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Properties

Label 16.0.720...625.1
Degree $16$
Signature $[0, 8]$
Discriminant $7.201\times 10^{19}$
Root discriminant \(17.42\)
Ramified primes $5,11,101$
Class number $1$
Class group trivial
Galois group $(C_2^2\times C_4^2):D_8$ (as 16T1276)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 11*x^14 - 17*x^13 + 40*x^12 - 37*x^11 + 66*x^10 + 25*x^9 - 30*x^8 + 254*x^7 - 180*x^6 + 443*x^5 - 120*x^4 + 127*x^3 + 575*x^2 - 527*x + 551)
 
Copy content gp:K = bnfinit(y^16 - 3*y^15 + 11*y^14 - 17*y^13 + 40*y^12 - 37*y^11 + 66*y^10 + 25*y^9 - 30*y^8 + 254*y^7 - 180*y^6 + 443*y^5 - 120*y^4 + 127*y^3 + 575*y^2 - 527*y + 551, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 11*x^14 - 17*x^13 + 40*x^12 - 37*x^11 + 66*x^10 + 25*x^9 - 30*x^8 + 254*x^7 - 180*x^6 + 443*x^5 - 120*x^4 + 127*x^3 + 575*x^2 - 527*x + 551);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 3*x^15 + 11*x^14 - 17*x^13 + 40*x^12 - 37*x^11 + 66*x^10 + 25*x^9 - 30*x^8 + 254*x^7 - 180*x^6 + 443*x^5 - 120*x^4 + 127*x^3 + 575*x^2 - 527*x + 551)
 

