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Properties

Label 21.5.211...000.1
Degree $21$
Signature $(5, 8)$
Discriminant $2.114\times 10^{32}$
Root discriminant \(34.62\)
Ramified primes $2,3,5,113$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_3^6:A_7.C_2$ (as 21T129)

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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 - 16*x^18 + 100*x^15 - 302*x^12 + 421*x^9 - 162*x^6 - 88*x^3 - 8)
 
Copy content gp:K = bnfinit(y^21 - 16*y^18 + 100*y^15 - 302*y^12 + 421*y^9 - 162*y^6 - 88*y^3 - 8, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 16*x^18 + 100*x^15 - 302*x^12 + 421*x^9 - 162*x^6 - 88*x^3 - 8);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - 16*x^18 + 100*x^15 - 302*x^12 + 421*x^9 - 162*x^6 - 88*x^3 - 8)
 

\( x^{21} - 16x^{18} + 100x^{15} - 302x^{12} + 421x^{9} - 162x^{6} - 88x^{3} - 8 \) 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:  $(5, 8)$
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:   \(211442929780702787875897344000000\) \(\medspace = 2^{24}\cdot 3^{18}\cdot 5^{6}\cdot 113^{6}\) 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:  \(34.62\)
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:  not computed
Ramified primes:   \(2\), \(3\), \(5\), \(113\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
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.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{4}$, $\frac{1}{6}a^{17}+\frac{1}{6}a^{16}+\frac{1}{6}a^{15}-\frac{1}{3}a^{14}-\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{6}a^{5}+\frac{1}{6}a^{4}+\frac{1}{6}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{12}a^{18}+\frac{1}{3}a^{12}+\frac{1}{6}a^{9}-\frac{1}{4}a^{6}-\frac{1}{2}a^{3}-\frac{1}{3}$, $\frac{1}{36}a^{19}-\frac{1}{36}a^{18}-\frac{1}{6}a^{16}+\frac{1}{6}a^{15}+\frac{1}{9}a^{13}-\frac{1}{9}a^{12}-\frac{5}{18}a^{10}+\frac{5}{18}a^{9}-\frac{1}{12}a^{7}+\frac{1}{12}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{2}{9}a-\frac{2}{9}$, $\frac{1}{36}a^{20}-\frac{1}{36}a^{18}+\frac{1}{6}a^{16}-\frac{1}{6}a^{15}-\frac{2}{9}a^{14}-\frac{1}{3}a^{13}-\frac{4}{9}a^{12}+\frac{1}{18}a^{11}+\frac{1}{3}a^{10}-\frac{7}{18}a^{9}-\frac{5}{12}a^{8}-\frac{1}{3}a^{7}-\frac{1}{4}a^{6}-\frac{1}{6}a^{5}+\frac{1}{6}a^{4}-\frac{1}{9}a^{2}-\frac{1}{3}a+\frac{4}{9}$ 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:  Not computed
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:  $12$
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:   $\frac{4}{3}a^{18}+\frac{43}{2}a^{15}-\frac{409}{3}a^{12}+\frac{1270}{3}a^{9}-629a^{6}+\frac{637}{2}a^{3}+\frac{202}{3}$, $\frac{25}{12}a^{18}-\frac{67}{2}a^{15}+\frac{634}{3}a^{12}-\frac{3905}{6}a^{9}+\frac{3819}{4}a^{6}-472a^{3}-\frac{307}{3}$, $\frac{19}{12}a^{18}+\frac{51}{2}a^{15}-\frac{484}{3}a^{12}+\frac{2993}{6}a^{9}-\frac{2941}{4}a^{6}+365a^{3}+\frac{241}{3}$, $a^{18}+\frac{33}{2}a^{15}-107a^{12}+339a^{9}-512a^{6}+\frac{521}{2}a^{3}+56$, $\frac{7}{12}a^{18}-9a^{15}+\frac{163}{3}a^{12}-\frac{959}{6}a^{9}+\frac{893}{4}a^{6}-\frac{209}{2}a^{3}-\frac{70}{3}$, $\frac{757}{36}a^{20}+\frac{217}{36}a^{18}+\frac{1019}{3}a^{17}-\frac{583}{6}a^{15}-\frac{19396}{9}a^{14}+\frac{5536}{9}a^{12}+\frac{120317}{18}a^{11}-\frac{34259}{18}a^{9}-\frac{118727}{12}a^{8}+\frac{33719}{12}a^{6}+\frac{29735}{6}a^{5}-\frac{4210}{3}a^{3}+\frac{9637}{9}a^{2}-\frac{2734}{9}$, $\frac{20}{3}a^{20}-\frac{7}{6}a^{19}-\frac{7}{6}a^{18}+\frac{647}{6}a^{17}+\frac{113}{6}a^{16}+\frac{113}{6}a^{15}-\frac{2056}{3}a^{14}-\frac{358}{3}a^{13}-\frac{358}{3}a^{12}+\frac{6391}{3}a^{11}+\frac{1108}{3}a^{10}+\frac{1108}{3}a^{9}-\frac{9491}{3}a^{8}-\frac{3271}{6}a^{7}-\frac{3271}{6}a^{6}+\frac{9575}{6}a^{5}+\frac{1637}{6}a^{4}+\frac{1637}{6}a^{3}+339a^{2}+56a+56$, $\frac{14}{9}a^{20}-\frac{55}{12}a^{19}-\frac{13}{18}a^{18}-\frac{151}{6}a^{17}+\frac{445}{6}a^{16}+12a^{15}+\frac{1439}{9}a^{14}-\frac{1415}{3}a^{13}-\frac{704}{9}a^{12}-\frac{4466}{9}a^{11}+\frac{8803}{6}a^{10}+\frac{2237}{9}a^{9}+734a^{8}-\frac{26155}{12}a^{7}-\frac{2257}{6}a^{6}-\frac{733}{2}a^{5}+\frac{3296}{3}a^{4}+\frac{580}{3}a^{3}-\frac{719}{9}a^{2}+233a+\frac{359}{9}$, $\frac{85}{36}a^{20}+\frac{145}{36}a^{19}+\frac{41}{12}a^{18}+\frac{229}{6}a^{17}-\frac{391}{6}a^{16}-55a^{15}-\frac{2182}{9}a^{14}+\frac{3727}{9}a^{13}+\frac{1043}{3}a^{12}+\frac{13565}{18}a^{11}-\frac{23153}{18}a^{10}-\frac{6445}{6}a^{9}-\frac{13451}{12}a^{8}+\frac{22883}{12}a^{7}+\frac{6335}{4}a^{6}+\frac{1714}{3}a^{5}-\frac{2875}{3}a^{4}-\frac{1583}{2}a^{3}+\frac{1018}{9}a^{2}-\frac{1825}{9}a-\frac{500}{3}$, $\frac{47}{18}a^{20}-\frac{19}{36}a^{18}-\frac{253}{6}a^{17}+\frac{26}{3}a^{15}+\frac{2407}{9}a^{14}-\frac{505}{9}a^{12}-\frac{7462}{9}a^{11}+\frac{3209}{18}a^{9}+\frac{7357}{6}a^{8}-\frac{3257}{12}a^{6}-\frac{3671}{6}a^{5}+\frac{845}{6}a^{3}-\frac{1216}{9}a^{2}+\frac{268}{9}$, $\frac{103}{18}a^{20}+\frac{103}{36}a^{19}+\frac{103}{36}a^{18}+\frac{278}{3}a^{17}-\frac{139}{3}a^{16}-\frac{139}{3}a^{15}-\frac{5306}{9}a^{14}+\frac{2653}{9}a^{13}+\frac{2653}{9}a^{12}+\frac{16505}{9}a^{11}-\frac{16505}{18}a^{10}-\frac{16505}{18}a^{9}-\frac{5447}{2}a^{8}+\frac{5447}{4}a^{7}+\frac{5447}{4}a^{6}+1371a^{5}-\frac{1371}{2}a^{4}-\frac{1371}{2}a^{3}+\frac{2633}{9}a^{2}-\frac{1312}{9}a-\frac{1312}{9}$, $\frac{13}{9}a^{20}+\frac{7}{18}a^{19}-\frac{5}{12}a^{18}-\frac{70}{3}a^{17}-\frac{13}{2}a^{16}+\frac{19}{3}a^{15}+\frac{1333}{9}a^{14}+\frac{386}{9}a^{13}-\frac{112}{3}a^{12}-\frac{4138}{9}a^{11}-\frac{1253}{9}a^{10}+\frac{635}{6}a^{9}+681a^{8}+\frac{1309}{6}a^{7}-\frac{1673}{12}a^{6}-339a^{5}-\frac{719}{6}a^{4}+\frac{353}{6}a^{3}-\frac{721}{9}a^{2}-\frac{203}{9}a+12$ 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:  \( 178510738.2852738 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{8}\cdot 178510738.2852738 \cdot 1}{2\cdot\sqrt{211442929780702787875897344000000}}\cr\approx \mathstrut & 0.477118978288736 \end{aligned}\] (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 - 16*x^18 + 100*x^15 - 302*x^12 + 421*x^9 - 162*x^6 - 88*x^3 - 8) 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 - 16*x^18 + 100*x^15 - 302*x^12 + 421*x^9 - 162*x^6 - 88*x^3 - 8, 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 - 16*x^18 + 100*x^15 - 302*x^12 + 421*x^9 - 162*x^6 - 88*x^3 - 8); 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 - 16*x^18 + 100*x^15 - 302*x^12 + 421*x^9 - 162*x^6 - 88*x^3 - 8); 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

