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Properties

Label 21.21.1038187830...8329.2
Degree $21$
Signature $[21, 0]$
Discriminant $7^{38}\cdot 19^{14}$
Root discriminant $240.84$
Ramified primes $7, 19$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_{21}$ (as 21T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![24521880853, 170427535873, 369969955635, 236480949201, -155322145538, -239196844705, -36501637424, 60756994317, 24629746463, -4292878786, -3677713942, -136995908, 236399408, 29273335, -7387297, -1365336, 108276, 28462, -574, -273, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 273*x^19 - 574*x^18 + 28462*x^17 + 108276*x^16 - 1365336*x^15 - 7387297*x^14 + 29273335*x^13 + 236399408*x^12 - 136995908*x^11 - 3677713942*x^10 - 4292878786*x^9 + 24629746463*x^8 + 60756994317*x^7 - 36501637424*x^6 - 239196844705*x^5 - 155322145538*x^4 + 236480949201*x^3 + 369969955635*x^2 + 170427535873*x + 24521880853)
 
gp: K = bnfinit(x^21 - 273*x^19 - 574*x^18 + 28462*x^17 + 108276*x^16 - 1365336*x^15 - 7387297*x^14 + 29273335*x^13 + 236399408*x^12 - 136995908*x^11 - 3677713942*x^10 - 4292878786*x^9 + 24629746463*x^8 + 60756994317*x^7 - 36501637424*x^6 - 239196844705*x^5 - 155322145538*x^4 + 236480949201*x^3 + 369969955635*x^2 + 170427535873*x + 24521880853, 1)
 

Normalized defining polynomial

\( x^{21} - 273 x^{19} - 574 x^{18} + 28462 x^{17} + 108276 x^{16} - 1365336 x^{15} - 7387297 x^{14} + 29273335 x^{13} + 236399408 x^{12} - 136995908 x^{11} - 3677713942 x^{10} - 4292878786 x^{9} + 24629746463 x^{8} + 60756994317 x^{7} - 36501637424 x^{6} - 239196844705 x^{5} - 155322145538 x^{4} + 236480949201 x^{3} + 369969955635 x^{2} + 170427535873 x + 24521880853 \)

