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Properties

Label 21.21.1080723365...6857.1
Degree $21$
Signature $[21, 0]$
Discriminant $3^{3}\cdot 7^{14}\cdot 8388019^{3}$
Root discriminant $41.75$
Ramified primes $3, 7, 8388019$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T56

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 4, 64, -105, -870, 1193, 4570, -5756, -11462, 13283, 14554, -15389, -9346, 8911, 3163, -2679, -578, 423, 54, -33, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 - 33*x^19 + 54*x^18 + 423*x^17 - 578*x^16 - 2679*x^15 + 3163*x^14 + 8911*x^13 - 9346*x^12 - 15389*x^11 + 14554*x^10 + 13283*x^9 - 11462*x^8 - 5756*x^7 + 4570*x^6 + 1193*x^5 - 870*x^4 - 105*x^3 + 64*x^2 + 4*x - 1)
 
gp: K = bnfinit(x^21 - 2*x^20 - 33*x^19 + 54*x^18 + 423*x^17 - 578*x^16 - 2679*x^15 + 3163*x^14 + 8911*x^13 - 9346*x^12 - 15389*x^11 + 14554*x^10 + 13283*x^9 - 11462*x^8 - 5756*x^7 + 4570*x^6 + 1193*x^5 - 870*x^4 - 105*x^3 + 64*x^2 + 4*x - 1, 1)
 

Normalized defining polynomial

\( x^{21} - 2 x^{20} - 33 x^{19} + 54 x^{18} + 423 x^{17} - 578 x^{16} - 2679 x^{15} + 3163 x^{14} + 8911 x^{13} - 9346 x^{12} - 15389 x^{11} + 14554 x^{10} + 13283 x^{9} - 11462 x^{8} - 5756 x^{7} + 4570 x^{6} + 1193 x^{5} - 870 x^{4} - 105 x^{3} + 64 x^{2} + 4 x - 1 \)

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:  \(10807233650734185519504921698416857=3^{3}\cdot 7^{14}\cdot 8388019^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.75$
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:  $3, 7, 8388019$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{11} a^{18} - \frac{5}{11} a^{17} - \frac{2}{11} a^{16} + \frac{4}{11} a^{15} - \frac{5}{11} a^{14} + \frac{3}{11} a^{13} + \frac{1}{11} a^{12} - \frac{3}{11} a^{11} - \frac{1}{11} a^{10} + \frac{5}{11} a^{9} - \frac{4}{11} a^{8} + \frac{3}{11} a^{7} - \frac{2}{11} a^{6} + \frac{2}{11} a^{5} - \frac{2}{11} a^{4} - \frac{5}{11} a^{3} + \frac{3}{11} a^{2} + \frac{5}{11} a - \frac{5}{11}$, $\frac{1}{55} a^{19} + \frac{2}{55} a^{18} - \frac{4}{55} a^{17} - \frac{2}{11} a^{16} - \frac{21}{55} a^{15} + \frac{23}{55} a^{14} + \frac{2}{5} a^{13} + \frac{4}{55} a^{12} - \frac{1}{5} a^{11} + \frac{9}{55} a^{10} - \frac{24}{55} a^{9} - \frac{3}{55} a^{8} - \frac{3}{55} a^{7} - \frac{12}{55} a^{6} + \frac{23}{55} a^{5} + \frac{5}{11} a^{4} + \frac{1}{55} a^{3} + \frac{4}{55} a^{2} - \frac{3}{55} a - \frac{24}{55}$, $\frac{1}{329049939854663225} a^{20} - \frac{460999440738811}{65809987970932645} a^{19} + \frac{540020951614612}{29913630895878475} a^{18} + \frac{28683973585153833}{329049939854663225} a^{17} + \frac{73205944478625074}{329049939854663225} a^{16} - \frac{2598479787869401}{13161997594186529} a^{15} - \frac{99019479847791204}{329049939854663225} a^{14} - \frac{4341973471755166}{13161997594186529} a^{13} - \frac{132110828539382664}{329049939854663225} a^{12} + \frac{148581048302700096}{329049939854663225} a^{11} + \frac{30775646956122348}{329049939854663225} a^{10} + \frac{5185986127900272}{65809987970932645} a^{9} + \frac{78703653399164953}{329049939854663225} a^{8} + \frac{60616308765257204}{329049939854663225} a^{7} + \frac{12665104420437307}{329049939854663225} a^{6} + \frac{28852783585681749}{329049939854663225} a^{5} + \frac{10185792800201896}{329049939854663225} a^{4} - \frac{162416908714196383}{329049939854663225} a^{3} + \frac{130729875695492794}{329049939854663225} a^{2} + \frac{64644924925164082}{329049939854663225} a - \frac{71913747407602867}{329049939854663225}$

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:  \( 16439548502.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T56:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15120
The 45 conjugacy class representatives for t21n56
Character table for t21n56 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.7.25164057.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 42 sibling: data not computed

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 ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{3}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ $15{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ $21$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{9}$ $15{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ $15{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
7Data not computed
8388019Data not computed