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Properties

Label 21.3.82255205352...4375.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,5^{9}\cdot 11^{7}\cdot 43^{10}$
Root discriminant $26.58$
Ramified primes $5, 11, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times D_7$ (as 21T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![211, -185, 903, -1825, 3001, -4212, 5112, -5448, 5990, -6067, 5129, -3572, 2243, -1230, 444, 72, -254, 212, -110, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 9*x^20 + 39*x^19 - 110*x^18 + 212*x^17 - 254*x^16 + 72*x^15 + 444*x^14 - 1230*x^13 + 2243*x^12 - 3572*x^11 + 5129*x^10 - 6067*x^9 + 5990*x^8 - 5448*x^7 + 5112*x^6 - 4212*x^5 + 3001*x^4 - 1825*x^3 + 903*x^2 - 185*x + 211)
 
gp: K = bnfinit(x^21 - 9*x^20 + 39*x^19 - 110*x^18 + 212*x^17 - 254*x^16 + 72*x^15 + 444*x^14 - 1230*x^13 + 2243*x^12 - 3572*x^11 + 5129*x^10 - 6067*x^9 + 5990*x^8 - 5448*x^7 + 5112*x^6 - 4212*x^5 + 3001*x^4 - 1825*x^3 + 903*x^2 - 185*x + 211, 1)
 

Normalized defining polynomial

\( x^{21} - 9 x^{20} + 39 x^{19} - 110 x^{18} + 212 x^{17} - 254 x^{16} + 72 x^{15} + 444 x^{14} - 1230 x^{13} + 2243 x^{12} - 3572 x^{11} + 5129 x^{10} - 6067 x^{9} + 5990 x^{8} - 5448 x^{7} + 5112 x^{6} - 4212 x^{5} + 3001 x^{4} - 1825 x^{3} + 903 x^{2} - 185 x + 211 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-822552053520401820057771484375=-\,5^{9}\cdot 11^{7}\cdot 43^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.58$
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:  $5, 11, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{15} - \frac{2}{5} a^{11} + \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{16} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{17} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{425} a^{19} - \frac{4}{85} a^{18} + \frac{4}{85} a^{17} + \frac{6}{85} a^{16} + \frac{32}{425} a^{15} - \frac{41}{425} a^{14} + \frac{3}{85} a^{13} - \frac{37}{425} a^{12} + \frac{97}{425} a^{11} + \frac{159}{425} a^{10} - \frac{114}{425} a^{9} + \frac{87}{425} a^{8} - \frac{178}{425} a^{7} + \frac{3}{17} a^{6} - \frac{111}{425} a^{5} - \frac{202}{425} a^{4} - \frac{26}{425} a^{3} - \frac{23}{85} a^{2} - \frac{126}{425} a + \frac{24}{425}$, $\frac{1}{4640345384605574372118625} a^{20} + \frac{2528619623901381403243}{4640345384605574372118625} a^{19} + \frac{2202664042817640774410}{37122763076844594976949} a^{18} + \frac{58932463378998047270093}{928069076921114874423725} a^{17} + \frac{128900419088232983603017}{4640345384605574372118625} a^{16} + \frac{60211905892358196194746}{928069076921114874423725} a^{15} + \frac{221850231024042169837007}{4640345384605574372118625} a^{14} + \frac{373235336968325587362658}{4640345384605574372118625} a^{13} - \frac{14851078457487188572439}{4640345384605574372118625} a^{12} - \frac{158354061032679730145772}{928069076921114874423725} a^{11} - \frac{1783031192177400331104367}{4640345384605574372118625} a^{10} + \frac{388297401799808124809139}{928069076921114874423725} a^{9} + \frac{1580813328271850947314048}{4640345384605574372118625} a^{8} + \frac{204920996562053539803261}{4640345384605574372118625} a^{7} - \frac{852785440919100574126051}{4640345384605574372118625} a^{6} - \frac{402308084759162315992483}{928069076921114874423725} a^{5} - \frac{1699899462969184400498542}{4640345384605574372118625} a^{4} - \frac{1515136341669582906206983}{4640345384605574372118625} a^{3} - \frac{2305563769494805788683891}{4640345384605574372118625} a^{2} + \frac{755112549033558620407921}{4640345384605574372118625} a + \frac{8098895557200653805746}{272961493212092610124625}$

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:  $11$
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:  \( 7898710.55939 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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

3.3.473.1, 7.1.9938375.1

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 $21$ ${\href{/LocalNumberField/3.14.0.1}{14} }{,}\,{\href{/LocalNumberField/3.7.0.1}{7} }$ R $21$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/37.14.0.1}{14} }{,}\,{\href{/LocalNumberField/37.7.0.1}{7} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.7.0.1}{7} }$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ $21$

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
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.7.0.1$x^{7} - x + 4$$1$$7$$0$$C_7$$[\ ]^{7}$
11.14.7.2$x^{14} - 1771561 x^{2} + 77948684$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.2.1.1$x^{2} - 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.1$x^{2} - 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.1$x^{2} - 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.1$x^{2} - 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.1$x^{2} - 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.1$x^{2} - 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.1$x^{2} - 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.1$x^{2} - 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.1$x^{2} - 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.1$x^{2} - 43$$2$$1$$1$$C_2$$[\ ]_{2}$