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Properties

Label 21.17.272...269.1
Degree $21$
Signature $[17, 2]$
Discriminant $2.726\times 10^{39}$
Root discriminant \(75.49\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{21}$ (as 21T164)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 - 19*x^19 + 34*x^18 + 151*x^17 - 233*x^16 - 653*x^15 + 831*x^14 + 1676*x^13 - 1665*x^12 - 2626*x^11 + 1895*x^10 + 2517*x^9 - 1184*x^8 - 1458*x^7 + 363*x^6 + 489*x^5 - 33*x^4 - 86*x^3 - 6*x^2 + 6*x + 1)
 
Copy content gp:K = bnfinit(y^21 - 2*y^20 - 19*y^19 + 34*y^18 + 151*y^17 - 233*y^16 - 653*y^15 + 831*y^14 + 1676*y^13 - 1665*y^12 - 2626*y^11 + 1895*y^10 + 2517*y^9 - 1184*y^8 - 1458*y^7 + 363*y^6 + 489*y^5 - 33*y^4 - 86*y^3 - 6*y^2 + 6*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 2*x^20 - 19*x^19 + 34*x^18 + 151*x^17 - 233*x^16 - 653*x^15 + 831*x^14 + 1676*x^13 - 1665*x^12 - 2626*x^11 + 1895*x^10 + 2517*x^9 - 1184*x^8 - 1458*x^7 + 363*x^6 + 489*x^5 - 33*x^4 - 86*x^3 - 6*x^2 + 6*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - 2*x^20 - 19*x^19 + 34*x^18 + 151*x^17 - 233*x^16 - 653*x^15 + 831*x^14 + 1676*x^13 - 1665*x^12 - 2626*x^11 + 1895*x^10 + 2517*x^9 - 1184*x^8 - 1458*x^7 + 363*x^6 + 489*x^5 - 33*x^4 - 86*x^3 - 6*x^2 + 6*x + 1)
 

\( x^{21} - 2 x^{20} - 19 x^{19} + 34 x^{18} + 151 x^{17} - 233 x^{16} - 653 x^{15} + 831 x^{14} + 1676 x^{13} + \cdots + 1 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $21$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[17, 2]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(2725960527801648753674435845823096440269\) \(\medspace = 3\cdot 683\cdot 1240271\cdot 39356833\cdot 49784320121\cdot 547454828827\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(75.49\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $3^{1/2}683^{1/2}1240271^{1/2}39356833^{1/2}49784320121^{1/2}547454828827^{1/2}\approx 5.2210731921719394e+19$
Ramified primes:   \(3\), \(683\), \(1240271\), \(39356833\), \(49784320121\), \(547454828827\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  $\Q(\sqrt{27259\!\cdots\!40269}$)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $18$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $a$, $a-1$, $a^{20}-2a^{19}-19a^{18}+34a^{17}+151a^{16}-233a^{15}-653a^{14}+831a^{13}+1676a^{12}-1665a^{11}-2626a^{10}+1895a^{9}+2517a^{8}-1184a^{7}-1458a^{6}+363a^{5}+489a^{4}-33a^{3}-86a^{2}-5a+7$, $a^{20}-2a^{19}-19a^{18}+34a^{17}+151a^{16}-233a^{15}-653a^{14}+831a^{13}+1676a^{12}-1665a^{11}-2626a^{10}+1895a^{9}+2517a^{8}-1184a^{7}-1458a^{6}+363a^{5}+489a^{4}-34a^{3}-85a^{2}-3a+4$, $a^{19}-a^{18}-20a^{17}+14a^{16}+165a^{15}-68a^{14}-721a^{13}+110a^{12}+1786a^{11}+121a^{10}-2505a^{9}-610a^{8}+1907a^{7}+723a^{6}-735a^{5}-372a^{4}+117a^{3}+84a^{2}-3a-7$, $4a^{20}-9a^{19}-74a^{18}+155a^{17}+570a^{16}-1083a^{15}-2379a^{14}+3977a^{13}+5873a^{12}-8336a^{11}-8839a^{10}+10206a^{9}+8173a^{8}-7253a^{7}-4648a^{6}+2910a^{5}+1592a^{4}-620a^{3}-306a^{2}+58a+25$, $79a^{20