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Properties

Label 21.21.5010020897...6881.1
Degree $21$
Signature $[21, 0]$
Discriminant $7^{14}\cdot 127^{18}$
Root discriminant $232.63$
Ramified primes $7, 127$
Class number $1$ (GRH)
Class group Trivial (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![-585814321, -387607364, 3268069590, 618720777, -5893987054, 269281123, 4442293901, -740975509, -1526878622, 339303301, 269546118, -67235959, -26070729, 6835412, 1422996, -378976, -43140, 11408, 666, -172, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^20 - 172*x^19 + 666*x^18 + 11408*x^17 - 43140*x^16 - 378976*x^15 + 1422996*x^14 + 6835412*x^13 - 26070729*x^12 - 67235959*x^11 + 269546118*x^10 + 339303301*x^9 - 1526878622*x^8 - 740975509*x^7 + 4442293901*x^6 + 269281123*x^5 - 5893987054*x^4 + 618720777*x^3 + 3268069590*x^2 - 387607364*x - 585814321)
 
gp: K = bnfinit(x^21 - 4*x^20 - 172*x^19 + 666*x^18 + 11408*x^17 - 43140*x^16 - 378976*x^15 + 1422996*x^14 + 6835412*x^13 - 26070729*x^12 - 67235959*x^11 + 269546118*x^10 + 339303301*x^9 - 1526878622*x^8 - 740975509*x^7 + 4442293901*x^6 + 269281123*x^5 - 5893987054*x^4 + 618720777*x^3 + 3268069590*x^2 - 387607364*x - 585814321, 1)
 

Normalized defining polynomial

\( x^{21} - 4 x^{20} - 172 x^{19} + 666 x^{18} + 11408 x^{17} - 43140 x^{16} - 378976 x^{15} + 1422996 x^{14} + 6835412 x^{13} - 26070729 x^{12} - 67235959 x^{11} + 269546118 x^{10} + 339303301 x^{9} - 1526878622 x^{8} - 740975509 x^{7} + 4442293901 x^{6} + 269281123 x^{5} - 5893987054 x^{4} + 618720777 x^{3} + 3268069590 x^{2} - 387607364 x - 585814321 \)

