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Properties

Label 21.21.3569422997...4801.1
Degree $21$
Signature $[21, 0]$
Discriminant $337^{20}$
Root discriminant $255.43$
Ramified prime $337$
Class number $1$ (GRH)
Class group Trivial (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![7333121, -12506848, -159561140, 545440112, -444734752, -370577748, 682493117, -85146670, -301059030, 115833483, 53760107, -30765519, -4575964, 3811310, 195358, -253705, -4485, 9107, 75, -160, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 160*x^19 + 75*x^18 + 9107*x^17 - 4485*x^16 - 253705*x^15 + 195358*x^14 + 3811310*x^13 - 4575964*x^12 - 30765519*x^11 + 53760107*x^10 + 115833483*x^9 - 301059030*x^8 - 85146670*x^7 + 682493117*x^6 - 370577748*x^5 - 444734752*x^4 + 545440112*x^3 - 159561140*x^2 - 12506848*x + 7333121)
 
gp: K = bnfinit(x^21 - x^20 - 160*x^19 + 75*x^18 + 9107*x^17 - 4485*x^16 - 253705*x^15 + 195358*x^14 + 3811310*x^13 - 4575964*x^12 - 30765519*x^11 + 53760107*x^10 + 115833483*x^9 - 301059030*x^8 - 85146670*x^7 + 682493117*x^6 - 370577748*x^5 - 444734752*x^4 + 545440112*x^3 - 159561140*x^2 - 12506848*x + 7333121, 1)
 

Normalized defining polynomial

\( x^{21} - x^{20} - 160 x^{19} + 75 x^{18} + 9107 x^{17} - 4485 x^{16} - 253705 x^{15} + 195358 x^{14} + 3811310 x^{13} - 4575964 x^{12} - 30765519 x^{11} + 53760107 x^{10} + 115833483 x^{9} - 301059030 x^{8} - 85146670 x^{7} + 682493117 x^{6} - 370577748 x^{5} - 444734752 x^{4} + 545440112 x^{3} - 159561140 x^{2} - 12506848 x + 7333121 \)

