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Properties

Label 21.1.11841910006...2512.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{26}\cdot 3^{19}\cdot 7^{21}\cdot 43^{7}$
Root discriminant $156.30$
Ramified primes $2, 3, 7, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times F_7$ (as 21T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4362447, 2233119, -3684954, 10378536, -9280761, 6082125, -1556478, 517684, -779296, 782026, -392630, 199136, -85652, 33726, -14550, 5208, -1589, 539, -140, 28, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 28*x^19 - 140*x^18 + 539*x^17 - 1589*x^16 + 5208*x^15 - 14550*x^14 + 33726*x^13 - 85652*x^12 + 199136*x^11 - 392630*x^10 + 782026*x^9 - 779296*x^8 + 517684*x^7 - 1556478*x^6 + 6082125*x^5 - 9280761*x^4 + 10378536*x^3 - 3684954*x^2 + 2233119*x - 4362447)
 
gp: K = bnfinit(x^21 - 7*x^20 + 28*x^19 - 140*x^18 + 539*x^17 - 1589*x^16 + 5208*x^15 - 14550*x^14 + 33726*x^13 - 85652*x^12 + 199136*x^11 - 392630*x^10 + 782026*x^9 - 779296*x^8 + 517684*x^7 - 1556478*x^6 + 6082125*x^5 - 9280761*x^4 + 10378536*x^3 - 3684954*x^2 + 2233119*x - 4362447, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 28 x^{19} - 140 x^{18} + 539 x^{17} - 1589 x^{16} + 5208 x^{15} - 14550 x^{14} + 33726 x^{13} - 85652 x^{12} + 199136 x^{11} - 392630 x^{10} + 782026 x^{9} - 779296 x^{8} + 517684 x^{7} - 1556478 x^{6} + 6082125 x^{5} - 9280761 x^{4} + 10378536 x^{3} - 3684954 x^{2} + 2233119 x - 4362447 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11841910006204867301303769538659846667225792512=2^{26}\cdot 3^{19}\cdot 7^{21}\cdot 43^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $156.30$
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:  $2, 3, 7, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{12} a^{14} + \frac{1}{12} a^{13} + \frac{1}{6} a^{11} - \frac{1}{12} a^{10} - \frac{1}{4} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{12} a^{15} - \frac{1}{12} a^{13} - \frac{1}{12} a^{12} - \frac{1}{4} a^{11} - \frac{1}{6} a^{10} + \frac{1}{12} a^{9} - \frac{1}{4} a^{8} - \frac{1}{12} a^{7} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{24} a^{16} - \frac{1}{4} a^{8} + \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{8}$, $\frac{1}{24} a^{17} - \frac{1}{4} a^{9} + \frac{1}{6} a^{8} - \frac{1}{2} a^{3} + \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{24} a^{18} - \frac{1}{4} a^{10} + \frac{1}{6} a^{9} - \frac{1}{2} a^{4} + \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{72} a^{19} + \frac{1}{72} a^{16} + \frac{1}{12} a^{13} - \frac{1}{12} a^{12} - \frac{1}{12} a^{11} - \frac{1}{9} a^{10} - \frac{1}{12} a^{9} + \frac{1}{6} a^{8} + \frac{1}{18} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{24} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{5}{24}$, $\frac{1}{1520755930648126484090267990474865941007146463309917187413206152} a^{20} + \frac{2272410110259223310709059113984032089650112329522215713644619}{380188982662031621022566997618716485251786615827479296853301538} a^{19} + \frac{3904859411059372383206426680829656605799387630353132223962995}{506918643549375494696755996824955313669048821103305729137735384} a^{18} - \frac{10655327141283046111204989156537798920016107388161921578707703}{760377965324063242045133995237432970503573231654958593706603076} a^{17} + \frac{23791417742595661927778289800589186142897088523062240367621281}{1520755930648126484090267990474865941007146463309917187413206152} a^{16} + \frac{503716986005943088332889814395990512936788533051196057158895}{253459321774687747348377998412477656834524410551652864568867692} a^{15} + \frac{1086816524667184191385199071705933694999625060584636042578537}{63364830443671936837094499603119414208631102637913216142216923} a^{14} + \frac{8903886410468735870019099134601852331577784151179422323004765}{253459321774687747348377998412477656834524410551652864568867692} a^{13} - \frac{13369241693096887065696930274478352708404364528174416636641525}{126729660887343873674188999206238828417262205275826432284433846} a^{12} + \frac{162078999304776547651520991566099447132390325208617264528605645}{760377965324063242045133995237432970503573231654958593706603076} a^{11} + \frac{153604337269068736704525752520920237879235334591336204881033703}{760377965324063242045133995237432970503573231654958593706603076} a^{10} + \frac{10241991823194719623471322853151016128487432816587150428735951}{84486440591562582449459332804159218944841470183884288189622564} a^{9} - \frac{20934961285774543783293783357043247181112953238370857345563598}{190094491331015810511283498809358242625893307913739648426650769} a^{8} + \frac{53548023182662715920238418949306701532407608624518330362448831}{760377965324063242045133995237432970503573231654958593706603076} a^{7} + \frac{18852725271421982144632280861686138349739543129313450127072705}{42243220295781291224729666402079609472420735091942144094811282} a^{6} + \frac{23778455656305397578583615764016954256987656814253475064488855}{84486440591562582449459332804159218944841470183884288189622564} a^{5} + \frac{228867478343856646670813919585287365540763365960003820886202757}{506918643549375494696755996824955313669048821103305729137735384} a^{4} - \frac{125277782052924169231642408949284003627331566579259123539123705}{253459321774687747348377998412477656834524410551652864568867692} a^{3} - \frac{39721936278852001065668665186765654341024560080433309041781343}{168972881183125164898918665608318437889682940367768576379245128} a^{2} - \frac{9311306645440951778923173457571895719381258747944833247234609}{126729660887343873674188999206238828417262205275826432284433846} a + \frac{149192563170174900136292941534261478848472917334505636075997957}{506918643549375494696755996824955313669048821103305729137735384}$

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:  $10$
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:  \( 2613750744265613.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_7$ (as 21T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 252
The 21 conjugacy class representatives for $S_3\times F_7$
Character table for $S_3\times F_7$ is not computed

Intermediate fields

3.1.516.1, 7.1.38423222208.8

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ $21$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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
$2$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.14.20.8$x^{14} + 2 x^{13} + 2 x^{11} + 2 x^{8} + 2 x^{7} + 2 x^{6} + 2$$14$$1$$20$$(C_7:C_3) \times C_2$$[2]_{7}^{3}$
$3$3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
3.14.13.2$x^{14} + 3$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
7Data not computed
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$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}$