802416485211836355382255010938050365201}{706469768723470806757003341613613512343948439483882303095}$

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:  $20$
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:  \( 4244799371179891.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{21}$ (as 21T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 21
The 21 conjugacy class representatives for $C_{21}$
Character table for $C_{21}$ is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.7.128100283921.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $21$ $21$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{7}$ R $21$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ $21$ $21$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ $21$ $21$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ $21$ $21$ $21$

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
7Data not computed
$71$71.7.6.1$x^{7} - 71$$7$$1$$6$$C_7$$[\ ]_{7}$
71.7.6.1$x^{7} - 71$$7$$1$$6$$C_7$$[\ ]_{7}$
71.7.6.1$x^{7} - 71$$7$$1$$6$$C_7$$[\ ]_{7}$