Properties

Label 18.0.533...000.2
Degree $18$
Signature $(0, 9)$
Discriminant $-5.334\times 10^{47}$
Root discriminant \(448.23\)
Ramified primes $2,3,5,137$
Class number $3$ (GRH)
Class group [3] (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

Related objects

Downloads

Learn more

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^18 - 8*x^15 + 35344*x^12 + 529920*x^9 + 381715200*x^6 - 933120000*x^3 + 1259712000000)
 
Copy content gp:K = bnfinit(y^18 - 8*y^15 + 35344*y^12 + 529920*y^9 + 381715200*y^6 - 933120000*y^3 + 1259712000000, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 8*x^15 + 35344*x^12 + 529920*x^9 + 381715200*x^6 - 933120000*x^3 + 1259712000000);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 8*x^15 + 35344*x^12 + 529920*x^9 + 381715200*x^6 - 933120000*x^3 + 1259712000000)
 

\( x^{18} - 8x^{15} + 35344x^{12} + 529920x^{9} + 381715200x^{6} - 933120000x^{3} + 1259712000000 \) 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:  $18$
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:  $(0, 9)$
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:   \(-533384837621025615313834850942347141675200000000\) \(\medspace = -\,2^{12}\cdot 3^{27}\cdot 5^{8}\cdot 137^{12}\) 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:  \(448.23\)
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:  $2^{2/3}3^{31/18}5^{2/3}137^{2/3}\approx 818.1932124074161$
Ramified primes:   \(2\), \(3\), \(5\), \(137\) 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:  \(\Q(\sqrt{-3}) \)
$\Aut(K/\Q)$:   $C_6$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\sqrt{-3}) \)

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

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{4}a^{3}$, $\frac{1}{8}a^{4}-\frac{1}{2}a$, $\frac{1}{8}a^{5}$, $\frac{1}{16}a^{6}$, $\frac{1}{32}a^{7}-\frac{1}{2}a$, $\frac{1}{192}a^{8}-\frac{1}{24}a^{5}+\frac{1}{12}a^{2}$, $\frac{1}{1920}a^{9}+\frac{13}{480}a^{6}-\frac{11}{120}a^{3}-\frac{1}{2}$, $\frac{1}{11520}a^{10}+\frac{43}{2880}a^{7}-\frac{41}{720}a^{4}-\frac{1}{4}a$, $\frac{1}{11520}a^{11}-\frac{1}{1440}a^{8}-\frac{41}{720}a^{5}$, $\frac{1}{21081600}a^{12}+\frac{853}{5270400}a^{9}-\frac{10301}{1317600}a^{6}-\frac{125}{1464}a^{3}+\frac{2}{61}$, $\frac{1}{126489600}a^{13}+\frac{853}{31622400}a^{10}-\frac{92651}{7905600}a^{7}+\frac{241}{8784}a^{4}-\frac{181}{366}a$, $\frac{1}{3794688000}a^{14}+\frac{3881}{118584000}a^{11}-\frac{7379}{59292000}a^{8}+\frac{14297}{658800}a^{5}-\frac{2741}{21960}a^{2}$, $\frac{1}{164044362240000}a^{15}-\frac{82657}{3728280960000}a^{12}-\frac{327853483}{5126386320000}a^{9}+\frac{189704791}{14239962000}a^{6}+\frac{3725849}{86302800}a^{3}+\frac{151259}{351604}$, $\frac{1}{492133086720000}a^{16}+\frac{721}{1398105360000}a^{13}+\frac{173984809}{30758317920000}a^{10}-\frac{1243832201}{170879544000}a^{7}+\frac{2707331}{64727100}a^{4}+\frac{332825}{1054812}a$, $\frac{1}{14\cdots 00}a^{17}+\frac{721}{41943160800000}a^{14}+\frac{4360486523}{115343692200000}a^{11}-\frac{337389793}{160199572500}a^{8}+\frac{13346489}{7767252000}a^{5}+\frac{561967}{15822180}a^{2}$ 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:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

