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Properties

Label 21.21.3739247391...8401.1
Degree $21$
Signature $[21, 0]$
Discriminant $379^{20}$
Root discriminant $285.66$
Ramified prime $379$
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![-206124161, 2278571400, 1522403319, -15613111936, -7395125885, 14227077020, 5900535983, -5285574412, -1796606075, 1071723493, 274701966, -131326631, -23230778, 9994129, 1104481, -466147, -28085, 12681, 325, -180, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 180*x^19 + 325*x^18 + 12681*x^17 - 28085*x^16 - 466147*x^15 + 1104481*x^14 + 9994129*x^13 - 23230778*x^12 - 131326631*x^11 + 274701966*x^10 + 1071723493*x^9 - 1796606075*x^8 - 5285574412*x^7 + 5900535983*x^6 + 14227077020*x^5 - 7395125885*x^4 - 15613111936*x^3 + 1522403319*x^2 + 2278571400*x - 206124161)
 
gp: K = bnfinit(x^21 - x^20 - 180*x^19 + 325*x^18 + 12681*x^17 - 28085*x^16 - 466147*x^15 + 1104481*x^14 + 9994129*x^13 - 23230778*x^12 - 131326631*x^11 + 274701966*x^10 + 1071723493*x^9 - 1796606075*x^8 - 5285574412*x^7 + 5900535983*x^6 + 14227077020*x^5 - 7395125885*x^4 - 15613111936*x^3 + 1522403319*x^2 + 2278571400*x - 206124161, 1)
 

Normalized defining polynomial

\( x^{21} - x^{20} - 180 x^{19} + 325 x^{18} + 12681 x^{17} - 28085 x^{16} - 466147 x^{15} + 1104481 x^{14} + 9994129 x^{13} - 23230778 x^{12} - 131326631 x^{11} + 274701966 x^{10} + 1071723493 x^{9} - 1796606075 x^{8} - 5285574412 x^{7} + 5900535983 x^{6} + 14227077020 x^{5} - 7395125885 x^{4} - 15613111936 x^{3} + 1522403319 x^{2} + 2278571400 x - 206124161 \)

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:  \(3739247391481032835088920564931572276509155285348401=379^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $285.66$
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:  $379$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(379\)
Dirichlet character group:    $\lbrace$$\chi_{379}(1,·)$, $\chi_{379}(322,·)$, $\chi_{379}(195,·)$, $\chi_{379}(5,·)$, $\chi_{379}(327,·)$, $\chi_{379}(138,·)$, $\chi_{379}(311,·)$, $\chi_{379}(76,·)$, $\chi_{379}(86,·)$, $\chi_{379}(217,·)$, $\chi_{379}(216,·)$, $\chi_{379}(25,·)$, $\chi_{379}(91,·)$, $\chi_{379}(93,·)$, $\chi_{379}(94,·)$, $\chi_{379}(39,·)$, $\chi_{379}(51,·)$, $\chi_{379}(246,·)$, $\chi_{379}(119,·)$, $\chi_{379}(125,·)$, $\chi_{379}(255,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{1249839824223437945588631433613532396655462394489911292040531747393355782006345134859959} a^{20} - \frac{522120908793096745621686047758171968642451890795592948181730859633448903374789216987073}{1249839824223437945588631433613532396655462394489911292040531747393355782006345134859959} a^{19} + \frac{132125432055272501767435141066648326492159674919529682176511955103727086723870889797384}{1249839824223437945588631433613532396655462394489911292040531747393355782006345134859959} a^{18} + \frac{314917731910058259811504957913406408003978858860515160976849022256883485297220993284823}{1249839824223437945588631433613532396655462394489911292040531747393355782006345134859959} a^{17} - \frac{272238409816137217596788420880069284086718643275762385114144582984144856605065771125760}{1249839824223437945588631433613532396655462394489911292040531747393355782006345134859959} a^{16} - \frac{283698736525186532516400251800109279863703206111830353623102947144339202925607645741263}{1249839824223437945588631433613532396655462394489911292040531747393355782006345134859959} a^{15} + \frac{102108527735458074217712611144884820536947188503906304294953351494334376195445251346360}{1249839824223437945588631433613532396655462394489911292040531747393355782006345134859959} a^{14} + \frac{260471125768159213055858326494911193110155245103667638890355416591702977530499482380958}{1249839824223437945588631433613532396655462394489911292040531747393355782006345134859959} a^{13} + \frac{440263221923060777084954113188833121350182319992528388475929438571265542224007097227367}{1249839824223437945588631433613532396655462394489911292040531747393355782006345134859959} a^{12} - \frac{225422717563517825883183861392478526025353493516355387506639660334862204334960322645970}{1249839824223437945588631433613532396655462394489911292040531747393355782006345134859959} a^{11} + \frac{58308337900414418897883865979966086128209228393249310079528826921375556646859779880925}{1249839824223437945588631433613532396655462394489911292040531747393355782006345134859959} a^{10} - \frac{38295818737090774442696583798961614468132509297211347985128072320392922642698202026097}{1249839824223437945588631433613532396655462394489911292040531747393355782006345134859959} a^{9} - \frac{330736041480758721432243009467916423809870553059031513773950400264886344406187813337016}{1249839824223437945588631433613532396655462394489911292040531747393355782006345134859959} a^{8} - \frac{350004215698005723944306393961037895686613678170070043079136691721303775212190383152994}{1249839824223437945588631433613532396655462394489911292040531747393355782006345134859959} a^{7} - \frac{273262302024920482338922660945659581237164871860345407687463696683385913395033725246083}{1249839824223437945588631433613532396655462394489911292040531747393355782006345134859959} a^{6} - \frac{430782869354541774815750620511197907002103159195322982942010426854490821788804022118878}{1249839824223437945588631433613532396655462394489911292040531747393355782006345134859959} a^{5} - \frac{229257373424696737720095067739495896059808039378547117651709383813648933347884785138642}{1249839824223437945588631433613532396655462394489911292040531747393355782006345134859959} a^{4} + \frac{557142655734813690511282572246099775449831180745148227106107704645704746076569836827414}{1249839824223437945588631433613532396655462394489911292040531747393355782006345134859959} a^{3} - \frac{110778992177276233757233059392738110161872260523442021724797058679684914714782918548167}{1249839824223437945588631433613532396655462394489911292040531747393355782006345134859959} a^{2} + \frac{266442363655005274488491734344260789049564172546273585507765917361950545058391477877675}{1249839824223437945588631433613532396655462394489911292040531747393355782006345134859959} a - \frac{76612890913394085388791818039631861239657906792544230630916198275473848497771668480332}{1249839824223437945588631433613532396655462394489911292040531747393355782006345134859959}$

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:  \( 6959071739608064000 \) (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

3.3.143641.1, 7.7.2963706958323721.1

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 $21$ $21$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ $21$ $21$ $21$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ $21$ $21$ $21$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{3}$

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
379Data not computed