\( x^{16} - 3 x^{15} + 11 x^{14} - 17 x^{13} + 40 x^{12} - 37 x^{11} + 66 x^{10} + 25 x^{9} - 30 x^{8} + \cdots + 551 \) 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:  $16$
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:  $[0, 8]$
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:   \(72011464084359765625\) \(\medspace = 5^{8}\cdot 11^{6}\cdot 101^{4}\) 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:  \(17.42\)
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:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $5^{1/2}11^{3/4}101^{1/2}\approx 135.73448607639935$
Ramified primes:   \(5\), \(11\), \(101\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{10\cdots 83}a^{15}+\frac{26\cdots 84}{10\cdots 83}a^{14}-\frac{17\cdots 91}{36\cdots 61}a^{13}-\frac{36\cdots 32}{10\cdots 83}a^{12}-\frac{286958796087661}{10\cdots 83}a^{11}-\frac{253359399792233}{10\cdots 83}a^{10}+\frac{19\cdots 75}{10\cdots 83}a^{9}-\frac{50\cdots 56}{10\cdots 83}a^{8}-\frac{25\cdots 38}{10\cdots 83}a^{7}+\frac{18\cdots 93}{10\cdots 83}a^{6}-\frac{24\cdots 94}{10\cdots 83}a^{5}+\frac{480208826706678}{36\cdots 61}a^{4}+\frac{13\cdots 62}{36\cdots 61}a^{3}-\frac{37\cdots 27}{10\cdots 83}a^{2}-\frac{318177385578827}{36\cdots 61}a+\frac{48\cdots 05}{10\cdots 83}$ 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 $1$
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 $1$
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:  $7$
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:   $\frac{47979255297860}{36\cdots 61}a^{15}-\frac{120958518260070}{36\cdots 61}a^{14}+\frac{399804389921063}{36\cdots 61}a^{13}-\frac{502051535357252}{36\cdots 61}a^{12}+\frac{12\cdots 57}{36\cdots 61}a^{11}-\frac{831880711837355}{36\cdots 61}a^{10}+\frac{12\cdots 40}{36\cdots 61}a^{9}+\frac{21\cdots 20}{36\cdots 61}a^{8}-\frac{28\cdots 10}{36\cdots 61}a^{7}+\frac{74\cdots 89}{36\cdots 61}a^{6}-\frac{63\cdots 16}{36\cdots 61}a^{5}+\frac{87\cdots 63}{36\cdots 61}a^{4}-\frac{23\cdots 46}{36\cdots 61}a^{3}-\frac{10\cdots 07}{36\cdots 61}a^{2}+\frac{16\cdots 74}{36\cdots 61}a-\frac{17\cdots 25}{36\cdots 61}$, $\frac{30586511643670}{10\cdots 83}a^{15}-\frac{78054623996522}{10\cdots 83}a^{14}+\frac{90724487133625}{36\cdots 61}a^{13}-\frac{299611260405146}{10\cdots 83}a^{12}+\frac{815202652411454}{10\cdots 83}a^{11}-\frac{284677564458977}{10\cdots 83}a^{10}+\frac{11\cdots 06}{10\cdots 83}a^{9}+\frac{22\cdots 00}{10\cdots 83}a^{8}-\frac{786676490855035}{10\cdots 83}a^{7}+\frac{65\cdots 22}{10\cdots 83}a^{6}-\frac{162404509232258}{10\cdots 83}a^{5}+\frac{29\cdots 33}{36\cdots 61}a^{4}+\frac{13\cdots 37}{36\cdots 61}a^{3}-\frac{49\cdots 09}{10\cdots 83}a^{2}+\frac{74\cdots 72}{36\cdots 61}a-\frac{10\cdots 18}{10\cdots 83}$, $\frac{107277548059100}{10\cdots 83}a^{15}-\frac{240135622884094}{10\cdots 83}a^{14}+\frac{288937832673033}{36\cdots 61}a^{13}-\frac{10\cdots 35}{10\cdots 83}a^{12}+\frac{27\cdots 90}{10\cdots 83}a^{11}-\frac{19\cdots 33}{10\cdots 83}a^{10}+\frac{35\cdots 89}{10\cdots 83}a^{9}+\frac{30\cdots 15}{10\cdots 83}a^{8}-\frac{36\cdots 12}{10\cdots 83}a^{7}+\frac{12\cdots 79}{10\cdots 83}a^{6}-\frac{12\cdots 09}{10\cdots 83}a^{5}+\frac{55\cdots 70}{36\cdots 61}a^{4}-\frac{37\cdots 81}{36\cdots 61}a^{3}-\frac{18\cdots 05}{10\cdots 83}a^{2}+\frac{73\cdots 20}{36\cdots 61}a-\frac{27\cdots 86}{10\cdots 83}$, $\frac{31818207762202}{10\cdots 83}a^{15}+\frac{91748843426512}{10\cdots 83}a^{14}-\frac{47468176795687}{36\cdots 61}a^{13}+\frac{926366920577245}{10\cdots 83}a^{12}-\frac{780298202451331}{10\cdots 83}a^{11}+\frac{34\cdots 61}{10\cdots 83}a^{10}-\frac{28\cdots 99}{10\cdots 83}a^{9}+\frac{73\cdots 45}{10\cdots 83}a^{8}-\frac{180370042484188}{10\cdots 83}a^{7}+\frac{46\cdots 05}{10\cdots 83}a^{6}+\frac{76\cdots 65}{10\cdots 83}a^{5}-\frac{17\cdots 91}{36\cdots 61}a^{4}+\frac{83\cdots 31}{36\cdots 61}a^{3}-\frac{17\cdots 00}{10\cdots 83}a^{2}+\frac{37\cdots 16}{36\cdots 61}a+\frac{29\cdots 04}{10\cdots 83}$, $\frac{16102848837530}{36\cdots 61}a^{15}-\frac{89957202445965}{36\cdots 61}a^{14}+\frac{240341399606892}{36\cdots 61}a^{13}-\frac{576550597394398}{36\cdots 61}a^{12}+\frac{721155447124820}{36\cdots 61}a^{11}-\frac{14\cdots 95}{36\cdots 61}a^{10}+\frac{597887391079088}{36\cdots 61}a^{9}-\frac{11\cdots 13}{36\cdots 61}a^{8}-\frac{47\cdots 52}{36\cdots 61}a^{7}+\frac{26\cdots 14}{36\cdots 61}a^{6}-\frac{11\cdots 98}{36\cdots 61}a^{5}+\frac{16\cdots 88}{36\cdots 61}a^{4}-\frac{16\cdots 09}{36\cdots 61}a^{3}-\frac{11\cdots 60}{36\cdots 61}a^{2}+\frac{89\cdots 90}{36\cdots 61}a-\frac{32\cdots 54}{36\cdots 61}$, $\frac{181390130974232}{10\cdots 83}a^{15}-\frac{224857934845513}{10\cdots 83}a^{14}+\frac{314521262730990}{36\cdots 61}a^{13}-\frac{232744493506930}{10\cdots 83}a^{12}+\frac{27\cdots 06}{10\cdots 83}a^{11}+\frac{16\cdots 32}{10\cdots 83}a^{10}+\frac{27\cdots 71}{10\cdots 83}a^{9}+\frac{12\cdots 30}{10\cdots 83}a^{8}+\frac{42\cdots 06}{10\cdots 83}a^{7}+\frac{21\cdots 42}{10\cdots 83}a^{6}+\frac{13\cdots 03}{10\cdots 83}a^{5}+\frac{92\cdots 52}{36\cdots 61}a^{4}+\frac{13\cdots 03}{36\cdots 61}a^{3}-\frac{170027350845619}{10\cdots 83}a^{2}+\frac{15\cdots 73}{36\cdots 61}a+\frac{40\cdots 99}{10\cdots 83}$, $\frac{2436107751500}{10\cdots 83}a^{15}+\frac{70029903883457}{10\cdots 83}a^{14}-\frac{36450997334773}{36\cdots 61}a^{13}+\frac{595315781090948}{10\cdots 83}a^{12}-\frac{412167345721286}{10\cdots 83}a^{11}+\frac{16\cdots 76}{10\cdots 83}a^{10}+\frac{399107311428644}{10\cdots 83}a^{9}+\frac{19\cdots 28}{10\cdots 83}a^{8}+\frac{51\cdots 54}{10\cdots 83}a^{7}+\frac{341789539344056}{10\cdots 83}a^{6}+\frac{12\cdots 50}{10\cdots 83}a^{5}+\frac{220160614762115}{36\cdots 61}a^{4}+\frac{49\cdots 61}{36\cdots 61}a^{3}+\frac{74\cdots 99}{10\cdots 83}a^{2}-\frac{17\cdots 46}{36\cdots 61}a+\frac{21\cdots 02}{10\cdots 83}$ Copy content Toggle raw display
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:  \( 1992.93411513 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1992.93411513 \cdot 1}{2\cdot\sqrt{72011464084359765625}}\cr\approx \mathstrut & 0.285233855498 \end{aligned}\]

