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Properties

Label 21.21.3919222256...7961.2
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![27487873, 287915376, 909084162, 449118201, -2657813739, -4505239431, -635242124, 3830424663, 3072449295, 311371348, -508309275, -155865357, 27946195, 14998752, -438534, -671202, -9597, 16005, 366, -198, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 198*x^19 + 366*x^18 + 16005*x^17 - 9597*x^16 - 671202*x^15 - 438534*x^14 + 14998752*x^13 + 27946195*x^12 - 155865357*x^11 - 508309275*x^10 + 311371348*x^9 + 3072449295*x^8 + 3830424663*x^7 - 635242124*x^6 - 4505239431*x^5 - 2657813739*x^4 + 449118201*x^3 + 909084162*x^2 + 287915376*x + 27487873)
 
gp: K = bnfinit(x^21 - 3*x^20 - 198*x^19 + 366*x^18 + 16005*x^17 - 9597*x^16 - 671202*x^15 - 438534*x^14 + 14998752*x^13 + 27946195*x^12 - 155865357*x^11 - 508309275*x^10 + 311371348*x^9 + 3072449295*x^8 + 3830424663*x^7 - 635242124*x^6 - 4505239431*x^5 - 2657813739*x^4 + 449118201*x^3 + 909084162*x^2 + 287915376*x + 27487873, 1)
 

Normalized defining polynomial

\( x^{21} - 3 x^{20} - 198 x^{19} + 366 x^{18} + 16005 x^{17} - 9597 x^{16} - 671202 x^{15} - 438534 x^{14} + 14998752 x^{13} + 27946195 x^{12} - 155865357 x^{11} - 508309275 x^{10} + 311371348 x^{9} + 3072449295 x^{8} + 3830424663 x^{7} - 635242124 x^{6} - 4505239431 x^{5} - 2657813739 x^{4} + 449118201 x^{3} + 909084162 x^{2} + 287915376 x + 27487873 \)