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:  \(50100208978759111971843370304908187034231525356881=7^{14}\cdot 127^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $232.63$
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:  $7, 127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(889=7\cdot 127\)
Dirichlet character group:    $\lbrace$$\chi_{889}(128,·)$, $\chi_{889}(1,·)$, $\chi_{889}(2,·)$, $\chi_{889}(4,·)$, $\chi_{889}(389,·)$, $\chi_{889}(135,·)$, $\chi_{889}(8,·)$, $\chi_{889}(778,·)$, $\chi_{889}(256,·)$, $\chi_{889}(16,·)$, $\chi_{889}(270,·)$, $\chi_{889}(512,·)$, $\chi_{889}(667,·)$, $\chi_{889}(540,·)$, $\chi_{889}(32,·)$, $\chi_{889}(64,·)$, $\chi_{889}(191,·)$, $\chi_{889}(764,·)$, $\chi_{889}(445,·)$, $\chi_{889}(382,·)$, $\chi_{889}(639,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{19} a^{15} + \frac{8}{19} a^{14} + \frac{8}{19} a^{13} + \frac{8}{19} a^{12} - \frac{2}{19} a^{11} + \frac{8}{19} a^{10} - \frac{2}{19} a^{9} + \frac{7}{19} a^{8} - \frac{8}{19} a^{7} + \frac{5}{19} a^{6} + \frac{6}{19} a^{5} - \frac{9}{19} a^{4} + \frac{6}{19} a^{3} - \frac{1}{19} a^{2} - \frac{9}{19} a + \frac{9}{19}$, $\frac{1}{19} a^{16} + \frac{1}{19} a^{14} + \frac{1}{19} a^{13} - \frac{9}{19} a^{12} + \frac{5}{19} a^{11} - \frac{9}{19} a^{10} + \frac{4}{19} a^{9} - \frac{7}{19} a^{8} - \frac{7}{19} a^{7} + \frac{4}{19} a^{6} + \frac{2}{19} a^{4} + \frac{8}{19} a^{3} - \frac{1}{19} a^{2} + \frac{5}{19} a + \frac{4}{19}$, $\frac{1}{19} a^{17} - \frac{7}{19} a^{14} + \frac{2}{19} a^{13} - \frac{3}{19} a^{12} - \frac{7}{19} a^{11} - \frac{4}{19} a^{10} - \frac{5}{19} a^{9} + \frac{5}{19} a^{8} - \frac{7}{19} a^{7} - \frac{5}{19} a^{6} - \frac{4}{19} a^{5} - \frac{2}{19} a^{4} - \frac{7}{19} a^{3} + \frac{6}{19} a^{2} - \frac{6}{19} a - \frac{9}{19}$, $\frac{1}{13357} a^{18} + \frac{116}{13357} a^{17} + \frac{280}{13357} a^{16} - \frac{53}{13357} a^{15} - \frac{2304}{13357} a^{14} + \frac{84}{13357} a^{13} - \frac{5162}{13357} a^{12} + \frac{3298}{13357} a^{11} + \frac{5649}{13357} a^{10} + \frac{6033}{13357} a^{9} - \frac{455}{13357} a^{8} + \frac{1999}{13357} a^{7} + \frac{2244}{13357} a^{6} + \frac{1}{361} a^{5} - \frac{2096}{13357} a^{4} + \frac{1462}{13357} a^{3} + \frac{319}{703} a^{2} + \frac{5175}{13357} a - \frac{1307}{13357}$, $\frac{1}{13357} a^{19} + \frac{181}{13357} a^{17} - \frac{195}{13357} a^{16} + \frac{329}{13357} a^{15} + \frac{4426}{13357} a^{14} + \frac{2669}{13357} a^{13} + \frac{2431}{13357} a^{12} + \frac{149}{361} a^{11} + \frac{6648}{13357} a^{10} - \frac{153}{703} a^{9} + \frac{4163}{13357} a^{8} - \frac{462}{13357} a^{7} - \frac{1563}{13357} a^{6} - \frac{764}{13357} a^{5} - \frac{6373}{13357} a^{4} - \frac{6059}{13357} a^{3} - \frac{5446}{13357} a^{2} + \frac{5785}{13357} a - \frac{4454}{13357}$, $\frac{1}{14054575991794839538689655529317273711512677620824967685706372635933680491695393} a^{20} - \frac{142268352674372348695954780776986026140387397558858314818172798402049353345}{14054575991794839538689655529317273711512677620824967685706372635933680491695393} a^{19} - \frac{402130852498238644341085822486219896515099082744577493400359798488331869777}{14054575991794839538689655529317273711512677620824967685706372635933680491695393} a^{18} - \frac{278916206722580914785544789760918375111931221430846890628525901331941515132297}{14054575991794839538689655529317273711512677620824967685706372635933680491695393} a^{17} + \frac{282796526583854305682690278257457140088461114782273303146512239606506039004939}{14054575991794839538689655529317273711512677620824967685706372635933680491695393} a^{16} - \frac{65944241177762895899958919002906123321442307199481772481095089200962503130626}{14054575991794839538689655529317273711512677620824967685706372635933680491695393} a^{15} - \frac{5201849845885656284274121072906104781410267552088304078882208801451526863556596}{14054575991794839538689655529317273711512677620824967685706372635933680491695393} a^{14} + \frac{6821504046819955346659852341746217642148661041991295431523877748647974863073337}{14054575991794839538689655529317273711512677620824967685706372635933680491695393} a^{13} - \frac{4984667039221296279373299012361758347814110075361808314762315125149839211249313}{14054575991794839538689655529317273711512677620824967685706372635933680491695393} a^{12} + \frac{86069686060117870831635994766410738380185667385722653608543277439260302623846}{379853405183644311856477176468034424635477773535809937451523584754964337613389} a^{11} + \frac{4250663028680648395204701682527622332107451038403231326850010537252316526118876}{14054575991794839538689655529317273711512677620824967685706372635933680491695393} a^{10} + \frac{516049463345155979277524023234433530367198596053179442648832374928054546687950}{14054575991794839538689655529317273711512677620824967685706372635933680491695393} a^{9} + \frac{6156756360825667622910388857082556722828912653114787233435995119160595575235339}{14054575991794839538689655529317273711512677620824967685706372635933680491695393} a^{8} - \frac{1871745535659141151166619120310385798705534343850712855716773314023203252447095}{14054575991794839538689655529317273711512677620824967685706372635933680491695393} a^{7} + \frac{2450143108566838224389563849156211179430545262698225598169586161634648585161878}{14054575991794839538689655529317273711512677620824967685706372635933680491695393} a^{6} + \frac{2828758974744094335091486896838493980903439339991818799140548159520330134150722}{14054575991794839538689655529317273711512677620824967685706372635933680491695393} a^{5} - \frac{4979660484305967902501077850650169919993213875746842050678561182237342238986255}{14054575991794839538689655529317273711512677620824967685706372635933680491695393} a^{4} + \frac{4992033091810892850241466376735843390447284722837187135935187376098322071965516}{14054575991794839538689655529317273711512677620824967685706372635933680491695393} a^{3} + \frac{3974686777351231133047528406391256037869892755320766830749228335393734624777478}{14054575991794839538689655529317273711512677620824967685706372635933680491695393} a^{2} + \frac{2315439933337560619421493047542760753116268035223479390849781506139723939859005}{14054575991794839538689655529317273711512677620824967685706372635933680491695393} a - \frac{5287970565431306964452931113540470406237232410311248723154146244545670553572002}{14054575991794839538689655529317273711512677620824967685706372635933680491695393}$

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:  \( 688733692772026100 \) (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

\(\Q(\zeta_{7})^+\), 7.7.4195872914689.1

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 $21$ $21$ $21$ R $21$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ $21$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ $21$ $21$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
$127$127.7.6.1$x^{7} - 127$$7$$1$$6$$C_7$$[\ ]_{7}$
127.7.6.1$x^{7} - 127$$7$$1$$6$$C_7$$[\ ]_{7}$
127.7.6.1$x^{7} - 127$$7$$1$$6$$C_7$$[\ ]_{7}$