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Properties

Label 16.4.527...609.12
Degree $16$
Signature $[4, 6]$
Discriminant $5.270\times 10^{25}$
Root discriminant \(40.51\)
Ramified primes $17,67$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_4^2.(C_2\times C_4)$ (as 16T362)

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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^16 - 3*x^15 + 10*x^14 - 36*x^13 - 45*x^12 + 334*x^11 - 90*x^10 - 918*x^9 + 1057*x^8 + 305*x^7 - 3257*x^6 + 7824*x^5 - 9426*x^4 + 4926*x^3 + 102*x^2 - 1076*x + 293)
 
Copy content gp:K = bnfinit(y^16 - 3*y^15 + 10*y^14 - 36*y^13 - 45*y^12 + 334*y^11 - 90*y^10 - 918*y^9 + 1057*y^8 + 305*y^7 - 3257*y^6 + 7824*y^5 - 9426*y^4 + 4926*y^3 + 102*y^2 - 1076*y + 293, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 10*x^14 - 36*x^13 - 45*x^12 + 334*x^11 - 90*x^10 - 918*x^9 + 1057*x^8 + 305*x^7 - 3257*x^6 + 7824*x^5 - 9426*x^4 + 4926*x^3 + 102*x^2 - 1076*x + 293);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 3*x^15 + 10*x^14 - 36*x^13 - 45*x^12 + 334*x^11 - 90*x^10 - 918*x^9 + 1057*x^8 + 305*x^7 - 3257*x^6 + 7824*x^5 - 9426*x^4 + 4926*x^3 + 102*x^2 - 1076*x + 293)
 