}-219a^{19}-1336a^{18}+3725a^{17}+9131a^{16}-25580a^{15}-32457a^{14}+91517a^{13}+64399a^{12}-183968a^{11}-72035a^{10}+210266a^{9}+46254a^{8}-134074a^{7}-19965a^{6}+46377a^{5}+6692a^{4}-8123a^{3}-1394a^{2}+556a+117$, $47a^{20}-118a^{19}-834a^{18}+2024a^{17}+6093a^{16}-14063a^{15}-23794a^{14}+51196a^{13}+54077a^{12}-105747a^{11}-73658a^{10}+126200a^{9}+61077a^{8}-85901a^{7}-31662a^{6}+32209a^{5}+10164a^{4}-6086a^{3}-1831a^{2}+444a+135$, $39a^{20}-108a^{19}-656a^{18}+1825a^{17}+4453a^{16}-12416a^{15}-15700a^{14}+43816a^{13}+30909a^{12}-86277a^{11}-34615a^{10}+95548a^{9}+23174a^{8}-58207a^{7}-11260a^{6}+19126a^{5}+4065a^{4}-3230a^{3}-770a^{2}+208a+54$, $42a^{20}-113a^{19}-722a^{18}+1929a^{17}+5052a^{16}-13315a^{15}-18595a^{14}+48006a^{13}+38939a^{12}-97685a^{11}-47467a^{10}+113889a^{9}+34572a^{8}-74972a^{7}-16540a^{6}+27142a^{5}+5488a^{4}-5046a^{3}-1094a^{2}+377a+91$, $40a^{20}-110a^{19}-697a^{18}+1916a^{17}+4979a^{16}-13599a^{15}-18943a^{14}+50979a^{13}+41915a^{12}-109596a^{11}-56047a^{10}+137972a^{9}+47035a^{8}-100494a^{7}-26331a^{6}+40692a^{5}+9838a^{4}-8272a^{3}-2095a^{2}+645a+174$, $51a^{20}-118a^{19}-930a^{18}+2023a^{17}+7026a^{16}-14041a^{15}-28555a^{14}+51031a^{13}+67892a^{12}-105160a^{11}-96719a^{10}+125012a^{9}+82478a^{8}-84252a^{7}-41857a^{6}+30662a^{5}+12336a^{4}-5446a^{3}-1976a^{2}+371a+130$, $30a^{20}-82a^{19}-519a^{18}+1418a^{17}+3663a^{16}-9966a^{15}-13644a^{14}+36862a^{13}+29043a^{12}-77805a^{11}-36142a^{10}+95542a^{9}+26787a^{8}-67366a^{7}-12789a^{6}+26240a^{5}+4309a^{4}-5118a^{3}-935a^{2}+381a+83$, $22a^{20}-44a^{19}-410a^{18}+732a^{17}+3173a^{16}-4860a^{15}-13214a^{14}+16518a^{13}+32082a^{12}-30647a^{11}-46173a^{10}+30632a^{9}+38757a^{8}-15080a^{7}-18244a^{6}+2681a^{5}+4351a^{4}+157a^{3}-422a^{2}-52a+6$, $187a^{20}-439a^{19}-3402a^{18}+7546a^{17}+25638a^{16}-52575a^{15}-103971a^{14}+192162a^{13}+246949a^{12}-399378a^{11}-352463a^{10}+481090a^{9}+302968a^{8}-331161a^{7}-156599a^{6}+124751a^{5}+47561a^{4}-23366a^{3}-7860a^{2}+1695a+527$, $25a^{20}-65a^{19}-425a^{18}+1082a^{17}+2923a^{16}-7196a^{15}-10455a^{14}+24506a^{13}+20793a^{12}-45471a^{11}-22910a^{10}+45194a^{9}+13521a^{8}-22061a^{7}-4242a^{6}+4296a^{5}+490a^{4}-57a^{3}+95a^{2}-49a-22$, $45a^{20}-111a^{19}-799a^{18}+1893a^{17}+5837a^{16}-13044a^{15}-22753a^{14}+46910a^{13}+51409a^{12}-95127a^{11}-69011a^{10}+110342a^{9}+55436a^{8}-71836a^{7}-27093a^{6}+25163a^{5}+7930a^{4}-4316a^{3}-1271a^{2}+280a+83$, $6a^{20}-21a^{19}-85a^{18}+345a^{17}+414a^{16}-2250a^{15}-551a^{14}+7414a^{13}-1923a^{12}-12908a^{11}+8079a^{10}+11081a^{9}-11620a^{8}-3370a^{7}+7740a^{6}-576a^{5}-2550a^{4}+433a^{3}+416a^{2}-52a-27$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 5743982678470 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{17}\cdot(2\pi)^{2}\cdot 5743982678470 \cdot 1}{2\cdot\sqrt{2725960527801648753674435845823096440269}}\cr\approx \mathstrut & 0.284638084067667 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 - 19*x^19 + 34*x^18 + 151*x^17 - 233*x^16 - 