Properties

Label 21.9.332...609.1
Degree $21$
Signature $[9, 6]$
Discriminant $3.327\times 10^{27}$
Root discriminant $20.44$
Ramified primes $3, 53$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group 21T44

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 + 12*x^19 - 2*x^18 - 27*x^17 + 36*x^16 - 39*x^15 + 180*x^14 - 396*x^13 + 183*x^12 + 558*x^11 - 738*x^10 - 339*x^9 + 1035*x^8 + 27*x^7 - 936*x^6 + 27*x^5 + 621*x^4 + 99*x^3 - 189*x^2 - 81*x - 9)
 
gp: K = bnfinit(x^21 - 6*x^20 + 12*x^19 - 2*x^18 - 27*x^17 + 36*x^16 - 39*x^15 + 180*x^14 - 396*x^13 + 183*x^12 + 558*x^11 - 738*x^10 - 339*x^9 + 1035*x^8 + 27*x^7 - 936*x^6 + 27*x^5 + 621*x^4 + 99*x^3 - 189*x^2 - 81*x - 9, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9, -81, -189, 99, 621, 27, -936, 27, 1035, -339, -738, 558, 183, -396, 180, -39, 36, -27, -2, 12, -6, 1]);
 

\( x^{21} - 6 x^{20} + 12 x^{19} - 2 x^{18} - 27 x^{17} + 36 x^{16} - 39 x^{15} + 180 x^{14} - 396 x^{13} + 183 x^{12} + 558 x^{11} - 738 x^{10} - 339 x^{9} + 1035 x^{8} + 27 x^{7} - 936 x^{6} + 27 x^{5} + 621 x^{4} + 99 x^{3} - 189 x^{2} - 81 x - 9 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[9, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(3326751700248238687839567609\)\(\medspace = 3^{36}\cdot 53^{6}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $20.44$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 53$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $3$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{9}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{10}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9}$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{9}$, $\frac{1}{3} a^{19} + \frac{1}{3} a^{10}$, $\frac{1}{131267373453922011} a^{20} + \frac{3568031924046535}{131267373453922011} a^{19} - \frac{5399012776593289}{131267373453922011} a^{18} + \frac{6080177706115754}{43755791151307337} a^{17} - \frac{14379236418416138}{131267373453922011} a^{16} - \frac{4236816674436930}{43755791151307337} a^{15} - \frac{3562962154747100}{131267373453922011} a^{14} - \frac{601857998880935}{43755791151307337} a^{13} - \frac{20139852429904810}{131267373453922011} a^{12} - \frac{8521005840023245}{131267373453922011} a^{11} + \frac{15212291606741854}{43755791151307337} a^{10} + \frac{56103061540374208}{131267373453922011} a^{9} - \frac{8043809703017535}{43755791151307337} a^{8} + \frac{1392902357004468}{43755791151307337} a^{7} - \frac{12860479142177285}{43755791151307337} a^{6} + \frac{20537715197487697}{43755791151307337} a^{5} - \frac{365761297640640}{43755791151307337} a^{4} + \frac{17974934912327320}{43755791151307337} a^{3} - \frac{18733238645192621}{43755791151307337} a^{2} + \frac{1711371639092380}{43755791151307337} a + \frac{4546273266963611}{43755791151307337}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 967276.035305 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{9}\cdot(2\pi)^{6}\cdot 967276.035305 \cdot 1}{2\sqrt{3326751700248238687839567609}}\approx 0.264155349265$ (assuming GRH)

Galois group

21T44:

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

Intermediate fields

\(\Q(\zeta_{9})^+\), 7.3.18429849.1

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 $21$ $15{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ $21$ $21$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{3}$ $15{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ $21$ R $15{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
$53$53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$