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Properties

Label 21.21.5116022582...2761.1
Degree $21$
Signature $[21, 0]$
Discriminant $7^{38}\cdot 13^{14}$
Root discriminant $187.01$
Ramified primes $7, 13$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_{21}$ (as 21T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![36861817, -940891777, -4978400147, 4239732917, 16657769884, -1210816971, -12232226328, -1670183867, 3516028369, 825207138, -476613326, -143285772, 33155136, 12183605, -1214079, -563108, 22050, 14364, -154, -189, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 189*x^19 - 154*x^18 + 14364*x^17 + 22050*x^16 - 563108*x^15 - 1214079*x^14 + 12183605*x^13 + 33155136*x^12 - 143285772*x^11 - 476613326*x^10 + 825207138*x^9 + 3516028369*x^8 - 1670183867*x^7 - 12232226328*x^6 - 1210816971*x^5 + 16657769884*x^4 + 4239732917*x^3 - 4978400147*x^2 - 940891777*x + 36861817)
 
gp: K = bnfinit(x^21 - 189*x^19 - 154*x^18 + 14364*x^17 + 22050*x^16 - 563108*x^15 - 1214079*x^14 + 12183605*x^13 + 33155136*x^12 - 143285772*x^11 - 476613326*x^10 + 825207138*x^9 + 3516028369*x^8 - 1670183867*x^7 - 12232226328*x^6 - 1210816971*x^5 + 16657769884*x^4 + 4239732917*x^3 - 4978400147*x^2 - 940891777*x + 36861817, 1)
 

Normalized defining polynomial

\( x^{21} - 189 x^{19} - 154 x^{18} + 14364 x^{17} + 22050 x^{16} - 563108 x^{15} - 1214079 x^{14} + 12183605 x^{13} + 33155136 x^{12} - 143285772 x^{11} - 476613326 x^{10} + 825207138 x^{9} + 3516028369 x^{8} - 1670183867 x^{7} - 12232226328 x^{6} - 1210816971 x^{5} + 16657769884 x^{4} + 4239732917 x^{3} - 4978400147 x^{2} - 940891777 x + 36861817 \)

