↑

Properties

Label 21.21.2007211581...3921.1
Degree $21$
Signature $[21, 0]$
Discriminant $7^{36}\cdot 31^{14}$
Root discriminant $277.32$
Ramified primes $7, 31$
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![-5373811, -46336661, 109049556, 142672320, -285286645, -155083208, 311945270, 70295426, -173794943, -9150197, 52185714, -2454739, -8504538, 879809, 737571, -95998, -33873, 4781, 777, -112, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 112*x^19 + 777*x^18 + 4781*x^17 - 33873*x^16 - 95998*x^15 + 737571*x^14 + 879809*x^13 - 8504538*x^12 - 2454739*x^11 + 52185714*x^10 - 9150197*x^9 - 173794943*x^8 + 70295426*x^7 + 311945270*x^6 - 155083208*x^5 - 285286645*x^4 + 142672320*x^3 + 109049556*x^2 - 46336661*x - 5373811)
 
gp: K = bnfinit(x^21 - 7*x^20 - 112*x^19 + 777*x^18 + 4781*x^17 - 33873*x^16 - 95998*x^15 + 737571*x^14 + 879809*x^13 - 8504538*x^12 - 2454739*x^11 + 52185714*x^10 - 9150197*x^9 - 173794943*x^8 + 70295426*x^7 + 311945270*x^6 - 155083208*x^5 - 285286645*x^4 + 142672320*x^3 + 109049556*x^2 - 46336661*x - 5373811, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 112 x^{19} + 777 x^{18} + 4781 x^{17} - 33873 x^{16} - 95998 x^{15} + 737571 x^{14} + 879809 x^{13} - 8504538 x^{12} - 2454739 x^{11} + 52185714 x^{10} - 9150197 x^{9} - 173794943 x^{8} + 70295426 x^{7} + 311945270 x^{6} - 155083208 x^{5} - 285286645 x^{4} + 142672320 x^{3} + 109049556 x^{2} - 46336661 x - 5373811 \)

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:  \(2007211581611376969912329733491048520227523013813921=7^{36}\cdot 31^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $277.32$
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:  $7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1519=7^{2}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{1519}(1,·)$, $\chi_{1519}(645,·)$, $\chi_{1519}(904,·)$, $\chi_{1519}(652,·)$, $\chi_{1519}(1296,·)$, $\chi_{1519}(211,·)$, $\chi_{1519}(470,·)$, $\chi_{1519}(1303,·)$, $\chi_{1519}(218,·)$, $\chi_{1519}(862,·)$, $\chi_{1519}(1121,·)$, $\chi_{1519}(36,·)$, $\chi_{1519}(869,·)$, $\chi_{1519}(1513,·)$, $\chi_{1519}(428,·)$, $\chi_{1519}(687,·)$, $\chi_{1519}(435,·)$, $\chi_{1519}(1079,·)$, $\chi_{1519}(1338,·)$, $\chi_{1519}(253,·)$, $\chi_{1519}(1086,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{1178} a^{18} + \frac{61}{589} a^{17} - \frac{60}{589} a^{16} - \frac{119}{589} a^{15} - \frac{68}{589} a^{14} + \frac{11}{38} a^{13} + \frac{91}{1178} a^{12} + \frac{223}{589} a^{11} + \frac{581}{1178} a^{10} + \frac{218}{589} a^{9} - \frac{509}{1178} a^{8} + \frac{565}{1178} a^{7} - \frac{27}{1178} a^{6} - \frac{1}{1178} a^{5} - \frac{165}{1178} a^{4} - \frac{1}{589} a^{3} + \frac{111}{589} a^{2} - \frac{581}{1178} a - \frac{61}{1178}$, $\frac{1}{36518} a^{19} - \frac{7}{18259} a^{18} + \frac{2835}{18259} a^{17} - \frac{2177}{36518} a^{16} + \frac{5727}{36518} a^{15} - \frac{1478}{18259} a^{14} + \frac{5424}{18259} a^{13} + \frac{6404}{18259} a^{12} + \frac{8249}{36518} a^{11} + \frac{7830}{18259} a^{10} - \frac{4281}{18259} a^{9} + \frac{6177}{36518} a^{8} - \frac{8543}{36518} a^{7} - \frac{4938}{18259} a^{6} - \frac{5610}{18259} a^{5} + \frac{3562}{18259} a^{4} - \frac{377}{1922} a^{3} + \frac{8468}{18259} a^{2} - \frac{4683}{36518} a - \frac{4687}{18259}$, $\frac{1}{112081883590544341866415727712652342561797920283605292382154579966} a^{20} + \frac{608720678811987766151961242722358363710254411391877372285061}{56040941795272170933207863856326171280898960141802646191077289983} a^{19} - \frac{13078308741782058908055012171211863589902753308799321774522971}{112081883590544341866415727712652342561797920283605292382154579966} a^{18} + \frac{4992281025632885891456704119330398453421475937109651120010362662}{56040941795272170933207863856326171280898960141802646191077289983} a^{17} + \frac{11377223095486394287778729797333892677706400610176776507190876467}{56040941795272170933207863856326171280898960141802646191077289983} a^{16} - \frac{10856785862297986421038003807185694828673102758161835529112978065}{56040941795272170933207863856326171280898960141802646191077289983} a^{15} - \frac{294326270246354996852240692188177963846142058969130201084626535}{1807772315976521643006705285687941009061256778767827296486364193} a^{14} + \frac{16161358822309362846960945939832266334103305566540623616540332512}{56040941795272170933207863856326171280898960141802646191077289983} a^{13} + \frac{23447026388030187010849581675567662034564039023149024437045427403}{112081883590544341866415727712652342561797920283605292382154579966} a^{12} - \frac{2906956653037936927704391586231072928749468218116339010043932000}{56040941795272170933207863856326171280898960141802646191077289983} a^{11} + \frac{27802725660850022934048870946652437816018245633490483528079041800}{56040941795272170933207863856326171280898960141802646191077289983} a^{10} - \frac{36962627987238299683085128094521984683456833737327933517163418251}{112081883590544341866415727712652342561797920283605292382154579966} a^{9} - \frac{9530576146962033231032543684395938996417111635763406188753074931}{112081883590544341866415727712652342561797920283605292382154579966} a^{8} - \frac{4701426166532634400600232729565928327861903715472322641030293708}{56040941795272170933207863856326171280898960141802646191077289983} a^{7} - \frac{2985515587729463141442635728504245997697319509116575846552052517}{56040941795272170933207863856326171280898960141802646191077289983} a^{6} - \frac{22250580018965699750586043984787267805159745647747160652496828824}{56040941795272170933207863856326171280898960141802646191077289983} a^{5} + \frac{9971298820751758803465387913679219543366161990297268329383567192}{56040941795272170933207863856326171280898960141802646191077289983} a^{4} + \frac{28939694872632025524285612164906025086829922960206038213214917257}{112081883590544341866415727712652342561797920283605292382154579966} a^{3} + \frac{1786191073705674315825141699537139336503889358953678108782540561}{56040941795272170933207863856326171280898960141802646191077289983} a^{2} - \frac{695111171860538531796257729938016092107284811043934404067638917}{56040941795272170933207863856326171280898960141802646191077289983} a - \frac{18034798628915286914923886447190033226205407159038670361057047319}{112081883590544341866415727712652342561797920283605292382154579966}$

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:  \( 7030134349554524000 \) (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

3.3.961.1, 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{3}$ $21$ $21$ R $21$ $21$ $21$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ R $21$ $21$ $21$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ $21$ $21$

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
$31$31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$