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Properties

Label 21.21.8286495416...0529.1
Degree $21$
Signature $[21, 0]$
Discriminant $13^{14}\cdot 29^{18}$
Root discriminant $99.11$
Ramified primes $13, 29$
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![521, 19366, 109604, -403833, -915746, 1093353, 1951543, -1353475, -1910342, 952245, 1007842, -405653, -301449, 104984, 51216, -16074, -4786, 1382, 224, -60, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^20 - 60*x^19 + 224*x^18 + 1382*x^17 - 4786*x^16 - 16074*x^15 + 51216*x^14 + 104984*x^13 - 301449*x^12 - 405653*x^11 + 1007842*x^10 + 952245*x^9 - 1910342*x^8 - 1353475*x^7 + 1951543*x^6 + 1093353*x^5 - 915746*x^4 - 403833*x^3 + 109604*x^2 + 19366*x + 521)
 
gp: K = bnfinit(x^21 - 4*x^20 - 60*x^19 + 224*x^18 + 1382*x^17 - 4786*x^16 - 16074*x^15 + 51216*x^14 + 104984*x^13 - 301449*x^12 - 405653*x^11 + 1007842*x^10 + 952245*x^9 - 1910342*x^8 - 1353475*x^7 + 1951543*x^6 + 1093353*x^5 - 915746*x^4 - 403833*x^3 + 109604*x^2 + 19366*x + 521, 1)
 

Normalized defining polynomial

\( x^{21} - 4 x^{20} - 60 x^{19} + 224 x^{18} + 1382 x^{17} - 4786 x^{16} - 16074 x^{15} + 51216 x^{14} + 104984 x^{13} - 301449 x^{12} - 405653 x^{11} + 1007842 x^{10} + 952245 x^{9} - 1910342 x^{8} - 1353475 x^{7} + 1951543 x^{6} + 1093353 x^{5} - 915746 x^{4} - 403833 x^{3} + 109604 x^{2} + 19366 x + 521 \)

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:  \(828649541657828886975358787618681750730529=13^{14}\cdot 29^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.11$
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:  $13, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(377=13\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{377}(256,·)$, $\chi_{377}(1,·)$, $\chi_{377}(326,·)$, $\chi_{377}(74,·)$, $\chi_{377}(139,·)$, $\chi_{377}(204,·)$, $\chi_{377}(16,·)$, $\chi_{377}(81,·)$, $\chi_{377}(146,·)$, $\chi_{377}(339,·)$, $\chi_{377}(152,·)$, $\chi_{377}(94,·)$, $\chi_{377}(198,·)$, $\chi_{377}(165,·)$, $\chi_{377}(170,·)$, $\chi_{377}(107,·)$, $\chi_{377}(373,·)$, $\chi_{377}(248,·)$, $\chi_{377}(313,·)$, $\chi_{377}(315,·)$, $\chi_{377}(53,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2669} a^{18} - \frac{386}{2669} a^{17} + \frac{682}{2669} a^{16} - \frac{18}{2669} a^{15} - \frac{437}{2669} a^{14} - \frac{195}{2669} a^{13} + \frac{876}{2669} a^{12} + \frac{22}{157} a^{11} + \frac{236}{2669} a^{10} - \frac{41}{157} a^{9} - \frac{1325}{2669} a^{8} + \frac{50}{2669} a^{7} + \frac{1273}{2669} a^{6} - \frac{347}{2669} a^{5} - \frac{622}{2669} a^{4} - \frac{970}{2669} a^{3} + \frac{1112}{2669} a^{2} - \frac{251}{2669} a + \frac{60}{2669}$, $\frac{1}{2669} a^{19} + \frac{1150}{2669} a^{17} - \frac{997}{2669} a^{16} + \frac{622}{2669} a^{15} - \frac{730}{2669} a^{14} + \frac{338}{2669} a^{13} - \frac{453}{2669} a^{12} + \frac{474}{2669} a^{11} - \frac{347}{2669} a^{10} - \frac{798}{2669} a^{9} + \frac{1048}{2669} a^{8} - \frac{779}{2669} a^{7} - \frac{65}{2669} a^{6} - \frac{1114}{2669} a^{5} - \frac{852}{2669} a^{4} + \frac{352}{2669} a^{3} - \frac{728}{2669} a^{2} - \frac{742}{2669} a - \frac{861}{2669}$, $\frac{1}{2338962517181940080639413554061118698758553} a^{20} - \frac{106744440522588547938062076733762968563}{2338962517181940080639413554061118698758553} a^{19} + \frac{289901333401393063704142704615004057228}{2338962517181940080639413554061118698758553} a^{18} - \frac{919576789738071137467928084324577121044094}{2338962517181940080639413554061118698758553} a^{17} + \frac{177440928824095908898446258632171843496714}{2338962517181940080639413554061118698758553} a^{16} - \frac{557947294105534201285688780940904521906589}{2338962517181940080639413554061118698758553} a^{15} - \frac{52746818777133819188693357138975692057770}{2338962517181940080639413554061118698758553} a^{14} + \frac{11840568057426958115013731910583837134978}{2338962517181940080639413554061118698758553} a^{13} + \frac{848321808569779165720676728621938109165746}{2338962517181940080639413554061118698758553} a^{12} + \frac{906840788092266440321626840382719152416907}{2338962517181940080639413554061118698758553} a^{11} - \frac{737875740759164678728393226645347103234883}{2338962517181940080639413554061118698758553} a^{10} + \frac{825833021441943047182333437238294735329111}{2338962517181940080639413554061118698758553} a^{9} - \frac{875830631765236225467285824326494731785923}{2338962517181940080639413554061118698758553} a^{8} + \frac{264500667399009853987523539171124075857263}{2338962517181940080639413554061118698758553} a^{7} + \frac{219493872516988812893588600009423358965853}{2338962517181940080639413554061118698758553} a^{6} - \frac{775530062316662442747288651302740286737109}{2338962517181940080639413554061118698758553} a^{5} - \frac{1080137933855755060574040906918219278374447}{2338962517181940080639413554061118698758553} a^{4} - \frac{404629742545760335504304021355527335702539}{2338962517181940080639413554061118698758553} a^{3} + \frac{439923283549333821044313401269979508997241}{2338962517181940080639413554061118698758553} a^{2} - \frac{310967551305874757978652386837211612006644}{2338962517181940080639413554061118698758553} a - \frac{914851860526258191510528182630654588775805}{2338962517181940080639413554061118698758553}$

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:  \( 117685929270000 \) (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, 7.7.594823321.1

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 $21$ $21$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ $21$ $21$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ $21$ $21$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{7}$ $21$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
13Data not computed
29Data not computed