↑

Properties

Label 21.21.8085137733...7472.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 3^{28}\cdot 7^{21}\cdot 17^{6}$
Root discriminant $123.27$
Ramified primes $2, 3, 7, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T43

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![85544, -218484, -1614354, -908327, 4078368, 4361112, -3298680, -5090418, 1205064, 2897321, -214326, -950355, 17598, 189189, -528, -22925, 0, 1638, 0, -63, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 63*x^19 + 1638*x^17 - 22925*x^15 - 528*x^14 + 189189*x^13 + 17598*x^12 - 950355*x^11 - 214326*x^10 + 2897321*x^9 + 1205064*x^8 - 5090418*x^7 - 3298680*x^6 + 4361112*x^5 + 4078368*x^4 - 908327*x^3 - 1614354*x^2 - 218484*x + 85544)
 
gp: K = bnfinit(x^21 - 63*x^19 + 1638*x^17 - 22925*x^15 - 528*x^14 + 189189*x^13 + 17598*x^12 - 950355*x^11 - 214326*x^10 + 2897321*x^9 + 1205064*x^8 - 5090418*x^7 - 3298680*x^6 + 4361112*x^5 + 4078368*x^4 - 908327*x^3 - 1614354*x^2 - 218484*x + 85544, 1)
 

Normalized defining polynomial

\( x^{21} - 63 x^{19} + 1638 x^{17} - 22925 x^{15} - 528 x^{14} + 189189 x^{13} + 17598 x^{12} - 950355 x^{11} - 214326 x^{10} + 2897321 x^{9} + 1205064 x^{8} - 5090418 x^{7} - 3298680 x^{6} + 4361112 x^{5} + 4078368 x^{4} - 908327 x^{3} - 1614354 x^{2} - 218484 x + 85544 \)

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:  \(80851377332871149873039593691602344912617472=2^{18}\cdot 3^{28}\cdot 7^{21}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $123.27$
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:  $2, 3, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{3}$, $\frac{1}{34} a^{18} + \frac{5}{34} a^{16} + \frac{3}{17} a^{14} + \frac{4}{17} a^{12} + \frac{8}{17} a^{11} - \frac{2}{17} a^{10} - \frac{7}{17} a^{9} - \frac{2}{17} a^{8} + \frac{5}{17} a^{7} - \frac{3}{17} a^{6} + \frac{1}{17} a^{5} + \frac{11}{34} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{680} a^{19} - \frac{1}{85} a^{18} + \frac{1}{136} a^{17} + \frac{7}{170} a^{16} + \frac{37}{340} a^{15} + \frac{1}{34} a^{14} - \frac{9}{680} a^{13} + \frac{28}{85} a^{12} + \frac{21}{680} a^{11} + \frac{43}{340} a^{10} + \frac{261}{680} a^{9} - \frac{23}{68} a^{8} + \frac{61}{136} a^{7} - \frac{32}{85} a^{6} - \frac{9}{68} a^{5} - \frac{28}{85} a^{4} + \frac{3}{10} a^{3} - \frac{1}{2} a^{2} - \frac{3}{40} a + \frac{1}{20}$, $\frac{1}{53936939098047499321113333813108800} a^{20} - \frac{16131159100232312735241318890357}{26968469549023749660556666906554400} a^{19} + \frac{692967435166142512750312354957733}{53936939098047499321113333813108800} a^{18} - \frac{1208400056174936682960067236189881}{26968469549023749660556666906554400} a^{17} - \frac{2624107080870757750362694812413747}{26968469549023749660556666906554400} a^{16} - \frac{753412507480479749060240732969121}{13484234774511874830278333453277200} a^{15} + \frac{725610364970188571695531379780251}{53936939098047499321113333813108800} a^{14} - \frac{1639330474916259104278845411979871}{26968469549023749660556666906554400} a^{13} + \frac{12538856667408671662936361579177777}{53936939098047499321113333813108800} a^{12} + \frac{1002211118988664539111709821030281}{2696846954902374966055666690655440} a^{11} + \frac{2848652832711335723351468585352673}{10787387819609499864222666762621760} a^{10} - \frac{308481178571206722591572843229417}{6742117387255937415139166726638600} a^{9} - \frac{470182610180858724346660249216351}{2157477563921899972844533352524352} a^{8} + \frac{2243478719008084415040998190258807}{26968469549023749660556666906554400} a^{7} - \frac{1361030985628437953781006422257407}{26968469549023749660556666906554400} a^{6} - \frac{3721137922637543359219647130572771}{13484234774511874830278333453277200} a^{5} - \frac{1117996382741242401780880875846307}{3371058693627968707569583363319300} a^{4} - \frac{21371257136673843984984486044731}{99148785106704961987340687156450} a^{3} + \frac{958869329416368417479640399792457}{3172761123414558783594901989006400} a^{2} + \frac{62144755677516133515242975317937}{158638056170727939179745099450320} a - \frac{146610057707495181497797165176303}{793190280853639695898725497251600}$

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:  $20$
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:  \( 5102009942530000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T43:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6174
The 45 conjugacy class representatives for t21n43
Character table for t21n43 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\)

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

Sibling fields

Degree 21 siblings: data not computed
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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ $21$ $21$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ $21$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
7Data not computed
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
17.7.6.1$x^{7} - 17$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$