↑

Properties

Label 21.9.14883844215...5697.1
Degree $21$
Signature $[9, 6]$
Discriminant $3^{28}\cdot 89\cdot 7310212084990290584871593$
Root discriminant $81.84$
Ramified primes $3, 89, 7310212084990290584871593$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T159

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -18, -75, 138, 1047, 903, -2485, -3513, 2958, 5431, -2196, -4785, 1073, 2637, -312, -904, 48, 186, -3, -21, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 48*x^16 - 904*x^15 - 312*x^14 + 2637*x^13 + 1073*x^12 - 4785*x^11 - 2196*x^10 + 5431*x^9 + 2958*x^8 - 3513*x^7 - 2485*x^6 + 903*x^5 + 1047*x^4 + 138*x^3 - 75*x^2 - 18*x - 1)
 
gp: K = bnfinit(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 48*x^16 - 904*x^15 - 312*x^14 + 2637*x^13 + 1073*x^12 - 4785*x^11 - 2196*x^10 + 5431*x^9 + 2958*x^8 - 3513*x^7 - 2485*x^6 + 903*x^5 + 1047*x^4 + 138*x^3 - 75*x^2 - 18*x - 1, 1)
 

Normalized defining polynomial

\( x^{21} - 21 x^{19} - 3 x^{18} + 186 x^{17} + 48 x^{16} - 904 x^{15} - 312 x^{14} + 2637 x^{13} + 1073 x^{12} - 4785 x^{11} - 2196 x^{10} + 5431 x^{9} + 2958 x^{8} - 3513 x^{7} - 2485 x^{6} + 903 x^{5} + 1047 x^{4} + 138 x^{3} - 75 x^{2} - 18 x - 1 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14883844215636283411475070335254453235697=3^{28}\cdot 89\cdot 7310212084990290584871593\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.84$
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:  $3, 89, 7310212084990290584871593$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{2}{9} a^{3} - \frac{4}{9} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{13} - \frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{3} a^{6} + \frac{1}{9} a^{5} + \frac{2}{9} a^{4} + \frac{2}{9} a^{3} - \frac{1}{9} a^{2} + \frac{1}{3} a + \frac{2}{9}$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{7} - \frac{2}{9} a^{6} - \frac{1}{9} a^{5} - \frac{4}{9} a^{4} + \frac{1}{9} a^{3} + \frac{4}{9} a^{2} - \frac{1}{9} a + \frac{2}{9}$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{13} - \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{4}{9} a^{4} - \frac{4}{9} a^{3} + \frac{1}{9} a^{2} + \frac{2}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{19} - \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{9} + \frac{1}{9} a^{7} - \frac{4}{9} a^{6} + \frac{4}{9} a^{5} - \frac{4}{9} a^{4} + \frac{1}{3} a^{3} + \frac{1}{9} a - \frac{4}{9}$, $\frac{1}{9} a^{20} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{2}{9} a^{6} - \frac{4}{9} a^{5} + \frac{1}{3} a^{4} + \frac{1}{9} a^{3} + \frac{1}{3} a^{2} - \frac{4}{9} a - \frac{2}{9}$

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:  $14$
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:  \( 1896679467890 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T159:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 384072192000
The 1165 conjugacy class representatives for t21n159 are not computed
Character table for t21n159 is not computed

Intermediate fields

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

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

Sibling fields

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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ R $18{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $18{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ $18{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ $21$ $15{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }$ $15{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $21$ $18{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ $15{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.3.0.1$x^{3} - x + 7$$1$$3$$0$$C_3$$[\ ]^{3}$
89.3.0.1$x^{3} - x + 7$$1$$3$$0$$C_3$$[\ ]^{3}$
89.3.0.1$x^{3} - x + 7$$1$$3$$0$$C_3$$[\ ]^{3}$
7310212084990290584871593Data not computed