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Properties

Label 21.1.156...857.1
Degree $21$
Signature $[1, 10]$
Discriminant $1.561\times 10^{26}$
Root discriminant $17.67$
Ramified primes $23, 71$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_3\times D_7$ (as 21T8)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - x^19 + 2*x^18 + 5*x^17 + 2*x^16 - 10*x^15 - 2*x^14 + 23*x^13 + 19*x^12 - 11*x^11 - 28*x^10 + 8*x^9 + 47*x^8 + 23*x^7 - 13*x^6 - 24*x^5 - 7*x^4 + 11*x^3 + 6*x^2 - 1)
 
gp: K = bnfinit(x^21 - x^20 - x^19 + 2*x^18 + 5*x^17 + 2*x^16 - 10*x^15 - 2*x^14 + 23*x^13 + 19*x^12 - 11*x^11 - 28*x^10 + 8*x^9 + 47*x^8 + 23*x^7 - 13*x^6 - 24*x^5 - 7*x^4 + 11*x^3 + 6*x^2 - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 6, 11, -7, -24, -13, 23, 47, 8, -28, -11, 19, 23, -2, -10, 2, 5, 2, -1, -1, 1]);
 

\( x^{21} - x^{20} - x^{19} + 2 x^{18} + 5 x^{17} + 2 x^{16} - 10 x^{15} - 2 x^{14} + 23 x^{13} + 19 x^{12} - 11 x^{11} - 28 x^{10} + 8 x^{9} + 47 x^{8} + 23 x^{7} - 13 x^{6} - 24 x^{5} - 7 x^{4} + 11 x^{3} + 6 x^{2} - 1 \)

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

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(156106141952973043121291857\)\(\medspace = 23^{7}\cdot 71^{9}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $17.67$
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:  $23, 71$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{7} a^{17} + \frac{1}{7} a^{16} - \frac{3}{7} a^{15} - \frac{1}{7} a^{14} - \frac{3}{7} a^{13} + \frac{3}{7} a^{12} - \frac{1}{7} a^{11} + \frac{2}{7} a^{10} + \frac{1}{7} a^{9} + \frac{1}{7} a^{8} - \frac{2}{7} a^{7} - \frac{2}{7} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} - \frac{2}{7}$, $\frac{1}{7} a^{18} + \frac{3}{7} a^{16} + \frac{2}{7} a^{15} - \frac{2}{7} a^{14} - \frac{1}{7} a^{13} + \frac{3}{7} a^{12} + \frac{3}{7} a^{11} - \frac{1}{7} a^{10} - \frac{3}{7} a^{8} + \frac{3}{7} a^{6} + \frac{3}{7} a^{5} + \frac{3}{7} a^{4} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{19} - \frac{1}{7} a^{16} + \frac{2}{7} a^{14} - \frac{2}{7} a^{13} + \frac{1}{7} a^{12} + \frac{2}{7} a^{11} + \frac{1}{7} a^{10} + \frac{1}{7} a^{9} - \frac{3}{7} a^{8} + \frac{2}{7} a^{7} + \frac{2}{7} a^{6} + \frac{2}{7} a^{4} - \frac{2}{7} a^{2} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{67050907} a^{20} + \frac{619012}{9578701} a^{19} - \frac{2430104}{67050907} a^{18} - \frac{48420}{9578701} a^{17} + \frac{1154233}{9578701} a^{16} + \frac{25935584}{67050907} a^{15} + \frac{2144834}{9578701} a^{14} - \frac{21230934}{67050907} a^{13} + \frac{21553102}{67050907} a^{12} - \frac{1793716}{67050907} a^{11} + \frac{25671549}{67050907} a^{10} - \frac{19772818}{67050907} a^{9} + \frac{31783945}{67050907} a^{8} - \frac{1214318}{9578701} a^{7} - \frac{17854070}{67050907} a^{6} + \frac{28224926}{67050907} a^{5} - \frac{23307932}{67050907} a^{4} + \frac{24879307}{67050907} a^{3} - \frac{28098866}{67050907} a^{2} + \frac{7421087}{67050907} a - \frac{6164758}{67050907}$

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:  $10$
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:  \( 32527.0761402 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{10}\cdot 32527.0761402 \cdot 1}{2\sqrt{156106141952973043121291857}}\approx 0.249651232568$ (assuming GRH)

Galois group

$S_3\times D_7$ (as 21T8):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 84
The 15 conjugacy class representatives for $S_3\times D_7$
Character table for $S_3\times D_7$

Intermediate fields

3.1.23.1, 7.1.357911.1

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 $21$ $21$ ${\href{/LocalNumberField/5.14.0.1}{14} }{,}\,{\href{/LocalNumberField/5.7.0.1}{7} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }{,}\,{\href{/LocalNumberField/19.7.0.1}{7} }$ R $21$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }{,}\,{\href{/LocalNumberField/37.7.0.1}{7} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$71$71.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
71.6.3.2$x^{6} - 5041 x^{2} + 715822$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
71.6.3.2$x^{6} - 5041 x^{2} + 715822$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
71.6.3.2$x^{6} - 5041 x^{2} + 715822$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$