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Properties

Label 21.21.1145731432...0624.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{27}\cdot 3^{34}\cdot 13^{15}$
Root discriminant $90.20$
Ramified primes $2, 3, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times F_7$ (as 21T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-27, -891, -2349, 31698, 54675, -342792, 168453, 549828, -463293, -341226, 390069, 90396, -157734, -5715, 34362, -2034, -4095, 459, 249, -36, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 - 36*x^19 + 249*x^18 + 459*x^17 - 4095*x^16 - 2034*x^15 + 34362*x^14 - 5715*x^13 - 157734*x^12 + 90396*x^11 + 390069*x^10 - 341226*x^9 - 463293*x^8 + 549828*x^7 + 168453*x^6 - 342792*x^5 + 54675*x^4 + 31698*x^3 - 2349*x^2 - 891*x - 27)
 
gp: K = bnfinit(x^21 - 6*x^20 - 36*x^19 + 249*x^18 + 459*x^17 - 4095*x^16 - 2034*x^15 + 34362*x^14 - 5715*x^13 - 157734*x^12 + 90396*x^11 + 390069*x^10 - 341226*x^9 - 463293*x^8 + 549828*x^7 + 168453*x^6 - 342792*x^5 + 54675*x^4 + 31698*x^3 - 2349*x^2 - 891*x - 27, 1)
 

Normalized defining polynomial

\( x^{21} - 6 x^{20} - 36 x^{19} + 249 x^{18} + 459 x^{17} - 4095 x^{16} - 2034 x^{15} + 34362 x^{14} - 5715 x^{13} - 157734 x^{12} + 90396 x^{11} + 390069 x^{10} - 341226 x^{9} - 463293 x^{8} + 549828 x^{7} + 168453 x^{6} - 342792 x^{5} + 54675 x^{4} + 31698 x^{3} - 2349 x^{2} - 891 x - 27 \)

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:  \(114573143280690357510711381762768711450624=2^{27}\cdot 3^{34}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.20$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{9} + \frac{1}{12} a^{8} + \frac{1}{12} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{10} + \frac{1}{12} a^{9} + \frac{1}{12} a^{8} - \frac{1}{6} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{36} a^{13} - \frac{1}{12} a^{10} + \frac{1}{12} a^{8} - \frac{1}{12} a^{7} + \frac{5}{12} a^{6} + \frac{1}{4} a^{3} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{36} a^{14} - \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{36} a^{15} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{36} a^{16} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{36} a^{17} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{36} a^{18} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{432} a^{19} - \frac{1}{144} a^{17} + \frac{1}{144} a^{16} - \frac{1}{72} a^{15} - \frac{1}{72} a^{12} + \frac{1}{48} a^{11} - \frac{1}{8} a^{10} + \frac{5}{48} a^{9} - \frac{1}{16} a^{8} + \frac{5}{48} a^{7} - \frac{1}{8} a^{6} - \frac{23}{48} a^{5} - \frac{1}{16} a^{4} + \frac{5}{16} a^{3} - \frac{1}{8} a^{2} - \frac{1}{16} a + \frac{5}{16}$, $\frac{1}{8887203101404904081328} a^{20} + \frac{2470082361482994253}{8887203101404904081328} a^{19} + \frac{29485820915405251823}{2962401033801634693776} a^{18} + \frac{939926908718303449}{185150064612602168361} a^{17} - \frac{1303300133227319387}{987467011267211564592} a^{16} - \frac{10899230454004277419}{1481200516900817346888} a^{15} - \frac{6504628638813793411}{740600258450408673444} a^{14} + \frac{4629973701289287317}{493733505633605782296} a^{13} - \frac{121115917259157037031}{2962401033801634693776} a^{12} - \frac{4249999133565733815}{329155670422403854864} a^{11} + \frac{124528272607192919711}{987467011267211564592} a^{10} - \frac{3379361475862918009}{164577835211201927432} a^{9} - \frac{68290991391319191943}{493733505633605782296} a^{8} - \frac{53169691470416484363}{329155670422403854864} a^{7} - \frac{483262487177104177861}{987467011267211564592} a^{6} + \frac{170876310246247053845}{493733505633605782296} a^{5} + \frac{6330304979603392571}{82288917605600963716} a^{4} + \frac{68499582336079895227}{329155670422403854864} a^{3} - \frac{65008941566663642787}{329155670422403854864} a^{2} - \frac{3989389812343774931}{20572229401400240929} a - \frac{83254871934947464903}{329155670422403854864}$

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:  \( 170702187428000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times F_7$ (as 21T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 126
The 21 conjugacy class representatives for $C_3\times F_7$
Character table for $C_3\times F_7$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 7.7.138584369664.1

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

Sibling fields

Degree 21 siblings: data not computed
Degree 42 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 R R $21$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
3Data not computed
13Data not computed