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Properties

Label 16.0.229...625.5
Degree $16$
Signature $[0, 8]$
Discriminant $2.294\times 10^{21}$
Root discriminant \(21.63\)
Ramified primes $3,5,29$
Class number $4$
Class group [2, 2]
Galois group $C_2^3:\OD_{16}$ (as 16T252)

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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 + 7*x^14 - 19*x^13 + 46*x^12 - 78*x^11 + 55*x^10 - 132*x^9 + 199*x^8 + 79*x^7 + 40*x^6 + 59*x^5 + 26*x^4 + 728*x^3 + 1273*x^2 + 659*x + 541)
 
Copy content gp:K = bnfinit(y^16 - 3*y^15 + 7*y^14 - 19*y^13 + 46*y^12 - 78*y^11 + 55*y^10 - 132*y^9 + 199*y^8 + 79*y^7 + 40*y^6 + 59*y^5 + 26*y^4 + 728*y^3 + 1273*y^2 + 659*y + 541, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 7*x^14 - 19*x^13 + 46*x^12 - 78*x^11 + 55*x^10 - 132*x^9 + 199*x^8 + 79*x^7 + 40*x^6 + 59*x^5 + 26*x^4 + 728*x^3 + 1273*x^2 + 659*x + 541);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 3*x^15 + 7*x^14 - 19*x^13 + 46*x^12 - 78*x^11 + 55*x^10 - 132*x^9 + 199*x^8 + 79*x^7 + 40*x^6 + 59*x^5 + 26*x^4 + 728*x^3 + 1273*x^2 + 659*x + 541)
 

\( x^{16} - 3 x^{15} + 7 x^{14} - 19 x^{13} + 46 x^{12} - 78 x^{11} + 55 x^{10} - 132 x^{9} + 199 x^{8} + \cdots + 541 \) 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:   \(2294177990240478515625\) \(\medspace = 3^{12}\cdot 5^{14}\cdot 29^{4}\) 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:  \(21.63\)
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:  $3^{3/4}5^{7/8}29^{1/2}\approx 50.19248452708817$
Ramified primes:   \(3\), \(5\), \(29\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2^2$
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.
Maximal CM subfield:  8.0.47897578125.1

