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Properties

Label 21.21.4814587615...3721.1
Degree $21$
Signature $[21, 0]$
Discriminant $3^{28}\cdot 29^{18}$
Root discriminant $77.56$
Ramified primes $3, 29$
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![289, 3264, -11880, -91517, -39813, 363423, 290104, -589995, -470445, 508726, 331947, -250395, -121339, 72072, 24246, -12066, -2619, 1131, 142, -54, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 54*x^19 + 142*x^18 + 1131*x^17 - 2619*x^16 - 12066*x^15 + 24246*x^14 + 72072*x^13 - 121339*x^12 - 250395*x^11 + 331947*x^10 + 508726*x^9 - 470445*x^8 - 589995*x^7 + 290104*x^6 + 363423*x^5 - 39813*x^4 - 91517*x^3 - 11880*x^2 + 3264*x + 289)
 
gp: K = bnfinit(x^21 - 3*x^20 - 54*x^19 + 142*x^18 + 1131*x^17 - 2619*x^16 - 12066*x^15 + 24246*x^14 + 72072*x^13 - 121339*x^12 - 250395*x^11 + 331947*x^10 + 508726*x^9 - 470445*x^8 - 589995*x^7 + 290104*x^6 + 363423*x^5 - 39813*x^4 - 91517*x^3 - 11880*x^2 + 3264*x + 289, 1)
 

Normalized defining polynomial

\( x^{21} - 3 x^{20} - 54 x^{19} + 142 x^{18} + 1131 x^{17} - 2619 x^{16} - 12066 x^{15} + 24246 x^{14} + 72072 x^{13} - 121339 x^{12} - 250395 x^{11} + 331947 x^{10} + 508726 x^{9} - 470445 x^{8} - 589995 x^{7} + 290104 x^{6} + 363423 x^{5} - 39813 x^{4} - 91517 x^{3} - 11880 x^{2} + 3264 x + 289 \)

