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Properties

Label 20.0.357...000.1
Degree $20$
Signature $[0, 10]$
Discriminant $3.575\times 10^{42}$
Root discriminant \(134.17\)
Ramified primes $2,5,601,1801$
Class number $41369600$ (GRH)
Class group [2, 2, 2, 160, 32320] (GRH)
Galois group $C_2^4:C_{20}$ (as 20T75)

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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^20 + 370*x^18 + 55620*x^16 + 4488850*x^14 + 214685475*x^12 + 6306436255*x^10 + 113790309300*x^8 + 1220914673775*x^6 + 7148125674525*x^4 + 18168127845025*x^2 + 5857959624005)
 
Copy content gp:K = bnfinit(y^20 + 370*y^18 + 55620*y^16 + 4488850*y^14 + 214685475*y^12 + 6306436255*y^10 + 113790309300*y^8 + 1220914673775*y^6 + 7148125674525*y^4 + 18168127845025*y^2 + 5857959624005, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 370*x^18 + 55620*x^16 + 4488850*x^14 + 214685475*x^12 + 6306436255*x^10 + 113790309300*x^8 + 1220914673775*x^6 + 7148125674525*x^4 + 18168127845025*x^2 + 5857959624005);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 + 370*x^18 + 55620*x^16 + 4488850*x^14 + 214685475*x^12 + 6306436255*x^10 + 113790309300*x^8 + 1220914673775*x^6 + 7148125674525*x^4 + 18168127845025*x^2 + 5857959624005)
 

\( x^{20} + 370 x^{18} + 55620 x^{16} + 4488850 x^{14} + 214685475 x^{12} + 6306436255 x^{10} + \cdots + 5857959624005 \) 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:  $20$
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:  $[0, 10]$
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:   \(3575414809573364257812500000000000000000000\) \(\medspace = 2^{20}\cdot 5^{35}\cdot 601^{2}\cdot 1801^{2}\) 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:  \(134.17\)
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:  $2^{15/8}5^{7/4}601^{1/2}1801^{1/2}\approx 63800.31777789046$
Ramified primes:   \(2\), \(5\), \(601\), \(1801\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_4$
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 a CM field.
Reflex fields:  unavailable$^{512}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{1043}a^{16}-\frac{20}{149}a^{14}+\frac{79}{1043}a^{12}+\frac{228}{1043}a^{10}+\frac{212}{1043}a^{8}+\frac{389}{1043}a^{6}-\frac{76}{1043}a^{4}-\frac{348}{1043}a^{2}+\frac{324}{1043}$, $\frac{1}{1043}a^{17}-\frac{20}{149}a^{15}+\frac{79}{1043}a^{13}+\frac{228}{1043}a^{11}+\frac{212}{1043}a^{9}+\frac{389}{1043}a^{7}-\frac{76}{1043}a^{5}-\frac{348}{1043}a^{3}+\frac{324}{1043}a$, $\frac{1}{13\cdots 93}a^{18}-\frac{60\cdots 56}{13\cdots 93}a^{16}-\frac{67\cdots 12}{13\cdots 93}a^{14}-\frac{61\cdots 18}{19\cdots 99}a^{12}+\frac{52\cdots 47}{13\cdots 93}a^{10}+\frac{10\cdots 30}{19\cdots 99}a^{8}-\frac{37\cdots 02}{19\cdots 99}a^{6}-\frac{70\cdots 77}{19\cdots 99}a^{4}-\frac{25\cdots 66}{13\cdots 93}a^{2}-\frac{55\cdots 50}{12\cdots 93}$, $\frac{1}{13\cdots 93}a^{19}-\frac{60\cdots 56}{13\cdots 93}a^{17}-\frac{67\cdots 12}{13\cdots 93}a^{15}-\frac{61\cdots 18}{19\cdots 99}a^{13}+\frac{52\cdots 47}{13\cdots 93}a^{11}+\frac{10\cdots 30}{19\cdots 99}a^{9}-\frac{37\cdots 02}{19\cdots 99}a^{7}-\frac{70\cdots 77}{19\cdots 99}a^{5}-\frac{25\cdots 66}{13\cdots 93}a^{3}-\frac{55\cdots 50}{12\cdots 93}a$ 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:  $C_{2}\times C_{2}\times C_{2}\times C_{160}\times C_{32320}$, which has order $41369600$ (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:  $C_{2}\times C_{2}\times C_{2}\times C_{160}\times C_{32320}$, which has order $41369600$ (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);
 