Ideal class group:  $C_{3}$, which has order $3$ (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:  $C_{3}$, which has order $3$ (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:  $8$
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:   \( -\frac{4277}{164044362240000} a^{15} + \frac{839}{3728280960000} a^{12} - \frac{2597717}{2563193160000} a^{9} - \frac{350929}{28479924000} a^{6} - \frac{1400863}{86302800} a^{3} + \frac{178009}{351604} \)  (order $6$) 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:   $\frac{52\cdots 81}{73\cdots 00}a^{17}+\frac{33\cdots 99}{492133086720000}a^{16}+\frac{37\cdots 93}{8202218112000}a^{15}+\frac{39\cdots 83}{167772643200000}a^{14}+\frac{53\cdots 41}{5592421440000}a^{13}+\frac{86\cdots 23}{372828096000}a^{12}+\frac{90\cdots 29}{461374768800000}a^{11}+\frac{11\cdots 13}{961197435000}a^{10}+\frac{39\cdots 39}{64079829000}a^{9}+\frac{71\cdots 69}{2563193160000}a^{8}+\frac{75\cdots 13}{85439772000}a^{7}+\frac{58\cdots 71}{1423996200}a^{6}+\frac{10\cdots 17}{1941813000}a^{5}+\frac{10\cdots 71}{258908400}a^{4}+\frac{25\cdots 03}{1078785}a^{3}+\frac{81\cdots 87}{7911090}a^{2}+\frac{16\cdots 31}{527406}a+\frac{19\cdots 03}{175802}$, $\frac{23\cdots 71}{14\cdots 00}a^{17}-\frac{53\cdots 11}{492133086720000}a^{16}-\frac{37\cdots 93}{8202218112000}a^{15}+\frac{16\cdots 53}{335545286400000}a^{14}+\frac{14\cdots 77}{11184842880000}a^{13}-\frac{86\cdots 23}{372828096000}a^{12}+\frac{11\cdots 29}{57671846100000}a^{11}+\frac{40\cdots 19}{7689579480000}a^{10}-\frac{39\cdots 39}{64079829000}a^{9}-\frac{72\cdots 83}{2563193160000}a^{8}+\frac{32\cdots 01}{175081500}a^{7}-\frac{58\cdots 71}{1423996200}a^{6}+\frac{10\cdots 69}{127332000}a^{5}+\frac{14\cdots 91}{258908400}a^{4}-\frac{25\cdots 03}{1078785}a^{3}-\frac{49\cdots 09}{31644360}a^{2}+\frac{72\cdots 05}{1054812}a-\frac{19\cdots 03}{175802}$, $\frac{497698413049}{123033271680000}a^{17}-\frac{43296279287}{1518929280000}a^{16}+\frac{8054361909503}{164044362240000}a^{15}+\frac{864239221249}{5592421440000}a^{14}-\frac{22216992571}{207126720000}a^{13}-\frac{570684862871}{3728280960000}a^{12}+\frac{16\cdots 87}{15379158960000}a^{11}-\frac{482051879970323}{569598480000}a^{10}+\frac{50\cdots 13}{2563193160000}a^{9}+\frac{607085579626069}{85439772000}a^{8}-\frac{235642978793203}{7119981000}a^{7}+\frac{827397615227881}{28479924000}a^{6}+\frac{103390870279363}{129454200}a^{5}-\frac{9341665726731}{1598200}a^{4}+\frac{18\cdots 57}{86302800}a^{3}-\frac{143050883219683}{7911090}a^{2}-\frac{49828164982919}{527406}a+\frac{137559613014775}{351604}$, $\frac{39372547498039}{14\cdots 00}a^{17}+\frac{43296279287}{1518929280000}a^{16}+\frac{102914948747}{8202218112000}a^{15}-\frac{79143319104373}{335545286400000}a^{14}+\frac{22216992571}{207126720000}a^{13}+\frac{237151838417}{372828096000}a^{12}+\frac{34\cdots 61}{57671846100000}a^{11}+\frac{482051879970323}{569598480000}a^{10}+\frac{140538541413649}{256319316000}a^{9}-\frac{10\cdots 27}{2563193160000}a^{8}+\frac{235642978793203}{7119981000}a^{7}+\frac{49160519806217}{711998100}a^{6}-\frac{833349784075999}{7767252000}a^{5}+\frac{9341665726731}{1598200}a^{4}+\frac{10176916719449}{2157570}a^{3}-\frac{44257844167135}{703208}a^{2}+\frac{49828164982919}{527406}a+\frac{233612627406313}{175802}$, $\frac{14\cdots 77}{164044362240000}a^{15}+\frac{91\cdots 61}{3728280960000}a^{12}+\frac{25\cdots 17}{2563193160000}a^{9}+\frac{25\cdots 89}{28479924000}a^{6}-\frac{15\cdots 97}{86302800}a^{3}+\frac{13\cdots 79}{351604}$, $\frac{49\cdots 57}{164044362240000}a^{15}-\frac{27\cdots 99}{3728280960000}a^{12}+\frac{20\cdots 47}{2563193160000}a^{9}+\frac{79\cdots 49}{28479924000}a^{6}+\frac{16\cdots 23}{86302800}a^{3}+\frac{73\cdots 35}{351604}$, $\frac{91\cdots 97}{1466864640000}a^{17}+\frac{14\cdots 99}{3844789740000}a^{16}+\frac{60\cdots 97}{3728280960000}a^{15}+\frac{19\cdots 51}{366716160000}a^{14}+\frac{14\cdots 21}{11184842880000}a^{13}+\frac{41\cdots 87}{1864140480000}a^{12}+\frac{11\cdots 63}{5729940000}a^{11}+\frac{17\cdots 99}{15379158960000}a^{10}+\frac{47\cdots 17}{954990000}a^{9}+\frac{17\cdots 17}{84888000}a^{8}+\frac{54\cdots 19}{85439772000}a^{7}+\frac{77\cdots 19}{647271000}a^{6}+\frac{37\cdots 41}{1697760}a^{5}+\frac{13\cdots 59}{129454200}a^{4}+\frac{86\cdots 23}{21575700}a^{3}+\frac{53\cdots 29}{47160}a^{2}+\frac{17\cdots 39}{1054812}a-\frac{11\cdots 23}{15982}$, $\frac{91\cdots 97}{1466864640000}a^{17}-\frac{10\cdots 51}{3417590880000}a^{16}-\frac{66\cdots 69}{7456561920000}a^{15}+\frac{19\cdots 51}{366716160000}a^{14}+\frac{17\cdots 37}{1242760320000}a^{13}-\frac{18\cdots 37}{1864140480000}a^{12}+\frac{11\cdots 63}{5729940000}a^{11}-\frac{32\cdots 19}{3417590880000}a^{10}-\frac{13\cdots 71}{466035120000}a^{9}+\frac{17\cdots 17}{84888000}a^{8}+\frac{12\cdots 33}{56959848000}a^{7}-\frac{93\cdots 51}{2589084000}a^{6}+\frac{37\cdots 41}{1697760}a^{5}-\frac{58\cdots 43}{9589200}a^{4}-\frac{18\cdots 37}{5393925}a^{3}+\frac{53\cdots 29}{47160}a^{2}+\frac{14\cdots 01}{263703}a-\frac{17\cdots 95}{7991}$ 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:  \( 159078827156081020 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 159078827156081020 \cdot 3}{6\cdot\sqrt{533384837621025615313834850942347141675200000000}}\cr\approx \mathstrut & 1.66219077225373 \end{aligned}\] (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^18 - 8*x^15 + 35344*x^12 + 529920*x^9 + 381715200*x^6 - 933120000*x^3 + 1259712000000) 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^18 - 8*x^15 + 35344*x^12 + 529920*x^9 + 381715200*x^6 - 933120000*x^3 + 1259712000000, 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^18 - 8*x^15 + 35344*x^12 + 529920*x^9 + 381715200*x^6 - 933120000*x^3 + 1259712000000); 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^18 - 8*x^15 + 35344*x^12 + 529920*x^9 + 381715200*x^6 - 933120000*x^3 + 1259712000000); 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