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$$\chi_{2709}(256,·)$, $\chi_{2709}(1,·)$, $\chi_{2709}(130,·)$, $\chi_{2709}(4,·)$, $\chi_{2709}(709,·)$, $\chi_{2709}(646,·)$, $\chi_{2709}(193,·)$, $\chi_{2709}(520,·)$, $\chi_{2709}(16,·)$, $\chi_{2709}(2584,·)$, $\chi_{2709}(772,·)$, $\chi_{2709}(2080,·)$, $\chi_{2709}(2209,·)$, $\chi_{2709}(1024,·)$, $\chi_{2709}(64,·)$, $\chi_{2709}(1387,·)$, $\chi_{2709}(1516,·)$, $\chi_{2709}(2032,·)$, $\chi_{2709}(379,·)$, $\chi_{2709}(508,·)$, $\chi_{2709}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{14} - \frac{1}{7} a^{13} + \frac{1}{7} a^{12} + \frac{2}{7} a^{11} - \frac{1}{7} a^{10} + \frac{3}{7} a^{9} + \frac{1}{7} a^{8} + \frac{2}{7} a^{7} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{16} - \frac{2}{7} a^{14} + \frac{2}{7} a^{13} + \frac{1}{7} a^{12} - \frac{3}{7} a^{11} - \frac{3}{7} a^{10} - \frac{2}{7} a^{9} + \frac{1}{7} a^{8} - \frac{3}{7} a^{7} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3}$, $\frac{1}{49} a^{17} + \frac{2}{49} a^{16} + \frac{2}{7} a^{14} + \frac{24}{49} a^{13} + \frac{1}{49} a^{12} + \frac{16}{49} a^{11} - \frac{24}{49} a^{10} + \frac{10}{49} a^{9} + \frac{22}{49} a^{8} - \frac{9}{49} a^{7} + \frac{19}{49} a^{6} - \frac{1}{7} a^{5} + \frac{22}{49} a^{4} - \frac{2}{7} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7} a$, $\frac{1}{49} a^{18} + \frac{3}{49} a^{16} + \frac{17}{49} a^{14} - \frac{19}{49} a^{13} + \frac{1}{7} a^{12} - \frac{1}{7} a^{11} + \frac{2}{49} a^{10} - \frac{5}{49} a^{9} - \frac{11}{49} a^{8} - \frac{12}{49} a^{7} + \frac{18}{49} a^{6} + \frac{1}{49} a^{5} + \frac{19}{49} a^{4} - \frac{2}{7} a^{3} + \frac{1}{7} a^{2} + \frac{3}{7} a$, $\frac{1}{49} a^{19} + \frac{1}{49} a^{16} + \frac{3}{49} a^{15} + \frac{9}{49} a^{14} + \frac{12}{49} a^{13} - \frac{17}{49} a^{12} + \frac{3}{49} a^{11} + \frac{11}{49} a^{10} + \frac{1}{49} a^{9} + \frac{13}{49} a^{8} - \frac{4}{49} a^{7} + \frac{1}{7} a^{6} + \frac{5}{49} a^{5} - \frac{3}{49} a^{4} - \frac{3}{7} a^{3} - \frac{2}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{935267833975523206845669576310263432561624930667065938703885068296478874942845609} a^{20} + \frac{311608154161877725995977110497823437350491992264156253474452952557996905032232}{935267833975523206845669576310263432561624930667065938703885068296478874942845609} a^{19} - \frac{2879607164705143989778294371371139986054728563868931046121047835678405110354924}{935267833975523206845669576310263432561624930667065938703885068296478874942845609} a^{18} + \frac{8086154570015809487747543989729420011646546296911035277039487626275070295087107}{935267833975523206845669576310263432561624930667065938703885068296478874942845609} a^{17} - \frac{2412923677277718689622135816359564436996727097267327622789192479466941036897665}{935267833975523206845669576310263432561624930667065938703885068296478874942845609} a^{16} - \frac{41059419005202141704728810521887905776339915494998031411567577854593258562905228}{935267833975523206845669576310263432561624930667065938703885068296478874942845609} a^{15} - \frac{269190967886227171720086438482683123559564288804374211392310079954654279979273854}{935267833975523206845669576310263432561624930667065938703885068296478874942845609} a^{14} - \frac{178574321820813923589217433508861209347335965917266710047400812012042591221062251}{935267833975523206845669576310263432561624930667065938703885068296478874942845609} a^{13} - \frac{148844218009976196537684771429509630593972696289190328702513355627343419771340337}{935267833975523206845669576310263432561624930667065938703885068296478874942845609} a^{12} + \frac{152175078945499815721716233493473934245083871547504394423452709220391971850704067}{935267833975523206845669576310263432561624930667065938703885068296478874942845609} a^{11} + \frac{201383073779944910164050460743513153291689769048559994093933965637993158732689135}{935267833975523206845669576310263432561624930667065938703885068296478874942845609} a^{10} + \frac{63934203188217302518988041737061643320192243733068488873281564945957826254244661}{133609690567931886692238510901466204651660704381009419814840724042354124991835087} a^{9} - \frac{22911139112248050975126567565908982643200222199315992204427544436279417938211167}{935267833975523206845669576310263432561624930667065938703885068296478874942845609} a^{8} + \frac{393076516037358294092760432898868764071295629042340297053705701384036243944676494}{935267833975523206845669576310263432561624930667065938703885068296478874942845609} a^{7} - \frac{199289778487198719155272911078419394193080349599752790715769364257461543239853105}{935267833975523206845669576310263432561624930667065938703885068296478874942845609} a^{6} - \frac{436677195293127285895660030010346965195199867389959665932738555094834037869813588}{935267833975523206845669576310263432561624930667065938703885068296478874942845609} a^{5} + \frac{99805814452521065651760321956185910140993121847252864268257959102141002668747024}{935267833975523206845669576310263432561624930667065938703885068296478874942845609} a^{4} - \frac{11402387536931305728983042826392596546499797798640458923398103237137261792196356}{133609690567931886692238510901466204651660704381009419814840724042354124991835087} a^{3} + \frac{34306628355846393333365695494914427946901915116018749357166270147488910029551004}{133609690567931886692238510901466204651660704381009419814840724042354124991835087} a^{2} + \frac{50539967040394844341542402584066128510111725088509384250217527789651221157585106}{133609690567931886692238510901466204651660704381009419814840724042354124991835087} a + \frac{149273724384406594612735507050023370473149537432645190241374936053320663003617}{19087098652561698098891215843066600664522957768715631402120103434622017855976441}$

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:  \( 232239644387987300000 \) (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

3.3.3969.2, 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 $21$ R ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ $21$ $21$ $21$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ $21$ $21$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ $21$ R $21$ $21$ $21$

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
3Data not computed
$7$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
43Data not computed