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Properties

Label 21.21.3919222256...7961.1
Degree $21$
Signature $[21, 0]$
Discriminant $3^{28}\cdot 7^{14}\cdot 43^{18}$
Root discriminant $397.81$
Ramified primes $3, 7, 43$
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![3324797, -96455373, -828477006, 1543507294, 3764283285, -2142757218, -5442055226, -254106018, 2667813231, 881813437, -349626786, -181736895, 15483570, 15300921, -7383, -665196, -16779, 15879, 415, -198, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 198*x^19 + 415*x^18 + 15879*x^17 - 16779*x^16 - 665196*x^15 - 7383*x^14 + 15300921*x^13 + 15483570*x^12 - 181736895*x^11 - 349626786*x^10 + 881813437*x^9 + 2667813231*x^8 - 254106018*x^7 - 5442055226*x^6 - 2142757218*x^5 + 3764283285*x^4 + 1543507294*x^3 - 828477006*x^2 - 96455373*x + 3324797)
 
gp: K = bnfinit(x^21 - 3*x^20 - 198*x^19 + 415*x^18 + 15879*x^17 - 16779*x^16 - 665196*x^15 - 7383*x^14 + 15300921*x^13 + 15483570*x^12 - 181736895*x^11 - 349626786*x^10 + 881813437*x^9 + 2667813231*x^8 - 254106018*x^7 - 5442055226*x^6 - 2142757218*x^5 + 3764283285*x^4 + 1543507294*x^3 - 828477006*x^2 - 96455373*x + 3324797, 1)
 

Normalized defining polynomial

\( x^{21} - 3 x^{20} - 198 x^{19} + 415 x^{18} + 15879 x^{17} - 16779 x^{16} - 665196 x^{15} - 7383 x^{14} + 15300921 x^{13} + 15483570 x^{12} - 181736895 x^{11} - 349626786 x^{10} + 881813437 x^{9} + 2667813231 x^{8} - 254106018 x^{7} - 5442055226 x^{6} - 2142757218 x^{5} + 3764283285 x^{4} + 1543507294 x^{3} - 828477006 x^{2} - 96455373 x + 3324797 \)