$C_3^2:S_3$ (as 18T24):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
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 solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt[3]{548})\) x3, 6.0.152182955952.1, 9.3.421657379721587708280000.2

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

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 9 siblings: data not computed
Degree 18 siblings: data not computed
Degree 27 sibling: data not computed
Minimal sibling: 9.3.5054766393380427000000.5

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 R ${\href{/padicField/7.3.0.1}{3} }^{6}$ ${\href{/padicField/11.2.0.1}{2} }^{9}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ ${\href{/padicField/17.2.0.1}{2} }^{9}$ ${\href{/padicField/19.3.0.1}{3} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }^{3}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.2.0.1}{2} }^{9}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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
\(2\) Copy content Toggle raw display 2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
\(3\) Copy content Toggle raw display 3.1.6.9a1.5$x^{6} + 3 x^{5} + 6 x^{4} + 12$$6$$1$$9$$S_3\times C_3$$$[\frac{3}{2}, 2]_{2}$$
3.1.6.9a1.5$x^{6} + 3 x^{5} + 6 x^{4} + 12$$6$$1$$9$$S_3\times C_3$$$[\frac{3}{2}, 2]_{2}$$
3.1.6.9a1.2$x^{6} + 6 x^{4} + 12$$6$$1$$9$$C_6$$$[2]_{2}$$
\(5\) Copy content Toggle raw display 5.2.3.4a1.1$x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 53 x + 8$$3$$2$$4$$S_3\times C_3$$$[\ ]_{3}^{6}$$
5.2.3.4a1.1$x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 53 x + 8$$3$$2$$4$$S_3\times C_3$$$[\ ]_{3}^{6}$$
5.6.1.0a1.1$x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
\(137\) Copy content Toggle raw display 137.6.3.12a1.2$x^{18} + 3 x^{16} + 348 x^{15} + 309 x^{14} + 705 x^{13} + 40990 x^{12} + 71358 x^{11} + 73992 x^{10} + 1635821 x^{9} + 4152708 x^{8} + 3745674 x^{7} + 1397178 x^{6} + 309798 x^{5} + 102681 x^{4} + 8667 x^{3} + 2835 x^{2} + 81 x + 164$$3$$6$$12$$S_3 \times C_3$$$[\ ]_{3}^{6}$$

Spectrum of ring of integers

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