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5}a^{11}+\frac{2}{5}a^{10}+\frac{2}{5}a^{9}+\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{2}{5}a^{4}+\frac{1}{5}a^{3}+\frac{1}{5}a^{2}-\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{12}-\frac{2}{5}a^{10}+\frac{2}{5}a^{9}-\frac{1}{5}a^{8}+\frac{2}{5}a^{7}+\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{2}{5}a^{4}-\frac{1}{5}a^{3}+\frac{1}{5}a^{2}+\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{13}+\frac{1}{5}a^{10}-\frac{2}{5}a^{9}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}+\frac{2}{5}a^{6}-\frac{2}{5}a^{4}-\frac{2}{5}a^{3}-\frac{2}{5}a^{2}+\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{25}a^{14}+\frac{1}{25}a^{13}+\frac{2}{25}a^{12}-\frac{12}{25}a^{10}+\frac{9}{25}a^{9}-\frac{6}{25}a^{8}-\frac{12}{25}a^{7}+\frac{2}{5}a^{6}-\frac{8}{25}a^{5}+\frac{8}{25}a^{4}-\frac{7}{25}a^{3}+\frac{1}{25}a^{2}+\frac{1}{5}a-\frac{11}{25}$, $\frac{1}{97\cdots 75}a^{15}+\frac{90\cdots 96}{97\cdots 75}a^{14}-\frac{66\cdots 98}{97\cdots 75}a^{13}+\frac{18\cdots 07}{19\cdots 55}a^{12}-\frac{87\cdots 32}{97\cdots 75}a^{11}+\frac{20\cdots 94}{97\cdots 75}a^{10}-\frac{18\cdots 61}{97\cdots 75}a^{9}+\frac{27\cdots 98}{97\cdots 75}a^{8}-\frac{11\cdots 69}{19\cdots 55}a^{7}+\frac{39\cdots 67}{97\cdots 75}a^{6}+\frac{19\cdots 73}{97\cdots 75}a^{5}-\frac{17\cdots 32}{97\cdots 75}a^{4}-\frac{16\cdots 89}{97\cdots 75}a^{3}-\frac{91\cdots 47}{19\cdots 55}a^{2}+\frac{10\cdots 64}{31\cdots 25}a-\frac{54\cdots 39}{19\cdots 55}$ 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:  $C_{2}\times C_{2}$, which has order $4$
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:  $C_{2}\times C_{2}$, which has order $4$
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:   $\frac{196956895747}{142623634607275}a^{15}-\frac{851232556477}{142623634607275}a^{14}+\frac{418423802757}{28524726921455}a^{13}-\frac{5186504291033}{142623634607275}a^{12}+\frac{13208446275606}{142623634607275}a^{11}-\frac{25677709131639}{142623634607275}a^{10}+\frac{26946821453642}{142623634607275}a^{9}-\frac{6378246767381}{28524726921455}a^{8}+\frac{67588934868528}{142623634607275}a^{7}-\frac{39404385511816}{142623634607275}a^{6}-\frac{31792439440367}{142623634607275}a^{5}+\frac{24593112320769}{142623634607275}a^{4}-\frac{1009140356722}{28524726921455}a^{3}+\frac{134495773280741}{142623634607275}a^{2}+\frac{31116610182768}{142623634607275}a-\frac{35546078503196}{142623634607275}$, $\frac{21\cdots 38}{97\cdots 75}a^{15}-\frac{62\cdots 62}{97\cdots 75}a^{14}+\frac{14\cdots 51}{97\cdots 75}a^{13}-\frac{15\cdots 68}{39\cdots 11}a^{12}+\frac{94\cdots 84}{97\cdots 75}a^{11}-\frac{16\cdots 03}{97\cdots 75}a^{10}+\frac{10\cdots 52}{97\cdots 75}a^{9}-\frac{27\cdots 66}{97\cdots 75}a^{8}+\frac{87\cdots 29}{19\cdots 55}a^{7}+\frac{13\cdots 81}{97\cdots 75}a^{6}+\frac{78\cdots 39}{97\cdots 75}a^{5}+\frac{29\cdots 64}{97\cdots 75}a^{4}+\frac{11\cdots 28}{97\cdots 75}a^{3}+\frac{30\cdots 01}{19\cdots 55}a^{2}+\frac{59\cdots 42}{31\cdots 25}a+\frac{18\cdots 26}{19\cdots 55}$, $\frac{25\cdots 72}{97\cdots 75}a^{15}-\frac{17\cdots 42}{97\cdots 75}a^{14}-\frac{30\cdots 48}{19\cdots 55}a^{13}+\frac{48\cdots 02}{97\cdots 75}a^{12}-\frac{13\cdots 19}{97\cdots 75}a^{11}+\frac{43\cdots 76}{97\cdots 75}a^{10}-\frac{11\cdots 93}{97\cdots 75}a^{9}+\frac{30\cdots 11}{19\cdots 55}a^{8}-\frac{16\cdots 42}{97\cdots 75}a^{7}+\frac{33\cdots 54}{97\cdots 75}a^{6}-\frac{22\cdots 87}{97\cdots 75}a^{5}-\frac{15\cdots 26}{97\cdots 75}a^{4}+\frac{12\cdots 98}{39\cdots 11}a^{3}-\frac{17\cdots 39}{97\cdots 75}a^{2}+\frac{43\cdots 23}{31\cdots 25}a-\frac{48\cdots 56}{97\cdots 75}$, $\frac{47\cdots 02}{19\cdots 55}a^{15}-\frac{64\cdots 04}{97\cdots 75}a^{14}+\frac{12\cdots 11}{97\cdots 75}a^{13}-\frac{34\cdots 98}{97\cdots 75}a^{12}+\frac{16\cdots 33}{19\cdots 55}a^{11}-\frac{11\cdots 27}{97\cdots 75}a^{10}-\frac{90\cdots 41}{97\cdots 75}a^{9}-\frac{12\cdots 11}{97\cdots 75}a^{8}+\frac{27\cdots 03}{97\cdots 75}a^{7}+\frac{95\cdots 67}{19\cdots 55}a^{6}-\frac{14\cdots 48}{97\cdots 75}a^{5}+\frac{14\cdots 38}{97\cdots 75}a^{4}+\frac{42\cdots 78}{97\cdots 75}a^{3}+\frac{13\cdots 66}{97\cdots 75}a^{2}+\frac{20\cdots 33}{63\cdots 05}a+\frac{14\cdots 49}{97\cdots 75}$, $\frac{71\cdots 53}{97\cdots 75}a^{15}-\frac{19\cdots 82}{97\cdots 75}a^{14}+\frac{31\cdots 91}{97\cdots 75}a^{13}-\frac{15\cdots 59}{19\cdots 55}a^{12}+\frac{20\cdots 24}{97\cdots 75}a^{11}-\frac{26\cdots 38}{97\cdots 75}a^{10}-\frac{25\cdots 78}{97\cdots 75}a^{9}-\frac{30\cdots 36}{97\cdots 75}a^{8}+\frac{20\cdots 49}{19\cdots 55}a^{7}+\frac{10\cdots 81}{97\cdots 75}a^{6}-\frac{16\cdots 06}{97\cdots 75}a^{5}-\frac{32\cdots 46}{97\cdots 75}a^{4}+\frac{10\cdots 68}{97\cdots 75}a^{3}+\frac{15\cdots 41}{39\cdots 11}a^{2}+\frac{34\cdots 27}{31\cdots 25}a+\frac{90\cdots 92}{19\cdots 55}$, $\frac{28\cdots 76}{97\cdots 75}a^{15}-\frac{12\cdots 83}{97\cdots 75}a^{14}+\frac{30\cdots 03}{97\cdots 75}a^{13}-\frac{77\cdots 53}{97\cdots 75}a^{12}+\frac{19\cdots 58}{97\cdots 75}a^{11}-\frac{37\cdots 88}{97\cdots 75}a^{10}+\frac{41\cdots 88}{97\cdots 75}a^{9}-\frac{52\cdots 88}{97\cdots 75}a^{8}+\frac{10\cdots 98}{97\cdots 75}a^{7}-\frac{51\cdots 08}{97\cdots 75}a^{6}-\frac{46\cdots 76}{19\cdots 55}a^{5}+\frac{62\cdots 21}{97\cdots 75}a^{4}+\frac{35\cdots 24}{97\cdots 75}a^{3}+\frac{16\cdots 11}{97\cdots 75}a^{2}+\frac{19\cdots 19}{31\cdots 25}a-\frac{35\cdots 56}{97\cdots 75}$, $\frac{62\cdots 62}{97\cdots 75}a^{15}-\frac{11\cdots 99}{97\cdots 75}a^{14}+\frac{29\cdots 18}{97\cdots 75}a^{13}-\frac{65\cdots 12}{97\cdots 75}a^{12}+\frac{15\cdots 61}{97\cdots 75}a^{11}-\frac{67\cdots 23}{19\cdots 55}a^{10}+\frac{42\cdots 49}{97\cdots 75}a^{9}+\frac{15\cdots 12}{97\cdots 75}a^{8}-\frac{13\cdots 58}{97\cdots 75}a^{7}+\frac{20\cdots 09}{97\cdots 75}a^{6}-\frac{21\cdots 81}{97\cdots 75}a^{5}+\frac{31\cdots 98}{97\cdots 75}a^{4}-\frac{36\cdots 51}{97\cdots 75}a^{3}+\frac{14\cdots 79}{97\cdots 75}a^{2}-\frac{67\cdots 17}{31\cdots 25}a-\frac{52\cdots 14}{97\cdots 75}$ 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:  \( 2578.17931228 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2578.17931228 \cdot 4}{2\cdot\sqrt{2294177990240478515625}}\cr\approx \mathstrut & 0.261498081689 \end{aligned}\]

