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Properties

Label 21.3.26160559119...0703.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 13^{18}$
Root discriminant $43.54$
Ramified primes $7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times D_7$ (as 21T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-197, -1859, -6766, -9224, 6041, 39606, 67348, 68396, 21246, 6174, -10884, -8526, -1196, -427, 660, -59, -58, -23, -34, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 21*x^19 - 34*x^18 - 23*x^17 - 58*x^16 - 59*x^15 + 660*x^14 - 427*x^13 - 1196*x^12 - 8526*x^11 - 10884*x^10 + 6174*x^9 + 21246*x^8 + 68396*x^7 + 67348*x^6 + 39606*x^5 + 6041*x^4 - 9224*x^3 - 6766*x^2 - 1859*x - 197)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 - 23*x^17 - 58*x^16 - 59*x^15 + 660*x^14 - 427*x^13 - 1196*x^12 - 8526*x^11 - 10884*x^10 + 6174*x^9 + 21246*x^8 + 68396*x^7 + 67348*x^6 + 39606*x^5 + 6041*x^4 - 9224*x^3 - 6766*x^2 - 1859*x - 197, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} - 23 x^{17} - 58 x^{16} - 59 x^{15} + 660 x^{14} - 427 x^{13} - 1196 x^{12} - 8526 x^{11} - 10884 x^{10} + 6174 x^{9} + 21246 x^{8} + 68396 x^{7} + 67348 x^{6} + 39606 x^{5} + 6041 x^{4} - 9224 x^{3} - 6766 x^{2} - 1859 x - 197 \)

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:  \(-26160559119874379648562947375700703=-\,7^{17}\cdot 13^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.54$
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:  $7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{13} a^{18} - \frac{2}{13} a^{17} - \frac{6}{13} a^{16} + \frac{4}{13} a^{15} + \frac{5}{13} a^{14} - \frac{6}{13} a^{13} + \frac{1}{13} a^{12} + \frac{1}{13} a^{11} - \frac{6}{13} a^{10} - \frac{5}{13} a^{8} + \frac{2}{13} a^{7} + \frac{3}{13} a^{6} - \frac{3}{13} a^{5} + \frac{5}{13} a^{4} + \frac{4}{13} a^{3} - \frac{3}{13} a^{2} + \frac{5}{13} a - \frac{3}{13}$, $\frac{1}{13} a^{19} + \frac{3}{13} a^{17} + \frac{5}{13} a^{16} + \frac{4}{13} a^{14} + \frac{2}{13} a^{13} + \frac{3}{13} a^{12} - \frac{4}{13} a^{11} + \frac{1}{13} a^{10} - \frac{5}{13} a^{9} + \frac{5}{13} a^{8} - \frac{6}{13} a^{7} + \frac{3}{13} a^{6} - \frac{1}{13} a^{5} + \frac{1}{13} a^{4} + \frac{5}{13} a^{3} - \frac{1}{13} a^{2} - \frac{6}{13} a - \frac{6}{13}$, $\frac{1}{8728484183488973500083225987418901314271780150617} a^{20} + \frac{206186179102524935913399212420976518492439748125}{8728484183488973500083225987418901314271780150617} a^{19} - \frac{217123057926102928076341759994523435589533366260}{8728484183488973500083225987418901314271780150617} a^{18} + \frac{2485282999112624764909867459578661267302433726752}{8728484183488973500083225987418901314271780150617} a^{17} - \frac{2049717614867957133495859999049385058072098366262}{8728484183488973500083225987418901314271780150617} a^{16} - \frac{260635711737820377778996852748109715946483768997}{8728484183488973500083225987418901314271780150617} a^{15} + \frac{3230086743770466296017396501511912720851817704170}{8728484183488973500083225987418901314271780150617} a^{14} - \frac{1037593761370734432166838776191519344518209523384}{8728484183488973500083225987418901314271780150617} a^{13} - \frac{29139645733351683015585967888698053992634348764}{212889858133877402441054292376070763762726345137} a^{12} - \frac{3917425866957165871373675776360954571811294409479}{8728484183488973500083225987418901314271780150617} a^{11} + \frac{360341192719311971398890849394243336748730203869}{8728484183488973500083225987418901314271780150617} a^{10} - \frac{2760825539535324990196770755911655859637233051099}{8728484183488973500083225987418901314271780150617} a^{9} - \frac{3958761303786076667890850065520306515066311681667}{8728484183488973500083225987418901314271780150617} a^{8} + \frac{2646839771330085376618970585126239169767547584568}{8728484183488973500083225987418901314271780150617} a^{7} + \frac{4163713763368778644248759951595367532721906681336}{8728484183488973500083225987418901314271780150617} a^{6} - \frac{137903915239731906703450616098749428495735753020}{8728484183488973500083225987418901314271780150617} a^{5} - \frac{3044412700288136509356731580539964212698402374721}{8728484183488973500083225987418901314271780150617} a^{4} + \frac{1360408169086025907394724495377542890227636128747}{8728484183488973500083225987418901314271780150617} a^{3} + \frac{3397214074909217255791209280142718582562852097}{671421860268382576929478922109146254943983088509} a^{2} - \frac{1099119051645533791193511544721699277918646151359}{8728484183488973500083225987418901314271780150617} a - \frac{9621143102150928689124079776584090352236610249}{44307026312126769035955461865070565047064873861}$

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:  \( 936748793.4645762 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times D_7$ (as 21T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 15 conjugacy class representatives for $C_3\times D_7$
Character table for $C_3\times D_7$

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.1.1655595487.1

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

Sibling fields

Galois closure: 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 $21$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R $21$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$13$13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$