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:  \(4814587615056751193058435502319478353721=3^{28}\cdot 29^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.56$
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:  $3, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(261=3^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{261}(256,·)$, $\chi_{261}(1,·)$, $\chi_{261}(7,·)$, $\chi_{261}(136,·)$, $\chi_{261}(139,·)$, $\chi_{261}(16,·)$, $\chi_{261}(82,·)$, $\chi_{261}(88,·)$, $\chi_{261}(25,·)$, $\chi_{261}(94,·)$, $\chi_{261}(223,·)$, $\chi_{261}(226,·)$, $\chi_{261}(103,·)$, $\chi_{261}(169,·)$, $\chi_{261}(199,·)$, $\chi_{261}(175,·)$, $\chi_{261}(112,·)$, $\chi_{261}(49,·)$, $\chi_{261}(52,·)$, $\chi_{261}(181,·)$, $\chi_{261}(190,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{17} a^{13} + \frac{2}{17} a^{11} - \frac{8}{17} a^{10} + \frac{7}{17} a^{9} + \frac{2}{17} a^{8} - \frac{1}{17} a^{7} - \frac{8}{17} a^{6} + \frac{3}{17} a^{5} - \frac{2}{17} a^{4} - \frac{1}{17} a^{3} - \frac{1}{17} a^{2} + \frac{6}{17} a$, $\frac{1}{17} a^{14} + \frac{2}{17} a^{12} - \frac{8}{17} a^{11} + \frac{7}{17} a^{10} + \frac{2}{17} a^{9} - \frac{1}{17} a^{8} - \frac{8}{17} a^{7} + \frac{3}{17} a^{6} - \frac{2}{17} a^{5} - \frac{1}{17} a^{4} - \frac{1}{17} a^{3} + \frac{6}{17} a^{2}$, $\frac{1}{17} a^{15} - \frac{8}{17} a^{12} + \frac{3}{17} a^{11} + \frac{1}{17} a^{10} + \frac{2}{17} a^{9} + \frac{5}{17} a^{8} + \frac{5}{17} a^{7} - \frac{3}{17} a^{6} - \frac{7}{17} a^{5} + \frac{3}{17} a^{4} + \frac{8}{17} a^{3} + \frac{2}{17} a^{2} + \frac{5}{17} a$, $\frac{1}{17} a^{16} + \frac{3}{17} a^{12} + \frac{6}{17} a^{10} - \frac{7}{17} a^{9} + \frac{4}{17} a^{8} + \frac{6}{17} a^{7} - \frac{3}{17} a^{6} - \frac{7}{17} a^{5} - \frac{8}{17} a^{4} - \frac{6}{17} a^{3} - \frac{3}{17} a^{2} - \frac{3}{17} a$, $\frac{1}{17} a^{17} - \frac{1}{17} a$, $\frac{1}{289} a^{18} + \frac{6}{289} a^{17} - \frac{7}{289} a^{16} + \frac{5}{289} a^{15} - \frac{4}{289} a^{14} - \frac{1}{289} a^{13} + \frac{101}{289} a^{12} + \frac{28}{289} a^{11} - \frac{108}{289} a^{10} - \frac{7}{289} a^{9} + \frac{101}{289} a^{8} + \frac{33}{289} a^{7} + \frac{70}{289} a^{6} + \frac{70}{289} a^{5} + \frac{9}{289} a^{4} + \frac{121}{289} a^{3} + \frac{126}{289} a^{2} - \frac{4}{17} a$, $\frac{1}{289} a^{19} + \frac{8}{289} a^{17} - \frac{4}{289} a^{16} + \frac{6}{289} a^{14} + \frac{5}{289} a^{13} + \frac{7}{17} a^{12} + \frac{47}{289} a^{11} - \frac{90}{289} a^{10} + \frac{109}{289} a^{9} + \frac{73}{289} a^{8} - \frac{26}{289} a^{7} - \frac{112}{289} a^{6} + \frac{14}{289} a^{5} - \frac{69}{289} a^{4} + \frac{97}{289} a^{3} - \frac{25}{289} a^{2} + \frac{4}{17} a$, $\frac{1}{25868591832567242447165180114767} a^{20} + \frac{44080777474058651215642036056}{25868591832567242447165180114767} a^{19} - \frac{29691303483750621108176082876}{25868591832567242447165180114767} a^{18} + \frac{605740634206734361101833712471}{25868591832567242447165180114767} a^{17} + \frac{541369516297095658990735191095}{25868591832567242447165180114767} a^{16} + \frac{675122617170149108144444388862}{25868591832567242447165180114767} a^{15} + \frac{373349194648755463698297829570}{25868591832567242447165180114767} a^{14} + \frac{246990417886384333589316773732}{25868591832567242447165180114767} a^{13} + \frac{3046038451476484985882847882808}{25868591832567242447165180114767} a^{12} + \frac{1389444622965649162953381668582}{25868591832567242447165180114767} a^{11} - \frac{2445005391295722299959424478131}{25868591832567242447165180114767} a^{10} + \frac{1387965008837562221639047260494}{25868591832567242447165180114767} a^{9} + \frac{5404695864881963443122245843595}{25868591832567242447165180114767} a^{8} - \frac{7074712923361484664987626200266}{25868591832567242447165180114767} a^{7} - \frac{3668368773144797376661635005402}{25868591832567242447165180114767} a^{6} + \frac{10158964056082137176614233128442}{25868591832567242447165180114767} a^{5} + \frac{12060365105884441313740348689913}{25868591832567242447165180114767} a^{4} - \frac{808611512761399862412719615007}{25868591832567242447165180114767} a^{3} - \frac{3688207106703257525852039904857}{25868591832567242447165180114767} a^{2} + \frac{2147789299507019562103351794}{1521681872503955438068540006751} a + \frac{33446454840682602666897759145}{89510698382585614004031765103}$

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:  \( 25788034339300 \) (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

\(\Q(\zeta_{9})^+\), 7.7.594823321.1

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 $21$ R $21$ $21$ $21$ $21$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{21}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{3}$ $21$ R $21$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{7}$ $21$ $21$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
3Data not computed
29Data not computed