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Properties

Label 21.21.8276699911...8729.1
Degree $21$
Signature $[21, 0]$
Discriminant $13^{14}\cdot 71^{18}$
Root discriminant $213.51$
Ramified primes $13, 71$
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![15389375, -102144250, 36273150, 300106465, -199051958, -315720051, 261729665, 143234027, -154124034, -23590055, 45520578, -1446343, -6986867, 893184, 557394, -100260, -23362, 4946, 488, -114, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^20 - 114*x^19 + 488*x^18 + 4946*x^17 - 23362*x^16 - 100260*x^15 + 557394*x^14 + 893184*x^13 - 6986867*x^12 - 1446343*x^11 + 45520578*x^10 - 23590055*x^9 - 154124034*x^8 + 143234027*x^7 + 261729665*x^6 - 315720051*x^5 - 199051958*x^4 + 300106465*x^3 + 36273150*x^2 - 102144250*x + 15389375)
 
gp: K = bnfinit(x^21 - 4*x^20 - 114*x^19 + 488*x^18 + 4946*x^17 - 23362*x^16 - 100260*x^15 + 557394*x^14 + 893184*x^13 - 6986867*x^12 - 1446343*x^11 + 45520578*x^10 - 23590055*x^9 - 154124034*x^8 + 143234027*x^7 + 261729665*x^6 - 315720051*x^5 - 199051958*x^4 + 300106465*x^3 + 36273150*x^2 - 102144250*x + 15389375, 1)
 

Normalized defining polynomial

\( x^{21} - 4 x^{20} - 114 x^{19} + 488 x^{18} + 4946 x^{17} - 23362 x^{16} - 100260 x^{15} + 557394 x^{14} + 893184 x^{13} - 6986867 x^{12} - 1446343 x^{11} + 45520578 x^{10} - 23590055 x^{9} - 154124034 x^{8} + 143234027 x^{7} + 261729665 x^{6} - 315720051 x^{5} - 199051958 x^{4} + 300106465 x^{3} + 36273150 x^{2} - 102144250 x + 15389375 \)