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^16 - 3*x^15 + 11*x^14 - 17*x^13 + 40*x^12 - 37*x^11 + 66*x^10 + 25*x^9 - 30*x^8 + 254*x^7 - 180*x^6 + 443*x^5 - 120*x^4 + 127*x^3 + 575*x^2 - 527*x + 551) 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^16 - 3*x^15 + 11*x^14 - 17*x^13 + 40*x^12 - 37*x^11 + 66*x^10 + 25*x^9 - 30*x^8 + 254*x^7 - 180*x^6 + 443*x^5 - 120*x^4 + 127*x^3 + 575*x^2 - 527*x + 551, 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^16 - 3*x^15 + 11*x^14 - 17*x^13 + 40*x^12 - 37*x^11 + 66*x^10 + 25*x^9 - 30*x^8 + 254*x^7 - 180*x^6 + 443*x^5 - 120*x^4 + 127*x^3 + 575*x^2 - 527*x + 551); 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 = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 + 11*x^14 - 17*x^13 + 40*x^12 - 37*x^11 + 66*x^10 + 25*x^9 - 30*x^8 + 254*x^7 - 180*x^6 + 443*x^5 - 120*x^4 + 127*x^3 + 575*x^2 - 527*x + 551); 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

$(C_2^2\times C_4^2):D_8$ (as 16T1276):

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); G, transitive_group_identification(G)
 
A solvable group of order 1024
The 43 conjugacy class representatives for $(C_2^2\times C_4^2):D_8$
Character table for $(C_2^2\times C_4^2):D_8$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 8.0.7638125.1

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(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.4.7273157872520336328125.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ R ${\href{/padicField/7.8.0.1}{8} }^{2}$ R ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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
\(5\) Copy content Toggle raw display 5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(11\) Copy content Toggle raw display 11.4.1.0a1.1$x^{4} + 8 x^{2} + 10 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
11.1.4.3a1.2$x^{4} + 22$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
11.4.1.0a1.1$x^{4} + 8 x^{2} + 10 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
11.1.4.3a1.2$x^{4} + 22$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
\(101\) Copy content Toggle raw display 101.2.1.0a1.1$x^{2} + 97 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
101.1.2.1a1.2$x^{2} + 202$$2$$1$$1$$C_2$$$[\ ]_{2}$$
101.2.1.0a1.1$x^{2} + 97 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
101.2.1.0a1.1$x^{2} + 97 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
101.1.2.1a1.2$x^{2} + 202$$2$$1$$1$$C_2$$$[\ ]_{2}$$
101.2.1.0a1.1$x^{2} + 97 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
101.2.2.2a1.2$x^{4} + 194 x^{3} + 9413 x^{2} + 388 x + 105$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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