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Properties

Label 21.21.4020583887...6809.1
Degree $21$
Signature $[21, 0]$
Discriminant $3^{36}\cdot 547^{6}$
Root discriminant $39.83$
Ramified primes $3, 547$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T44

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -45, -243, 692, 2628, -2763, -11469, 3222, 23868, 2313, -25767, -7965, 14828, 6687, -4401, -2529, 603, 450, -30, -36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 36*x^19 - 30*x^18 + 450*x^17 + 603*x^16 - 2529*x^15 - 4401*x^14 + 6687*x^13 + 14828*x^12 - 7965*x^11 - 25767*x^10 + 2313*x^9 + 23868*x^8 + 3222*x^7 - 11469*x^6 - 2763*x^5 + 2628*x^4 + 692*x^3 - 243*x^2 - 45*x + 9)
 
gp: K = bnfinit(x^21 - 36*x^19 - 30*x^18 + 450*x^17 + 603*x^16 - 2529*x^15 - 4401*x^14 + 6687*x^13 + 14828*x^12 - 7965*x^11 - 25767*x^10 + 2313*x^9 + 23868*x^8 + 3222*x^7 - 11469*x^6 - 2763*x^5 + 2628*x^4 + 692*x^3 - 243*x^2 - 45*x + 9, 1)
 

Normalized defining polynomial

\( x^{21} - 36 x^{19} - 30 x^{18} + 450 x^{17} + 603 x^{16} - 2529 x^{15} - 4401 x^{14} + 6687 x^{13} + 14828 x^{12} - 7965 x^{11} - 25767 x^{10} + 2313 x^{9} + 23868 x^{8} + 3222 x^{7} - 11469 x^{6} - 2763 x^{5} + 2628 x^{4} + 692 x^{3} - 243 x^{2} - 45 x + 9 \)

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:  \(4020583887700065748020343503016809=3^{36}\cdot 547^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.83$
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:  $3, 547$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{11} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{19} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{140295618326319} a^{20} - \frac{18055789013339}{140295618326319} a^{19} + \frac{2166363376594}{140295618326319} a^{18} - \frac{1998659569646}{46765206108773} a^{17} - \frac{6350754419801}{140295618326319} a^{16} + \frac{5450557757368}{46765206108773} a^{15} - \frac{10184473953161}{140295618326319} a^{14} - \frac{7457334432031}{140295618326319} a^{13} - \frac{4709263007867}{140295618326319} a^{12} + \frac{12831898002899}{46765206108773} a^{11} + \frac{42956233957424}{140295618326319} a^{10} - \frac{19127646010240}{46765206108773} a^{9} - \frac{65801362043770}{140295618326319} a^{8} - \frac{9639464130560}{140295618326319} a^{7} - \frac{14140419525568}{140295618326319} a^{6} - \frac{5112451243108}{140295618326319} a^{5} + \frac{3057322980087}{46765206108773} a^{4} - \frac{24686770872226}{140295618326319} a^{3} - \frac{19917097963372}{46765206108773} a^{2} + \frac{3080538315841}{46765206108773} a + \frac{2015438444676}{46765206108773}$

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:  \( 15121990516.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T44:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7560
The 27 conjugacy class representatives for t21n44
Character table for t21n44 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 7.7.1963110249.1

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

Sibling fields

Degree 45 siblings: 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 $21$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $15{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ $15{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{7}$ $21$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ $21$

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
3Data not computed
547Data not computed