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 + 7*x^14 - 19*x^13 + 46*x^12 - 78*x^11 + 55*x^10 - 132*x^9 + 199*x^8 + 79*x^7 + 40*x^6 + 59*x^5 + 26*x^4 + 728*x^3 + 1273*x^2 + 659*x + 541) 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 + 7*x^14 - 19*x^13 + 46*x^12 - 78*x^11 + 55*x^10 - 132*x^9 + 199*x^8 + 79*x^7 + 40*x^6 + 59*x^5 + 26*x^4 + 728*x^3 + 1273*x^2 + 659*x + 541, 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 + 7*x^14 - 19*x^13 + 46*x^12 - 78*x^11 + 55*x^10 - 132*x^9 + 199*x^8 + 79*x^7 + 40*x^6 + 59*x^5 + 26*x^4 + 728*x^3 + 1273*x^2 + 659*x + 541); 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 + 7*x^14 - 19*x^13 + 46*x^12 - 78*x^11 + 55*x^10 - 132*x^9 + 199*x^8 + 79*x^7 + 40*x^6 + 59*x^5 + 26*x^4 + 728*x^3 + 1273*x^2 + 659*x + 541); 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

$C_2^3:\OD_{16}$ (as 16T252):

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 128
The 32 conjugacy class representatives for $C_2^3:\OD_{16}$
Character table for $C_2^3:\OD_{16}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 8.0.47897578125.1, 8.4.56953125.1, 8.4.1064390625.2

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(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: 16.8.2294177990240478515625.4

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ R R ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ R ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$

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 3.4.4.12a1.1$x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 24 x^{12} + 48 x^{11} + 96 x^{10} + 64 x^{9} + 24 x^{8} + 96 x^{7} + 96 x^{6} + 32 x^{4} + 64 x^{3} + 3 x^{2} + 16$$4$$4$$12$$C_8: C_2$$$[\ ]_{4}^{4}$$
\(5\) Copy content Toggle raw display 5.1.8.7a1.4$x^{8} + 20$$8$$1$$7$$C_8:C_2$$$[\ ]_{8}^{2}$$
5.1.8.7a1.4$x^{8} + 20$$8$$1$$7$$C_8:C_2$$$[\ ]_{8}^{2}$$
\(29\) Copy content Toggle raw display 29.1.2.1a1.2$x^{2} + 58$$2$$1$$1$$C_2$$$[\ ]_{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.1.2.1a1.2$x^{2} + 58$$2$$1$$1$$C_2$$$[\ ]_{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.2.2a1.2$x^{4} + 48 x^{3} + 580 x^{2} + 96 x + 33$$2$$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)