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:  \(511602258272191839961280749067569616320187702761=7^{38}\cdot 13^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $187.01$
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:  $7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(637=7^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{637}(1,·)$, $\chi_{637}(263,·)$, $\chi_{637}(456,·)$, $\chi_{637}(9,·)$, $\chi_{637}(464,·)$, $\chi_{637}(81,·)$, $\chi_{637}(274,·)$, $\chi_{637}(536,·)$, $\chi_{637}(282,·)$, $\chi_{637}(92,·)$, $\chi_{637}(354,·)$, $\chi_{637}(547,·)$, $\chi_{637}(100,·)$, $\chi_{637}(555,·)$, $\chi_{637}(172,·)$, $\chi_{637}(365,·)$, $\chi_{637}(627,·)$, $\chi_{637}(373,·)$, $\chi_{637}(183,·)$, $\chi_{637}(445,·)$, $\chi_{637}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{19} a^{11} - \frac{8}{19} a^{10} - \frac{6}{19} a^{9} + \frac{5}{19} a^{8} - \frac{4}{19} a^{7} - \frac{8}{19} a^{6} + \frac{9}{19} a^{5} + \frac{3}{19} a^{4} + \frac{5}{19} a^{3} + \frac{8}{19} a^{2} - \frac{4}{19} a - \frac{9}{19}$, $\frac{1}{19} a^{12} + \frac{6}{19} a^{10} - \frac{5}{19} a^{9} - \frac{2}{19} a^{8} - \frac{2}{19} a^{7} + \frac{2}{19} a^{6} - \frac{1}{19} a^{5} - \frac{9}{19} a^{4} - \frac{9}{19} a^{3} + \frac{3}{19} a^{2} - \frac{3}{19} a + \frac{4}{19}$, $\frac{1}{19} a^{13} + \frac{5}{19} a^{10} - \frac{4}{19} a^{9} + \frac{6}{19} a^{8} + \frac{7}{19} a^{7} + \frac{9}{19} a^{6} - \frac{6}{19} a^{5} - \frac{8}{19} a^{4} - \frac{8}{19} a^{3} + \frac{6}{19} a^{2} + \frac{9}{19} a - \frac{3}{19}$, $\frac{1}{19} a^{14} - \frac{2}{19} a^{10} - \frac{2}{19} a^{9} + \frac{1}{19} a^{8} - \frac{9}{19} a^{7} - \frac{4}{19} a^{6} + \frac{4}{19} a^{5} - \frac{4}{19} a^{4} + \frac{7}{19} a^{2} - \frac{2}{19} a + \frac{7}{19}$, $\frac{1}{19} a^{15} + \frac{1}{19} a^{10} + \frac{8}{19} a^{9} + \frac{1}{19} a^{8} + \frac{7}{19} a^{7} + \frac{7}{19} a^{6} - \frac{5}{19} a^{5} + \frac{6}{19} a^{4} - \frac{2}{19} a^{3} - \frac{5}{19} a^{2} - \frac{1}{19} a + \frac{1}{19}$, $\frac{1}{19} a^{16} - \frac{3}{19} a^{10} + \frac{7}{19} a^{9} + \frac{2}{19} a^{8} - \frac{8}{19} a^{7} + \frac{3}{19} a^{6} - \frac{3}{19} a^{5} - \frac{5}{19} a^{4} + \frac{9}{19} a^{3} - \frac{9}{19} a^{2} + \frac{5}{19} a + \frac{9}{19}$, $\frac{1}{19} a^{17} + \frac{2}{19} a^{10} + \frac{3}{19} a^{9} + \frac{7}{19} a^{8} - \frac{9}{19} a^{7} - \frac{8}{19} a^{6} + \frac{3}{19} a^{5} - \frac{1}{19} a^{4} + \frac{6}{19} a^{3} - \frac{9}{19} a^{2} - \frac{3}{19} a - \frac{8}{19}$, $\frac{1}{19} a^{18} - \frac{1}{19}$, $\frac{1}{24187} a^{19} + \frac{540}{24187} a^{18} - \frac{447}{24187} a^{17} - \frac{492}{24187} a^{16} + \frac{405}{24187} a^{15} - \frac{111}{24187} a^{14} + \frac{101}{24187} a^{13} - \frac{175}{24187} a^{12} + \frac{33}{1273} a^{11} + \frac{3476}{24187} a^{10} - \frac{7654}{24187} a^{9} - \frac{2008}{24187} a^{8} - \frac{7}{361} a^{7} + \frac{3563}{24187} a^{6} + \frac{1092}{24187} a^{5} + \frac{8771}{24187} a^{4} + \frac{4741}{24187} a^{3} + \frac{9359}{24187} a^{2} + \frac{9042}{24187} a + \frac{3142}{24187}$, $\frac{1}{1295733093361191493489266412889165176597456708254098528253123161077499571724279} a^{20} + \frac{8731016957371127602729241935862292939278565358298242203340072553989423986}{1295733093361191493489266412889165176597456708254098528253123161077499571724279} a^{19} - \frac{30758073285793130514892381467403071764237168455023958279240126957957044717662}{1295733093361191493489266412889165176597456708254098528253123161077499571724279} a^{18} - \frac{28529162642526814251755374017716260641153019656617609223751842999159900134908}{1295733093361191493489266412889165176597456708254098528253123161077499571724279} a^{17} + \frac{24607517393687086941213861083883877070371007747554593117520462433794108507361}{1295733093361191493489266412889165176597456708254098528253123161077499571724279} a^{16} + \frac{19194850951950062817572518239561669961087462038369298879305143293748750059654}{1295733093361191493489266412889165176597456708254098528253123161077499571724279} a^{15} - \frac{5270253149926619917144357005965542268553377340342482343799096439837004067368}{1295733093361191493489266412889165176597456708254098528253123161077499571724279} a^{14} - \frac{4881991170732712421490924094730040798532589336298412066832220966021082093600}{1295733093361191493489266412889165176597456708254098528253123161077499571724279} a^{13} - \frac{33518794200393671557344478393345589498835550446952779798507207446283827278259}{1295733093361191493489266412889165176597456708254098528253123161077499571724279} a^{12} - \frac{17006367092017504844939054619286639600717547086739972967494049999893637934131}{1295733093361191493489266412889165176597456708254098528253123161077499571724279} a^{11} + \frac{164847972506348976047992153552291247147329333566099345140342748674158990231025}{1295733093361191493489266412889165176597456708254098528253123161077499571724279} a^{10} + \frac{155146467485480907588387981760713762299335385533602236489304598140318823525264}{1295733093361191493489266412889165176597456708254098528253123161077499571724279} a^{9} - \frac{484650598535265642238879025496145104405659804464875721385606689030875106847639}{1295733093361191493489266412889165176597456708254098528253123161077499571724279} a^{8} + \frac{633293228859404141312148233575300241989709808176881624132469453495179047083057}{1295733093361191493489266412889165176597456708254098528253123161077499571724279} a^{7} - \frac{33251395375418820427337287804508062337803373299053237673044534765302389029436}{68196478597957447025750863836271851399866142539689396223848587425131556406541} a^{6} - \frac{293377531774699186522219993036052578941986491293630380275592760789519688595719}{1295733093361191493489266412889165176597456708254098528253123161077499571724279} a^{5} - \frac{606273291347905163503473210230262514657238194487743972246992781251545222069600}{1295733093361191493489266412889165176597456708254098528253123161077499571724279} a^{4} + \frac{591873597453074643854224601022961394027287732522649456182805506450677568784292}{1295733093361191493489266412889165176597456708254098528253123161077499571724279} a^{3} + \frac{107935264884802564660448636376212466299252154217166280802593829858127215395679}{1295733093361191493489266412889165176597456708254098528253123161077499571724279} a^{2} + \frac{142631472645387855385779397269646395740255713159647822228639844596603339292842}{1295733093361191493489266412889165176597456708254098528253123161077499571724279} a - \frac{517657264491999500822778522437796061880465286141651507819107143894285871430058}{1295733093361191493489266412889165176597456708254098528253123161077499571724279}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 108700043286586450 \) (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

3.3.8281.2, 7.7.13841287201.1

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 $21$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{3}$ $21$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ R $21$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{21}$ $21$ $21$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ $21$ $21$ $21$ $21$ $21$ $21$

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
13Data not computed