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:  \(3919222256983996499704582149784632135677558495355107961=3^{28}\cdot 7^{14}\cdot 43^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $397.81$
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:  $3, 7, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2709=3^{2}\cdot 7\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{2709}(64,·)$, $\chi_{2709}(1,·)$, $\chi_{2709}(1411,·)$, $\chi_{2709}(1159,·)$, $\chi_{2709}(1096,·)$, $\chi_{2709}(1033,·)$, $\chi_{2709}(907,·)$, $\chi_{2709}(2578,·)$, $\chi_{2709}(2515,·)$, $\chi_{2709}(2452,·)$, $\chi_{2709}(2326,·)$, $\chi_{2709}(2584,·)$, $\chi_{2709}(1822,·)$, $\chi_{2709}(2080,·)$, $\chi_{2709}(1129,·)$, $\chi_{2709}(1387,·)$, $\chi_{2709}(403,·)$, $\chi_{2709}(121,·)$, $\chi_{2709}(2419,·)$, $\chi_{2709}(379,·)$, $\chi_{2709}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{14} a^{15} + \frac{1}{14} a^{14} - \frac{1}{14} a^{13} + \frac{1}{14} a^{12} + \frac{1}{7} a^{11} + \frac{3}{7} a^{10} - \frac{2}{7} a^{9} + \frac{1}{14} a^{8} + \frac{1}{7} a^{7} + \frac{3}{7} a^{6} - \frac{1}{14} a^{5} - \frac{1}{14} a^{4} - \frac{3}{7} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{14} a^{16} - \frac{1}{7} a^{14} + \frac{1}{7} a^{13} + \frac{1}{14} a^{12} + \frac{2}{7} a^{11} + \frac{2}{7} a^{10} + \frac{5}{14} a^{9} + \frac{1}{14} a^{8} + \frac{2}{7} a^{7} - \frac{1}{2} a^{6} - \frac{5}{14} a^{4} - \frac{1}{14} a^{3} - \frac{1}{2} a$, $\frac{1}{98} a^{17} + \frac{1}{49} a^{16} - \frac{11}{98} a^{13} - \frac{10}{49} a^{12} + \frac{1}{49} a^{11} + \frac{39}{98} a^{10} - \frac{11}{98} a^{9} - \frac{17}{49} a^{8} - \frac{23}{98} a^{7} - \frac{22}{49} a^{6} - \frac{3}{14} a^{5} - \frac{41}{98} a^{4} - \frac{2}{7} a^{3} - \frac{3}{14} a^{2} - \frac{2}{7} a$, $\frac{1}{98} a^{18} + \frac{3}{98} a^{16} + \frac{12}{49} a^{14} - \frac{33}{98} a^{13} + \frac{1}{7} a^{11} + \frac{37}{98} a^{10} + \frac{23}{98} a^{9} - \frac{23}{49} a^{8} - \frac{19}{98} a^{7} + \frac{9}{49} a^{6} + \frac{1}{98} a^{5} - \frac{15}{49} a^{4} - \frac{3}{14} a^{3} + \frac{1}{7} a^{2} - \frac{3}{7} a - \frac{1}{2}$, $\frac{1}{98} a^{19} + \frac{1}{98} a^{16} + \frac{3}{98} a^{15} - \frac{19}{98} a^{14} + \frac{19}{98} a^{13} + \frac{11}{98} a^{12} - \frac{16}{49} a^{11} + \frac{2}{49} a^{10} + \frac{4}{49} a^{9} - \frac{29}{98} a^{8} + \frac{12}{49} a^{7} - \frac{3}{7} a^{6} - \frac{22}{49} a^{5} + \frac{39}{98} a^{4} - \frac{2}{7} a^{3} - \frac{2}{7} a^{2} - \frac{1}{7} a - \frac{1}{2}$, $\frac{1}{487538475838852927773033618646842626543920494370189470112962511975327611011558714} a^{20} + \frac{159512307963130571788980467680318578921515114251110932193724681249966262687867}{34824176845632351983788115617631616181708606740727819293783036569666257929397051} a^{19} + \frac{306633200101197585971571085671328342005692307738128370873062661277317640579013}{487538475838852927773033618646842626543920494370189470112962511975327611011558714} a^{18} + \frac{85239513520826342620305386877649632143122814559845505392270416909509092354126}{34824176845632351983788115617631616181708606740727819293783036569666257929397051} a^{17} + \frac{13875269545032746066569072024798247836216504962687733593701552013424553295114985}{487538475838852927773033618646842626543920494370189470112962511975327611011558714} a^{16} + \frac{1514055155362856675405784844173950144588365292908429549388687979275515316678557}{243769237919426463886516809323421313271960247185094735056481255987663805505779357} a^{15} + \frac{2484558319134927226181947945377702359304283392162315695253267763430021000660413}{34824176845632351983788115617631616181708606740727819293783036569666257929397051} a^{14} + \frac{9719612124606152843434235870363339455471358452356175681332283605782768186663491}{34824176845632351983788115617631616181708606740727819293783036569666257929397051} a^{13} + \frac{78002286335770543629706330373809777163047647868109927598856031538095191677447997}{243769237919426463886516809323421313271960247185094735056481255987663805505779357} a^{12} + \frac{82419670228714134071449276981352150902304581334542482956716374431845157469071921}{243769237919426463886516809323421313271960247185094735056481255987663805505779357} a^{11} - \frac{62227128089818293219973423506072898516693632978234712686204581273723974773394250}{243769237919426463886516809323421313271960247185094735056481255987663805505779357} a^{10} - \frac{894313041585846341273536655297989847198541789221694343746790627643524414753901}{3085686555942107137804010244600269788252661356773351076664319696046377284883283} a^{9} + \frac{96224843704569877714293132841992162075274053754057898453190123181967022136806115}{243769237919426463886516809323421313271960247185094735056481255987663805505779357} a^{8} - \frac{49823060661639730044676772788193160838072578826534787455930951859722724311467897}{243769237919426463886516809323421313271960247185094735056481255987663805505779357} a^{7} - \frac{209467162030838510000371116596533164000471511616678693707102641537228605142251419}{487538475838852927773033618646842626543920494370189470112962511975327611011558714} a^{6} - \frac{7171148833612204781659840166311748281054056954687114489386500683404320459524289}{34824176845632351983788115617631616181708606740727819293783036569666257929397051} a^{5} - \frac{2314675851741499760380162551029971286028789128877579124766436643726757361267517}{9949764813037814852510890176466176051916744783065091226795153305618930836970586} a^{4} - \frac{6329974464663673669025181406564113461909730859549607081836512928595920735400897}{34824176845632351983788115617631616181708606740727819293783036569666257929397051} a^{3} - \frac{896969764935983386454244435254176470122774594172767950373502937339387859781064}{4974882406518907426255445088233088025958372391532545613397576652809465418485293} a^{2} + \frac{3641387124846430901576295793083770873272078818083599978812700360371432187580021}{9949764813037814852510890176466176051916744783065091226795153305618930836970586} a - \frac{1321621837534484423385037772855446721603216214898762312931216960559758479980109}{9949764813037814852510890176466176051916744783065091226795153305618930836970586}$

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:  \( 286692515494192970000 \) (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

3.3.3969.1, 7.7.6321363049.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{3}$ R $21$ R $21$ $21$ $21$ $21$ $21$ $21$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ $21$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{3}$

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
3Data not computed
$7$7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
43Data not computed