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:  \(356942299737980709598670980612578149070093175924801=337^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $255.43$
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:  $337$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(337\)
Dirichlet character group:    $\lbrace$$\chi_{337}(64,·)$, $\chi_{337}(1,·)$, $\chi_{337}(2,·)$, $\chi_{337}(4,·)$, $\chi_{337}(8,·)$, $\chi_{337}(128,·)$, $\chi_{337}(13,·)$, $\chi_{337}(256,·)$, $\chi_{337}(79,·)$, $\chi_{337}(16,·)$, $\chi_{337}(26,·)$, $\chi_{337}(158,·)$, $\chi_{337}(32,·)$, $\chi_{337}(208,·)$, $\chi_{337}(295,·)$, $\chi_{337}(104,·)$, $\chi_{337}(169,·)$, $\chi_{337}(175,·)$, $\chi_{337}(52,·)$, $\chi_{337}(316,·)$, $\chi_{337}(253,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{59} a^{17} + \frac{8}{59} a^{16} + \frac{9}{59} a^{15} - \frac{27}{59} a^{14} + \frac{29}{59} a^{13} - \frac{20}{59} a^{12} - \frac{10}{59} a^{11} + \frac{6}{59} a^{10} + \frac{17}{59} a^{9} + \frac{27}{59} a^{8} + \frac{9}{59} a^{7} - \frac{21}{59} a^{6} + \frac{27}{59} a^{5} + \frac{9}{59} a^{4} - \frac{22}{59} a^{3} + \frac{12}{59} a^{2} - \frac{8}{59} a + \frac{22}{59}$, $\frac{1}{59} a^{18} + \frac{4}{59} a^{16} + \frac{19}{59} a^{15} + \frac{9}{59} a^{14} - \frac{16}{59} a^{13} - \frac{27}{59} a^{12} + \frac{27}{59} a^{11} + \frac{28}{59} a^{10} + \frac{9}{59} a^{9} + \frac{29}{59} a^{8} + \frac{25}{59} a^{7} + \frac{18}{59} a^{6} + \frac{29}{59} a^{5} + \frac{24}{59} a^{4} + \frac{11}{59} a^{3} + \frac{14}{59} a^{2} + \frac{27}{59} a + \frac{1}{59}$, $\frac{1}{59} a^{19} - \frac{13}{59} a^{16} - \frac{27}{59} a^{15} - \frac{26}{59} a^{14} - \frac{25}{59} a^{13} - \frac{11}{59} a^{12} + \frac{9}{59} a^{11} - \frac{15}{59} a^{10} + \frac{20}{59} a^{9} - \frac{24}{59} a^{8} - \frac{18}{59} a^{7} - \frac{5}{59} a^{6} - \frac{25}{59} a^{5} - \frac{25}{59} a^{4} - \frac{16}{59} a^{3} - \frac{21}{59} a^{2} - \frac{26}{59} a - \frac{29}{59}$, $\frac{1}{1810654795414219711439761144265949364836609748887749507494168293407003} a^{20} + \frac{5385094613061184272359666136459815196924037186131992203010908437011}{1810654795414219711439761144265949364836609748887749507494168293407003} a^{19} - \frac{8779566554147965701284536761763681730509105632514923249847246060665}{1810654795414219711439761144265949364836609748887749507494168293407003} a^{18} + \frac{3312402767190802798664627591485192143819540488030680066692094133654}{1810654795414219711439761144265949364836609748887749507494168293407003} a^{17} - \frac{846106341037194618539994394248996391995694000729445461099812466122746}{1810654795414219711439761144265949364836609748887749507494168293407003} a^{16} + \frac{697868050730973818302156009396446883600902558078455706041947083792025}{1810654795414219711439761144265949364836609748887749507494168293407003} a^{15} - \frac{285315780476896941490934408336365556360150385341584378711381007351261}{1810654795414219711439761144265949364836609748887749507494168293407003} a^{14} + \frac{401697750940940912389125128688840582850425050394121343993888549501084}{1810654795414219711439761144265949364836609748887749507494168293407003} a^{13} + \frac{397191362255957100209127591855590236680677709000730428108638451947741}{1810654795414219711439761144265949364836609748887749507494168293407003} a^{12} - \frac{172711184019079963121548287769966115229186265948303241285839479839193}{1810654795414219711439761144265949364836609748887749507494168293407003} a^{11} - \frac{576173890414037285181618394798471098022076098741803809284025318212056}{1810654795414219711439761144265949364836609748887749507494168293407003} a^{10} - \frac{742977944044699786417798554852804934153611138603144280645829097210030}{1810654795414219711439761144265949364836609748887749507494168293407003} a^{9} + \frac{206371974115138038684666382092646332760808379190871627937453804422790}{1810654795414219711439761144265949364836609748887749507494168293407003} a^{8} - \frac{190843032569828450307039795776541607884169095073383196338468174150312}{1810654795414219711439761144265949364836609748887749507494168293407003} a^{7} + \frac{114227242021283017775881182556821988731362215841284410217560779733410}{1810654795414219711439761144265949364836609748887749507494168293407003} a^{6} + \frac{456812862030134796974650600155268494618478890040436608239191251463870}{1810654795414219711439761144265949364836609748887749507494168293407003} a^{5} - \frac{557090224116122133930460062223695313477250238736985856792885060612522}{1810654795414219711439761144265949364836609748887749507494168293407003} a^{4} - \frac{379335504307604106141765677918049834835306048557620467981781354732082}{1810654795414219711439761144265949364836609748887749507494168293407003} a^{3} - \frac{118786261088158854440676888921652292279487577284510399514816475379353}{1810654795414219711439761144265949364836609748887749507494168293407003} a^{2} + \frac{496683613132120124575436188464247321279753899952503256103163584438081}{1810654795414219711439761144265949364836609748887749507494168293407003} a - \frac{39007749806498226233636789657596783560919343875914320098480421726968}{1810654795414219711439761144265949364836609748887749507494168293407003}$

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:  \( 3025545659707238400 \) (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

3.3.113569.1, 7.7.1464803622199009.1

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 $21$ $21$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{3}$ $21$ $21$ $21$ $21$ $21$ $21$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{21}$

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
337Data not computed