$C_3^6:A_7.C_2$ (as 21T129):

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 non-solvable group of order 3674160
The 71 conjugacy class representatives for $C_3^6:A_7.C_2$
Character table for $C_3^6:A_7.C_2$

Intermediate fields

7.5.61291200.1

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 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 R R R ${\href{/padicField/7.7.0.1}{7} }^{3}$ ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.5.0.1}{5} }$ ${\href{/padicField/13.7.0.1}{7} }^{3}$ $18{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.7.0.1}{7} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ ${\href{/padicField/37.7.0.1}{7} }^{3}$ ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ ${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }$

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 2.4.6.9$x^{4} + 2 x^{3} + 6$$4$$1$$6$$D_{4}$$$[2, 2]^{2}$$
2.8.12.14$x^{8} + 4 x^{7} + 12 x^{6} + 22 x^{5} + 31 x^{4} + 30 x^{3} + 22 x^{2} + 10 x + 9$$4$$2$$12$$D_4$$$[2, 2]^{2}$$
2.9.6.1$x^{9} + 3 x^{7} + 3 x^{6} + 3 x^{5} + 6 x^{4} + 4 x^{3} + 3 x^{2} + 3 x + 3$$3$$3$$6$$S_3\times C_3$$$[\ ]_{3}^{6}$$
\(3\) Copy content Toggle raw display 3.6.3.2$x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
3.15.15.9$x^{15} + 6 x^{11} + 3 x^{10} + 12 x^{7} + 18 x^{6} + 3 x^{5} + 8 x^{3} + 24 x^{2} + 12 x + 4$$3$$5$$15$15T33$$[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{5}$$
\(5\) Copy content Toggle raw display 5.3.0.1$x^{3} + 3 x + 3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
5.6.0.1$x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
5.12.6.2$x^{12} + 2 x^{10} + 8 x^{9} + 3 x^{8} + 8 x^{7} + 22 x^{6} + 8 x^{5} + 5 x^{4} + 16 x^{3} + 4 x^{2} + 5 x + 4$$2$$6$$6$$C_{12}$$$[\ ]_{2}^{6}$$
\(113\) Copy content Toggle raw display $\Q_{113}$$x + 110$$1$$1$$0$Trivial$$[\ ]$$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$$[\ ]_{2}$$
113.2.0.1$x^{2} + 101 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$$[\ ]_{2}$$
113.4.2.1$x^{4} + 202 x^{3} + 10207 x^{2} + 606 x + 122$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
113.4.2.1$x^{4} + 202 x^{3} + 10207 x^{2} + 606 x + 122$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
113.6.0.1$x^{6} + x^{4} + 59 x^{3} + 30 x^{2} + 71 x + 3$$1$$6$$0$$C_6$$$[\ ]^{6}$$

Spectrum of ring of integers

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