Relative class number:   $41369600$ (assuming GRH)

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:  $9$
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:   $\frac{37\cdots 50}{17\cdots 57}a^{18}+\frac{13\cdots 30}{17\cdots 57}a^{16}+\frac{19\cdots 70}{17\cdots 57}a^{14}+\frac{20\cdots 60}{24\cdots 51}a^{12}+\frac{61\cdots 07}{17\cdots 57}a^{10}+\frac{22\cdots 10}{24\cdots 51}a^{8}+\frac{32\cdots 85}{24\cdots 51}a^{6}+\frac{24\cdots 00}{24\cdots 51}a^{4}+\frac{54\cdots 25}{17\cdots 57}a^{2}+\frac{17\cdots 14}{15\cdots 57}$, $\frac{93\cdots 52}{19\cdots 99}a^{18}+\frac{23\cdots 76}{13\cdots 93}a^{16}+\frac{48\cdots 48}{19\cdots 99}a^{14}+\frac{26\cdots 24}{13\cdots 93}a^{12}+\frac{11\cdots 40}{13\cdots 93}a^{10}+\frac{30\cdots 16}{13\cdots 93}a^{8}+\frac{47\cdots 68}{13\cdots 93}a^{6}+\frac{38\cdots 07}{13\cdots 93}a^{4}+\frac{12\cdots 88}{13\cdots 93}a^{2}+\frac{17\cdots 34}{12\cdots 93}$, $\frac{15\cdots 92}{13\cdots 93}a^{18}+\frac{80\cdots 99}{19\cdots 99}a^{16}+\frac{81\cdots 04}{13\cdots 93}a^{14}+\frac{63\cdots 80}{13\cdots 93}a^{12}+\frac{28\cdots 64}{13\cdots 93}a^{10}+\frac{74\cdots 84}{13\cdots 93}a^{8}+\frac{11\cdots 28}{13\cdots 93}a^{6}+\frac{88\cdots 08}{13\cdots 93}a^{4}+\frac{28\cdots 12}{13\cdots 93}a^{2}+\frac{11\cdots 38}{17\cdots 99}$, $\frac{34\cdots 86}{13\cdots 93}a^{18}+\frac{12\cdots 56}{13\cdots 93}a^{16}+\frac{17\cdots 89}{13\cdots 93}a^{14}+\frac{12\cdots 48}{13\cdots 93}a^{12}+\frac{54\cdots 23}{13\cdots 93}a^{10}+\frac{13\cdots 06}{13\cdots 93}a^{8}+\frac{18\cdots 86}{13\cdots 93}a^{6}+\frac{13\cdots 22}{13\cdots 93}a^{4}+\frac{40\cdots 23}{13\cdots 93}a^{2}+\frac{12\cdots 44}{12\cdots 93}$, $\frac{36\cdots 26}{13\cdots 93}a^{18}+\frac{18\cdots 49}{19\cdots 99}a^{16}+\frac{18\cdots 91}{13\cdots 93}a^{14}+\frac{13\cdots 50}{13\cdots 93}a^{12}+\frac{57\cdots 65}{13\cdots 93}a^{10}+\frac{14\cdots 14}{13\cdots 93}a^{8}+\frac{20\cdots 61}{13\cdots 93}a^{6}+\frac{14\cdots 33}{13\cdots 93}a^{4}+\frac{42\cdots 82}{13\cdots 93}a^{2}+\frac{14\cdots 98}{17\cdots 99}$, $\frac{20\cdots 60}{19\cdots 99}a^{18}+\frac{49\cdots 28}{13\cdots 93}a^{16}+\frac{94\cdots 60}{19\cdots 99}a^{14}+\frac{46\cdots 00}{13\cdots 93}a^{12}+\frac{18\cdots 86}{13\cdots 93}a^{10}+\frac{47\cdots 78}{13\cdots 93}a^{8}+\frac{72\cdots 52}{13\cdots 93}a^{6}+\frac{64\cdots 10}{13\cdots 93}a^{4}+\frac{23\cdots 47}{13\cdots 93}a^{2}-\frac{16\cdots 46}{12\cdots 93}$, $\frac{48\cdots 52}{13\cdots 93}a^{18}+\frac{17\cdots 52}{13\cdots 93}a^{16}+\frac{26\cdots 56}{13\cdots 93}a^{14}+\frac{21\cdots 60}{13\cdots 93}a^{12}+\frac{10\cdots 56}{13\cdots 93}a^{10}+\frac{27\cdots 45}{13\cdots 93}a^{8}+\frac{45\cdots 32}{13\cdots 93}a^{6}+\frac{38\cdots 20}{13\cdots 93}a^{4}+\frac{19\cdots 64}{19\cdots 99}a^{2}+\frac{56\cdots 86}{12\cdots 93}$, $\frac{85\cdots 80}{19\cdots 99}a^{18}+\frac{21\cdots 61}{13\cdots 93}a^{16}+\frac{43\cdots 01}{19\cdots 99}a^{14}+\frac{22\cdots 32}{13\cdots 93}a^{12}+\frac{96\cdots 98}{13\cdots 93}a^{10}+\frac{24\cdots 13}{13\cdots 93}a^{8}+\frac{34\cdots 13}{13\cdots 93}a^{6}+\frac{26\cdots 41}{13\cdots 93}a^{4}+\frac{81\cdots 72}{13\cdots 93}a^{2}+\frac{26\cdots 84}{12\cdots 93}$, $\frac{14\cdots 40}{13\cdots 93}a^{18}-\frac{92\cdots 20}{13\cdots 93}a^{16}-\frac{13\cdots 16}{13\cdots 93}a^{14}-\frac{20\cdots 67}{13\cdots 93}a^{12}-\frac{14\cdots 72}{13\cdots 93}a^{10}-\frac{36\cdots 14}{89\cdots 57}a^{8}-\frac{10\cdots 36}{13\cdots 93}a^{6}-\frac{89\cdots 15}{13\cdots 93}a^{4}-\frac{25\cdots 92}{13\cdots 93}a^{2}-\frac{61\cdots 14}{12\cdots 93}$ 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:  \( 161406.837641 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 161406.837641 \cdot 41369600}{2\cdot\sqrt{3575414809573364257812500000000000000000000}}\cr\approx \mathstrut & 0.169320179816 \end{aligned}\] (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^20 + 370*x^18 + 55620*x^16 + 4488850*x^14 + 214685475*x^12 + 6306436255*x^10 + 113790309300*x^8 + 1220914673775*x^6 + 7148125674525*x^4 + 18168127845025*x^2 + 5857959624005) 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^20 + 370*x^18 + 55620*x^16 + 4488850*x^14 + 214685475*x^12 + 6306436255*x^10 + 113790309300*x^8 + 1220914673775*x^6 + 7148125674525*x^4 + 18168127845025*x^2 + 5857959624005, 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^20 + 370*x^18 + 55620*x^16 + 4488850*x^14 + 214685475*x^12 + 6306436255*x^10 + 113790309300*x^8 + 1220914673775*x^6 + 7148125674525*x^4 + 18168127845025*x^2 + 5857959624005); 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^20 + 370*x^18 + 55620*x^16 + 4488850*x^14 + 214685475*x^12 + 6306436255*x^10 + 113790309300*x^8 + 1220914673775*x^6 + 7148125674525*x^4 + 18168127845025*x^2 + 5857959624005); 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

