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Properties

Label 21.21.1621725891...3457.1
Degree $21$
Signature $[21, 0]$
Discriminant $3^{28}\cdot 577^{9}$
Root discriminant $66.00$
Ramified primes $3, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times D_7$ (as 21T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 99, -2085, 13205, -15456, -64710, 111435, 117288, -207960, -94753, 179256, 34194, -82163, -3636, 21012, -955, -2967, 306, 213, -30, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 - 30*x^19 + 213*x^18 + 306*x^17 - 2967*x^16 - 955*x^15 + 21012*x^14 - 3636*x^13 - 82163*x^12 + 34194*x^11 + 179256*x^10 - 94753*x^9 - 207960*x^8 + 117288*x^7 + 111435*x^6 - 64710*x^5 - 15456*x^4 + 13205*x^3 - 2085*x^2 + 99*x - 1)
 
gp: K = bnfinit(x^21 - 6*x^20 - 30*x^19 + 213*x^18 + 306*x^17 - 2967*x^16 - 955*x^15 + 21012*x^14 - 3636*x^13 - 82163*x^12 + 34194*x^11 + 179256*x^10 - 94753*x^9 - 207960*x^8 + 117288*x^7 + 111435*x^6 - 64710*x^5 - 15456*x^4 + 13205*x^3 - 2085*x^2 + 99*x - 1, 1)
 

Normalized defining polynomial

\( x^{21} - 6 x^{20} - 30 x^{19} + 213 x^{18} + 306 x^{17} - 2967 x^{16} - 955 x^{15} + 21012 x^{14} - 3636 x^{13} - 82163 x^{12} + 34194 x^{11} + 179256 x^{10} - 94753 x^{9} - 207960 x^{8} + 117288 x^{7} + 111435 x^{6} - 64710 x^{5} - 15456 x^{4} + 13205 x^{3} - 2085 x^{2} + 99 x - 1 \)

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:  \(162172589188461288482391931972916633457=3^{28}\cdot 577^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.00$
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:  $3, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{610356691403163221781423451962979} a^{20} + \frac{120920033985251318200640443232069}{610356691403163221781423451962979} a^{19} - \frac{298940816663538278122471804757159}{610356691403163221781423451962979} a^{18} - \frac{193398901649460211138911239239728}{610356691403163221781423451962979} a^{17} + \frac{125858501012869838160191626859022}{610356691403163221781423451962979} a^{16} + \frac{233118050763135893850562544458541}{610356691403163221781423451962979} a^{15} - \frac{255670425106578197676254206352417}{610356691403163221781423451962979} a^{14} - \frac{180482529042977819404629332156620}{610356691403163221781423451962979} a^{13} - \frac{78695826916383973691890985523745}{610356691403163221781423451962979} a^{12} - \frac{176978586260430073586973604372892}{610356691403163221781423451962979} a^{11} + \frac{9568793064480240998153510747066}{610356691403163221781423451962979} a^{10} + \frac{174525912264644152566988387129446}{610356691403163221781423451962979} a^{9} - \frac{42532927069460084782808353602267}{610356691403163221781423451962979} a^{8} - \frac{155416912739475075923333222972229}{610356691403163221781423451962979} a^{7} - \frac{33924744733265734730278921277502}{610356691403163221781423451962979} a^{6} - \frac{271210860133578914828317149680482}{610356691403163221781423451962979} a^{5} + \frac{73437550921219518468509742018104}{610356691403163221781423451962979} a^{4} - \frac{134323852427311462149752053822608}{610356691403163221781423451962979} a^{3} - \frac{167288091694170259087152505783448}{610356691403163221781423451962979} a^{2} - \frac{30785918724921341463877406542799}{610356691403163221781423451962979} a - \frac{82859313644976441799548311679348}{610356691403163221781423451962979}$

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:  \( 1603475686170 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times D_7$ (as 21T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 15 conjugacy class representatives for $C_3\times D_7$
Character table for $C_3\times D_7$

Intermediate fields

\(\Q(\zeta_{9})^+\), 7.7.192100033.1

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ $21$

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
3Data not computed
577Data not computed