Properties

Label 21.3.396...527.1
Degree $21$
Signature $[3, 9]$
Discriminant $-3.961\times 10^{24}$
Root discriminant $14.84$
Ramified primes $13, 1801, 193327$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group 21T139

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 5*x^20 + 11*x^19 - 20*x^18 + 37*x^17 - 68*x^16 + 120*x^15 - 168*x^14 + 237*x^13 - 331*x^12 + 362*x^11 - 431*x^10 + 452*x^9 - 381*x^8 + 350*x^7 - 238*x^6 + 169*x^5 - 90*x^4 + 43*x^3 + 6*x^2 + 27)
 
gp: K = bnfinit(x^21 - 5*x^20 + 11*x^19 - 20*x^18 + 37*x^17 - 68*x^16 + 120*x^15 - 168*x^14 + 237*x^13 - 331*x^12 + 362*x^11 - 431*x^10 + 452*x^9 - 381*x^8 + 350*x^7 - 238*x^6 + 169*x^5 - 90*x^4 + 43*x^3 + 6*x^2 + 27, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27, 0, 6, 43, -90, 169, -238, 350, -381, 452, -431, 362, -331, 237, -168, 120, -68, 37, -20, 11, -5, 1]);
 

\(x^{21} - 5 x^{20} + 11 x^{19} - 20 x^{18} + 37 x^{17} - 68 x^{16} + 120 x^{15} - 168 x^{14} + 237 x^{13} - 331 x^{12} + 362 x^{11} - 431 x^{10} + 452 x^{9} - 381 x^{8} + 350 x^{7} - 238 x^{6} + 169 x^{5} - 90 x^{4} + 43 x^{3} + 6 x^{2} + 27\)  Toggle raw display

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

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[3, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-3960879820606443846053527\)\(\medspace = -\,13^{2}\cdot 1801^{2}\cdot 193327^{3}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $14.84$
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:  $13, 1801, 193327$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{3} a^{19} + \frac{1}{3} a^{18} - \frac{1}{3} a^{17} + \frac{1}{3} a^{16} + \frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{860153495355557217} a^{20} + \frac{181578608182792}{860153495355557217} a^{19} + \frac{52575980687782442}{860153495355557217} a^{18} + \frac{314489200548185155}{860153495355557217} a^{17} - \frac{397364443881064868}{860153495355557217} a^{16} + \frac{260490419820728083}{860153495355557217} a^{15} + \frac{40292578962412640}{95572610595061913} a^{14} + \frac{105710859436732222}{286717831785185739} a^{13} - \frac{44911611440406269}{286717831785185739} a^{12} - \frac{267143808405119254}{860153495355557217} a^{11} - \frac{412540822785471286}{860153495355557217} a^{10} - \frac{216459893359979384}{860153495355557217} a^{9} - \frac{129466399558311343}{860153495355557217} a^{8} - \frac{23402351702819640}{95572610595061913} a^{7} + \frac{160713392039199359}{860153495355557217} a^{6} - \frac{321587809763497516}{860153495355557217} a^{5} - \frac{368615916599978978}{860153495355557217} a^{4} + \frac{142990715126005727}{286717831785185739} a^{3} + \frac{196925523467236531}{860153495355557217} a^{2} - \frac{89655634718498909}{286717831785185739} a + \frac{31206378164293990}{95572610595061913}$  Toggle raw display

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:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
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:  \( 6158.71427835 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{3}\cdot(2\pi)^{9}\cdot 6158.71427835 \cdot 1}{2\sqrt{3960879820606443846053527}}\approx 0.188917865550$ (assuming GRH)

Galois group

21T139:

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 11022480
The 429 conjugacy class representatives for t21n139 are not computed
Character table for t21n139 is not computed

Intermediate fields

7.1.193327.1

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

Sibling fields

Degree 21 sibling: 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 $21$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{3}$ $15{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ $21$ R $18{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ $15{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ $15{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ $18{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ $18{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.9.0.1$x^{9} - 2 x + 2$$1$$9$$0$$C_9$$[\ ]^{9}$
1801Data not computed
193327Data not computed