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:  \(103818783062189717738091671292152377422379176428329=7^{38}\cdot 19^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $240.84$
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:  $7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(931=7^{2}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{931}(1,·)$, $\chi_{931}(900,·)$, $\chi_{931}(134,·)$, $\chi_{931}(267,·)$, $\chi_{931}(400,·)$, $\chi_{931}(533,·)$, $\chi_{931}(666,·)$, $\chi_{931}(30,·)$, $\chi_{931}(799,·)$, $\chi_{931}(163,·)$, $\chi_{931}(102,·)$, $\chi_{931}(296,·)$, $\chi_{931}(235,·)$, $\chi_{931}(429,·)$, $\chi_{931}(368,·)$, $\chi_{931}(562,·)$, $\chi_{931}(501,·)$, $\chi_{931}(695,·)$, $\chi_{931}(634,·)$, $\chi_{931}(828,·)$, $\chi_{931}(767,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{19} a^{18} + \frac{8}{19} a^{17} + \frac{4}{19} a^{16} - \frac{9}{19} a^{15} + \frac{6}{19} a^{13} + \frac{8}{19} a^{12} - \frac{9}{19} a^{11} - \frac{1}{19} a^{10} - \frac{5}{19} a^{9} + \frac{8}{19} a^{8} + \frac{5}{19} a^{7} + \frac{4}{19} a^{6} + \frac{2}{19} a^{5} - \frac{9}{19} a^{4} + \frac{6}{19} a^{3} + \frac{6}{19} a^{2} + \frac{6}{19} a + \frac{8}{19}$, $\frac{1}{2040956839} a^{19} + \frac{46717285}{2040956839} a^{18} - \frac{952954412}{2040956839} a^{17} - \frac{401982188}{2040956839} a^{16} + \frac{105929323}{2040956839} a^{15} + \frac{879552091}{2040956839} a^{14} + \frac{988227083}{2040956839} a^{13} - \frac{451830014}{2040956839} a^{12} - \frac{146695039}{2040956839} a^{11} + \frac{297287503}{2040956839} a^{10} + \frac{400953640}{2040956839} a^{9} + \frac{680907440}{2040956839} a^{8} + \frac{2990423}{6965723} a^{7} + \frac{974307257}{2040956839} a^{6} + \frac{899090215}{2040956839} a^{5} - \frac{982781938}{2040956839} a^{4} + \frac{105530667}{2040956839} a^{3} - \frac{179120189}{2040956839} a^{2} + \frac{139674087}{2040956839} a + \frac{89443629}{2040956839}$, $\frac{1}{28985389711145518678161852479440387639792103548187423687405038420326329752521139567832545561} a^{20} + \frac{4718441280581405720687833744176397033971707996427036808140367515685866539956157652}{28985389711145518678161852479440387639792103548187423687405038420326329752521139567832545561} a^{19} - \frac{85128703901632655512501482727215833204617990700648583650859702431131191165151865208565848}{28985389711145518678161852479440387639792103548187423687405038420326329752521139567832545561} a^{18} - \frac{11738043037766626617474463040767519360745645648754314862418562839920774937073715441756227114}{28985389711145518678161852479440387639792103548187423687405038420326329752521139567832545561} a^{17} - \frac{8645884545066782823326469969487607658186292872602177776747634735733747742529950364096789173}{28985389711145518678161852479440387639792103548187423687405038420326329752521139567832545561} a^{16} - \frac{7890190627888110856039757075794822232883506504991196587120962774378290322907620947872995953}{28985389711145518678161852479440387639792103548187423687405038420326329752521139567832545561} a^{15} + \frac{6948994011124113459705886082527122756244214621690244964125324902309790760753272167033029946}{28985389711145518678161852479440387639792103548187423687405038420326329752521139567832545561} a^{14} + \frac{8814547526088747992248256924821821793186188767805315360659696357231105432929483470129177715}{28985389711145518678161852479440387639792103548187423687405038420326329752521139567832545561} a^{13} + \frac{5718476987690329857892566824910348390372778529079738645421737841318196036766497579694219647}{28985389711145518678161852479440387639792103548187423687405038420326329752521139567832545561} a^{12} + \frac{7187376560238882113689005547911296461399943953063121999940660836082153908267673793144413730}{28985389711145518678161852479440387639792103548187423687405038420326329752521139567832545561} a^{11} - \frac{6171935420716614401835218973848246400315236732796305090238105383289874313617944748697605497}{28985389711145518678161852479440387639792103548187423687405038420326329752521139567832545561} a^{10} - \frac{8627013665433248296388476299504971656806978441362825245545510576132535776475539104378136047}{28985389711145518678161852479440387639792103548187423687405038420326329752521139567832545561} a^{9} + \frac{12190556011991120485938059509296844037743437778191552652837036242486894982221171718181915295}{28985389711145518678161852479440387639792103548187423687405038420326329752521139567832545561} a^{8} + \frac{9293133217073774219165601211479762788549064870592624981364252382923854797342557152182336477}{28985389711145518678161852479440387639792103548187423687405038420326329752521139567832545561} a^{7} + \frac{13667999833370444749985222092221346856796354284559895390708472418542632440164079216983626990}{28985389711145518678161852479440387639792103548187423687405038420326329752521139567832545561} a^{6} - \frac{7139232924741240083471412723567704544938557460207951236418369758335338879568504941923676807}{28985389711145518678161852479440387639792103548187423687405038420326329752521139567832545561} a^{5} - \frac{46325494098798745878049404112016512986800351877142205179399934726588685323120356327635211}{110210607266712998776280807906617443497308378510218341016749195514548782328977716987956447} a^{4} - \frac{4441914277079900546822813860186766677042489927624272829154277808123852206624565401925902806}{28985389711145518678161852479440387639792103548187423687405038420326329752521139567832545561} a^{3} + \frac{635783954324970389877930293614618694191133284970903443399070232495650455510236339587273764}{1525546826902395719903255393654757244199584397273022299337107285280333144869533661464870819} a^{2} + \frac{2895029751066536897738357256811185613516358243159706822341109828553350248804328988533696449}{28985389711145518678161852479440387639792103548187423687405038420326329752521139567832545561} a - \frac{9517634542189100726358805928377792756596655417056499623730496519596782646326003586979253054}{28985389711145518678161852479440387639792103548187423687405038420326329752521139567832545561}$

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:  \( 644113238376879900 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{21}$ (as 21T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 21
The 21 conjugacy class representatives for $C_{21}$
Character table for $C_{21}$ is not computed

Intermediate fields

3.3.17689.1, 7.7.13841287201.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $21$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{3}$ $21$ R $21$ $21$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{3}$ R ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ $21$ $21$ $21$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{3}$

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
7Data not computed
$19$19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$