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:  \(8276699911115832752222429511299889027510277198729=13^{14}\cdot 71^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $213.51$
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:  $13, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(923=13\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{923}(640,·)$, $\chi_{923}(1,·)$, $\chi_{923}(900,·)$, $\chi_{923}(261,·)$, $\chi_{923}(321,·)$, $\chi_{923}(456,·)$, $\chi_{923}(458,·)$, $\chi_{923}(588,·)$, $\chi_{923}(529,·)$, $\chi_{923}(659,·)$, $\chi_{923}(534,·)$, $\chi_{923}(471,·)$, $\chi_{923}(542,·)$, $\chi_{923}(243,·)$, $\chi_{923}(742,·)$, $\chi_{923}(872,·)$, $\chi_{923}(711,·)$, $\chi_{923}(172,·)$, $\chi_{923}(48,·)$, $\chi_{923}(755,·)$, $\chi_{923}(250,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{4}$, $\frac{1}{5} a^{9} - \frac{1}{5} a$, $\frac{1}{25} a^{10} - \frac{2}{25} a^{6} + \frac{1}{25} a^{2}$, $\frac{1}{25} a^{11} - \frac{2}{25} a^{7} + \frac{1}{25} a^{3}$, $\frac{1}{25} a^{12} - \frac{2}{25} a^{8} + \frac{1}{25} a^{4}$, $\frac{1}{125} a^{13} + \frac{1}{125} a^{12} + \frac{1}{125} a^{11} + \frac{1}{125} a^{10} - \frac{7}{125} a^{9} + \frac{8}{125} a^{8} + \frac{8}{125} a^{7} + \frac{8}{125} a^{6} - \frac{4}{125} a^{5} - \frac{9}{125} a^{4} + \frac{41}{125} a^{3} + \frac{41}{125} a^{2} + \frac{12}{25} a + \frac{2}{5}$, $\frac{1}{125} a^{14} + \frac{2}{125} a^{10} - \frac{2}{25} a^{9} - \frac{7}{125} a^{6} - \frac{1}{25} a^{5} + \frac{2}{5} a^{4} + \frac{4}{125} a^{2} + \frac{3}{25} a - \frac{2}{5}$, $\frac{1}{625} a^{15} + \frac{2}{625} a^{14} + \frac{2}{625} a^{13} + \frac{7}{625} a^{12} - \frac{6}{625} a^{11} - \frac{4}{625} a^{10} - \frac{9}{625} a^{9} + \frac{6}{625} a^{8} - \frac{46}{625} a^{7} - \frac{28}{625} a^{6} - \frac{18}{625} a^{5} - \frac{288}{625} a^{4} + \frac{26}{625} a^{3} + \frac{51}{125} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{625} a^{16} - \frac{2}{625} a^{14} - \frac{2}{625} a^{13} + \frac{3}{625} a^{11} - \frac{6}{625} a^{10} + \frac{59}{625} a^{9} - \frac{23}{625} a^{8} + \frac{24}{625} a^{7} - \frac{2}{625} a^{6} + \frac{18}{625} a^{5} - \frac{78}{625} a^{4} - \frac{2}{625} a^{3} + \frac{57}{125} a^{2} - \frac{12}{25} a + \frac{2}{5}$, $\frac{1}{625} a^{17} + \frac{2}{625} a^{14} - \frac{1}{625} a^{13} + \frac{12}{625} a^{12} + \frac{2}{625} a^{11} - \frac{4}{625} a^{10} - \frac{6}{625} a^{9} - \frac{4}{625} a^{8} - \frac{59}{625} a^{7} + \frac{22}{625} a^{6} + \frac{31}{625} a^{5} + \frac{92}{625} a^{4} + \frac{32}{625} a^{3} - \frac{9}{125} a^{2} - \frac{2}{25} a - \frac{1}{5}$, $\frac{1}{53125} a^{18} + \frac{42}{53125} a^{17} - \frac{22}{53125} a^{16} - \frac{41}{53125} a^{15} + \frac{26}{53125} a^{14} - \frac{12}{53125} a^{13} + \frac{118}{10625} a^{12} + \frac{507}{53125} a^{11} - \frac{28}{10625} a^{10} - \frac{2937}{53125} a^{9} - \frac{4524}{53125} a^{8} - \frac{2626}{53125} a^{7} + \frac{1763}{53125} a^{6} - \frac{5268}{53125} a^{5} + \frac{23906}{53125} a^{4} - \frac{2453}{10625} a^{3} + \frac{857}{2125} a^{2} - \frac{5}{17} a - \frac{3}{17}$, $\frac{1}{53125} a^{19} - \frac{1}{53125} a^{17} + \frac{33}{53125} a^{16} - \frac{37}{53125} a^{15} + \frac{171}{53125} a^{14} - \frac{11}{53125} a^{13} - \frac{48}{53125} a^{12} - \frac{779}{53125} a^{11} - \frac{882}{53125} a^{10} - \frac{39}{625} a^{9} + \frac{3357}{53125} a^{8} + \frac{69}{2125} a^{7} + \frac{3136}{53125} a^{6} + \frac{1127}{53125} a^{5} - \frac{16292}{53125} a^{4} - \frac{1897}{10625} a^{3} - \frac{58}{425} a^{2} + \frac{194}{425} a + \frac{7}{17}$, $\frac{1}{330412896443946624798117975406036924484375} a^{20} + \frac{356174058800390146111748145148698563}{66082579288789324959623595081207384896875} a^{19} - \frac{1953056166933707220111847736592416219}{330412896443946624798117975406036924484375} a^{18} - \frac{78055566150209055680886447754361039618}{330412896443946624798117975406036924484375} a^{17} - \frac{15587875047513287949814349330150004771}{330412896443946624798117975406036924484375} a^{16} + \frac{157188460769464856988552541302165984734}{330412896443946624798117975406036924484375} a^{15} + \frac{737330300574081258527344375573662707386}{330412896443946624798117975406036924484375} a^{14} + \frac{249811459356172662951375371594064444518}{330412896443946624798117975406036924484375} a^{13} - \frac{6159646730324161723119832356626144605394}{330412896443946624798117975406036924484375} a^{12} - \frac{5609653800504094104013697275556743878533}{330412896443946624798117975406036924484375} a^{11} + \frac{12164376265408076253933960353321931526}{13216515857757864991924719016241476979375} a^{10} + \frac{7248503347818363923782897694593885302733}{330412896443946624798117975406036924484375} a^{9} + \frac{26204636670550595363603512399680609455062}{330412896443946624798117975406036924484375} a^{8} + \frac{2896605318755150480264193917389485345619}{330412896443946624798117975406036924484375} a^{7} + \frac{2310727924991414185335657467017276602608}{330412896443946624798117975406036924484375} a^{6} + \frac{29538405364575800940113068842953727799067}{330412896443946624798117975406036924484375} a^{5} - \frac{59786729299017910146860648620287314486198}{330412896443946624798117975406036924484375} a^{4} + \frac{587427300911806963299271207942174852563}{66082579288789324959623595081207384896875} a^{3} + \frac{4489207627395401185438382531649663928021}{13216515857757864991924719016241476979375} a^{2} - \frac{585366464493152343102520648059902239794}{2643303171551572998384943803248295395875} a + \frac{22290019719328619926602531116177042973}{528660634310314599676988760649659079175}$

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:  \( 30398950792448647000 \) (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

3.3.169.1, 7.7.128100283921.1

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 $21$ $21$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{21}$ $21$ $21$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ $21$ $21$ $21$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{3}$ $21$ $21$ $21$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ $21$

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
13Data not computed
71Data not computed