653*x^15 + 831*x^14 + 1676*x^13 - 1665*x^12 - 2626*x^11 + 1895*x^10 + 2517*x^9 - 1184*x^8 - 1458*x^7 + 363*x^6 + 489*x^5 - 33*x^4 - 86*x^3 - 6*x^2 + 6*x + 1) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^21 - 2*x^20 - 19*x^19 + 34*x^18 + 151*x^17 - 233*x^16 - 653*x^15 + 831*x^14 + 1676*x^13 - 1665*x^12 - 2626*x^11 + 1895*x^10 + 2517*x^9 - 1184*x^8 - 1458*x^7 + 363*x^6 + 489*x^5 - 33*x^4 - 86*x^3 - 6*x^2 + 6*x + 1, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 2*x^20 - 19*x^19 + 34*x^18 + 151*x^17 - 233*x^16 - 653*x^15 + 831*x^14 + 1676*x^13 - 1665*x^12 - 2626*x^11 + 1895*x^10 + 2517*x^9 - 1184*x^8 - 1458*x^7 + 363*x^6 + 489*x^5 - 33*x^4 - 86*x^3 - 6*x^2 + 6*x + 1); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - 2*x^20 - 19*x^19 + 34*x^18 + 151*x^17 - 233*x^16 - 653*x^15 + 831*x^14 + 1676*x^13 - 1665*x^12 - 2626*x^11 + 1895*x^10 + 2517*x^9 - 1184*x^8 - 1458*x^7 + 363*x^6 + 489*x^5 - 33*x^4 - 86*x^3 - 6*x^2 + 6*x + 1); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$S_{21}$ (as 21T164):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A non-solvable group of order 51090942171709440000
The 792 conjugacy class representatives for $S_{21}$
Character table for $S_{21}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 42 sibling: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.11.0.1}{11} }{,}\,{\href{/padicField/2.10.0.1}{10} }$ R $21$ ${\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.10.0.1}{10} }$ ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}$ ${\href{/padicField/17.13.0.1}{13} }{,}\,{\href{/padicField/17.8.0.1}{8} }$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ $20{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ $16{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ $20{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.11.0.1}{11} }{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ $19{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.2.1.0a1.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
3.3.1.0a1.1$x^{3} + 2 x + 1$$1$$3$$0$$C_3$$$[\ ]^{3}$$
3.5.1.0a1.1$x^{5} + 2 x + 1$$1$$5$$0$$C_5$$$[\ ]^{5}$$
3.9.1.0a1.1$x^{9} + 2 x^{3} + 2 x^{2} + x + 1$$1$$9$$0$$C_9$$$[\ ]^{9}$$
\(683\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
Deg $7$$1$$7$$0$$C_7$$$[\ ]^{7}$$
\(1240271\) Copy content Toggle raw display $\Q_{1240271}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $15$$1$$15$$0$$C_{15}$$$[\ ]^{15}$$
\(39356833\) Copy content Toggle raw display $\Q_{39356833}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $5$$1$$5$$0$$C_5$$$[\ ]^{5}$$
Deg $11$$1$$11$$0$$C_{11}$$$[\ ]^{11}$$
\(49784320121\) Copy content Toggle raw display $\Q_{49784320121}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $9$$1$$9$$0$$C_9$$$[\ ]^{9}$$
Deg $9$$1$$9$$0$$C_9$$$[\ ]^{9}$$
\(547454828827\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $5$$1$$5$$0$$C_5$$$[\ ]^{5}$$
Deg $11$$1$$11$$0$$C_{11}$$$[\ ]^{11}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)