$C_2^4:C_{20}$ (as 20T75):

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 solvable group of order 320
The 32 conjugacy class representatives for $C_2^4:C_{20}$
Character table for $C_2^4:C_{20}$

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.390625.1, \(\Q(\zeta_{25})^+\)

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 20 siblings: data not computed
Degree 40 siblings: 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 $20$ R ${\href{/padicField/7.4.0.1}{4} }^{5}$ ${\href{/padicField/11.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/padicField/29.10.0.1}{10} }^{2}$ ${\href{/padicField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/padicField/59.10.0.1}{10} }^{2}$

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
\(2\) Copy content Toggle raw display 2.10.2.20a85.2$x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{16} + 4 x^{15} + 6 x^{14} + 4 x^{13} + 7 x^{12} + 8 x^{11} + 9 x^{10} + 8 x^{9} + 8 x^{8} + 8 x^{7} + 7 x^{6} + 8 x^{5} + 7 x^{4} + 6 x^{3} + 5 x^{2} + 6 x + 3$$2$$10$$20$20T75not computed
\(5\) Copy content Toggle raw display 5.1.20.35a1.1$x^{20} + 20 x^{16} + 5$$20$$1$$35$not computednot computed
\(601\) Copy content Toggle raw display $\Q_{601}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{601}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{601}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{601}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{601}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{601}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{601}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{601}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{601}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{601}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{601}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{601}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
\(1801\) Copy content Toggle raw display $\Q_{1801}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1801}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1801}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1801}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$

Spectrum of ring of integers

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