↑

Properties

Label 21.21.5629684989...3216.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 3^{28}\cdot 7^{26}$
Root discriminant $87.20$
Ramified primes $2, 3, 7$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_7:C_3$ (as 21T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-28779219, -391156857, -1588530069, -3073057932, -3090183579, -1276860228, 439003348, 700361361, 197166396, -81978176, -58140705, -1703289, 6432475, 1118670, -360885, -103334, 10290, 4599, -119, -105, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 105*x^19 - 119*x^18 + 4599*x^17 + 10290*x^16 - 103334*x^15 - 360885*x^14 + 1118670*x^13 + 6432475*x^12 - 1703289*x^11 - 58140705*x^10 - 81978176*x^9 + 197166396*x^8 + 700361361*x^7 + 439003348*x^6 - 1276860228*x^5 - 3090183579*x^4 - 3073057932*x^3 - 1588530069*x^2 - 391156857*x - 28779219)
 
gp: K = bnfinit(x^21 - 105*x^19 - 119*x^18 + 4599*x^17 + 10290*x^16 - 103334*x^15 - 360885*x^14 + 1118670*x^13 + 6432475*x^12 - 1703289*x^11 - 58140705*x^10 - 81978176*x^9 + 197166396*x^8 + 700361361*x^7 + 439003348*x^6 - 1276860228*x^5 - 3090183579*x^4 - 3073057932*x^3 - 1588530069*x^2 - 391156857*x - 28779219, 1)
 

Normalized defining polynomial

\( x^{21} - 105 x^{19} - 119 x^{18} + 4599 x^{17} + 10290 x^{16} - 103334 x^{15} - 360885 x^{14} + 1118670 x^{13} + 6432475 x^{12} - 1703289 x^{11} - 58140705 x^{10} - 81978176 x^{9} + 197166396 x^{8} + 700361361 x^{7} + 439003348 x^{6} - 1276860228 x^{5} - 3090183579 x^{4} - 3073057932 x^{3} - 1588530069 x^{2} - 391156857 x - 28779219 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(56296849895429213104110710203449262473216=2^{18}\cdot 3^{28}\cdot 7^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.20$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{4}$, $\frac{1}{21} a^{11} - \frac{1}{3} a^{5}$, $\frac{1}{21} a^{12} - \frac{1}{3} a^{6}$, $\frac{1}{21} a^{13} - \frac{1}{3} a^{7}$, $\frac{1}{21} a^{14} - \frac{1}{3} a^{8}$, $\frac{1}{21} a^{15} - \frac{1}{3} a^{3}$, $\frac{1}{63} a^{16} + \frac{1}{9} a^{10} + \frac{1}{3} a^{7} + \frac{4}{9} a^{4} - \frac{1}{3} a$, $\frac{1}{189} a^{17} + \frac{1}{63} a^{15} - \frac{1}{63} a^{14} - \frac{1}{63} a^{13} - \frac{2}{189} a^{11} - \frac{1}{9} a^{9} + \frac{2}{9} a^{8} + \frac{1}{9} a^{7} - \frac{5}{27} a^{5} - \frac{1}{3} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{567} a^{18} + \frac{1}{189} a^{16} + \frac{2}{189} a^{15} - \frac{4}{189} a^{14} - \frac{1}{63} a^{13} + \frac{1}{81} a^{12} - \frac{1}{63} a^{11} - \frac{4}{27} a^{10} - \frac{1}{27} a^{9} + \frac{4}{27} a^{8} + \frac{4}{9} a^{7} + \frac{40}{81} a^{6} + \frac{1}{9} a^{5} + \frac{11}{27} a^{3} + \frac{1}{9} a^{2}$, $\frac{1}{567} a^{19} - \frac{1}{189} a^{16} + \frac{2}{189} a^{15} - \frac{11}{567} a^{13} - \frac{1}{63} a^{12} + \frac{1}{189} a^{11} - \frac{4}{27} a^{10} - \frac{2}{27} a^{9} + \frac{2}{9} a^{8} + \frac{31}{81} a^{7} + \frac{1}{9} a^{6} + \frac{5}{27} a^{5} - \frac{1}{27} a^{4} + \frac{4}{9} a^{3} - \frac{2}{9} a^{2}$, $\frac{1}{1701355218047919981407766684470391} a^{20} + \frac{1141358790317069337183484383248}{1701355218047919981407766684470391} a^{19} - \frac{892360487833708136858658654689}{1701355218047919981407766684470391} a^{18} + \frac{703275787359527574512924253953}{567118406015973327135922228156797} a^{17} + \frac{3065243910641263001611581900751}{567118406015973327135922228156797} a^{16} + \frac{4050663639120508748914817857411}{189039468671991109045307409385599} a^{15} + \frac{34233552054441755111613429710995}{1701355218047919981407766684470391} a^{14} - \frac{22565835583363084886134489981555}{1701355218047919981407766684470391} a^{13} - \frac{3444324916226137809980846169023}{243050745435417140201109526352913} a^{12} + \frac{4322808439748158709215653155365}{567118406015973327135922228156797} a^{11} + \frac{12688375683016815341328582108571}{81016915145139046733703175450971} a^{10} + \frac{8637979327056790225757162492347}{81016915145139046733703175450971} a^{9} - \frac{37793807405882573102238104606}{243050745435417140201109526352913} a^{8} + \frac{18847421962863085011803181245051}{243050745435417140201109526352913} a^{7} + \frac{92478592837062419558724819227077}{243050745435417140201109526352913} a^{6} + \frac{4305915936563647628099735742959}{27005638381713015577901058483657} a^{5} - \frac{13211359267686933692418075207932}{81016915145139046733703175450971} a^{4} - \frac{19955372973206186897942683685404}{81016915145139046733703175450971} a^{3} - \frac{314307435556730403473381595647}{1000208828952333910292631795691} a^{2} - \frac{445979612775237259218654591070}{1000208828952333910292631795691} a + \frac{44057990948988595325070988564}{1000208828952333910292631795691}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}$, which has order $3$ (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:  $20$
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:  \( 218203272125000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7:C_3$ (as 21T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 21
The 5 conjugacy class representatives for $C_7:C_3$
Character table for $C_7:C_3$

Intermediate fields

3.3.3969.2, 7.7.2420662999104.1 x7

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

Sibling fields

Degree 7 sibling: 7.7.2420662999104.1

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 ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
2Data not computed
$3$3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
7Data not computed