\( x^{16} - 3 x^{15} + 10 x^{14} - 36 x^{13} - 45 x^{12} + 334 x^{11} - 90 x^{10} - 918 x^{9} + 1057 x^{8} + \cdots + 293 \) 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:  $[4, 6]$
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:   \(52703064995487500398531609\) \(\medspace = 17^{12}\cdot 67^{6}\) 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:  \(40.51\)
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:  $17^{3/4}67^{1/2}\approx 68.52895233095656$
Ramified primes:   \(17\), \(67\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11\cdots 11}a^{15}+\frac{22\cdots 11}{11\cdots 11}a^{14}-\frac{26\cdots 00}{11\cdots 11}a^{13}+\frac{40\cdots 22}{91\cdots 47}a^{12}-\frac{57\cdots 28}{11\cdots 11}a^{11}-\frac{42\cdots 04}{91\cdots 47}a^{10}-\frac{50\cdots 76}{11\cdots 11}a^{9}-\frac{40\cdots 42}{91\cdots 47}a^{8}-\frac{56\cdots 79}{11\cdots 11}a^{7}-\frac{56\cdots 54}{11\cdots 11}a^{6}+\frac{15\cdots 61}{11\cdots 11}a^{5}-\frac{27\cdots 90}{11\cdots 11}a^{4}-\frac{32\cdots 57}{11\cdots 11}a^{3}+\frac{51\cdots 46}{11\cdots 11}a^{2}+\frac{26\cdots 67}{11\cdots 11}a-\frac{57\cdots 44}{11\cdots 11}$ 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:  $C_{2}$, which has order $2$ (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:  $9$
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:   $\frac{16\cdots 50}{91\cdots 47}a^{15}-\frac{31\cdots 56}{91\cdots 47}a^{14}+\frac{12\cdots 97}{91\cdots 47}a^{13}-\frac{44\cdots 38}{91\cdots 47}a^{12}-\frac{11\cdots 12}{91\cdots 47}a^{11}+\frac{41\cdots 57}{91\cdots 47}a^{10}+\frac{28\cdots 59}{91\cdots 47}a^{9}-\frac{11\cdots 39}{91\cdots 47}a^{8}+\frac{47\cdots 26}{91\cdots 47}a^{7}+\frac{98\cdots 72}{91\cdots 47}a^{6}-\frac{42\cdots 08}{91\cdots 47}a^{5}+\frac{82\cdots 22}{91\cdots 47}a^{4}-\frac{66\cdots 83}{91\cdots 47}a^{3}+\frac{10\cdots 13}{91\cdots 47}a^{2}+\frac{12\cdots 56}{91\cdots 47}a-\frac{44\cdots 03}{91\cdots 47}$, $\frac{77\cdots 19}{91\cdots 47}a^{15}-\frac{13\cdots 08}{91\cdots 47}a^{14}+\frac{60\cdots 08}{91\cdots 47}a^{13}-\frac{20\cdots 61}{91\cdots 47}a^{12}-\frac{59\cdots 93}{91\cdots 47}a^{11}+\frac{18\cdots 11}{91\cdots 47}a^{10}+\frac{15\cdots 44}{91\cdots 47}a^{9}-\frac{52\cdots 63}{91\cdots 47}a^{8}+\frac{17\cdots 51}{91\cdots 47}a^{7}+\frac{45\cdots 68}{91\cdots 47}a^{6}-\frac{19\cdots 18}{91\cdots 47}a^{5}+\frac{36\cdots 61}{91\cdots 47}a^{4}-\frac{28\cdots 22}{91\cdots 47}a^{3}+\frac{37\cdots 43}{91\cdots 47}a^{2}+\frac{53\cdots 79}{91\cdots 47}a-\frac{18\cdots 79}{91\cdots 47}$, $\frac{84\cdots 31}{91\cdots 47}a^{15}-\frac{17\cdots 48}{91\cdots 47}a^{14}+\frac{68\cdots 89}{91\cdots 47}a^{13}-\frac{24\cdots 77}{91\cdots 47}a^{12}-\frac{59\cdots 19}{91\cdots 47}a^{11}+\frac{22\cdots 46}{91\cdots 47}a^{10}+\frac{12\cdots 15}{91\cdots 47}a^{9}-\frac{65\cdots 76}{91\cdots 47}a^{8}+\frac{29\cdots 75}{91\cdots 47}a^{7}+\frac{53\cdots 04}{91\cdots 47}a^{6}-\frac{22\cdots 90}{91\cdots 47}a^{5}+\frac{45\cdots 61}{91\cdots 47}a^{4}-\frac{38\cdots 61}{91\cdots 47}a^{3}+\frac{66\cdots 70}{91\cdots 47}a^{2}+\frac{67\cdots 77}{91\cdots 47}a-\frac{25\cdots 24}{91\cdots 47}$, $\frac{61\cdots 82}{91\cdots 47}a^{15}-\frac{13\cdots 84}{91\cdots 47}a^{14}+\frac{51\cdots 83}{91\cdots 47}a^{13}-\frac{18\cdots 88}{91\cdots 47}a^{12}-\frac{41\cdots 46}{91\cdots 47}a^{11}+\frac{17\cdots 65}{91\cdots 47}a^{10}+\frac{76\cdots 96}{91\cdots 47}a^{9}-\frac{50\cdots 01}{91\cdots 47}a^{8}+\frac{26\cdots 50}{91\cdots 47}a^{7}+\frac{38\cdots 80}{91\cdots 47}a^{6}-\frac{17\cdots 49}{91\cdots 47}a^{5}+\frac{35\cdots 55}{91\cdots 47}a^{4}-\frac{31\cdots 60}{91\cdots 47}a^{3}+\frac{67\cdots 41}{91\cdots 47}a^{2}+\frac{56\cdots 11}{91\cdots 47}a-\frac{24\cdots 55}{91\cdots 47}$, $a-1$, $\frac{66\cdots 45}{91\cdots 47}a^{15}-\frac{10\cdots 02}{91\cdots 47}a^{14}+\frac{48\cdots 17}{91\cdots 47}a^{13}-\frac{16\cdots 49}{91\cdots 47}a^{12}-\frac{56\cdots 10}{91\cdots 47}a^{11}+\frac{15\cdots 92}{91\cdots 47}a^{10}+\frac{18\cdots 49}{91\cdots 47}a^{9}-\frac{42\cdots 27}{91\cdots 47}a^{8}+\frac{15\cdots 42}{91\cdots 47}a^{7}+\frac{40\cdots 20}{91\cdots 47}a^{6}-\frac{15\cdots 57}{91\cdots 47}a^{5}+\frac{27\cdots 11}{91\cdots 47}a^{4}-\frac{15\cdots 90}{91\cdots 47}a^{3}-\frac{26\cdots 36}{91\cdots 47}a^{2}+\frac{35\cdots 31}{91\cdots 47}a-\frac{20\cdots 04}{91\cdots 47}$, $\frac{17\cdots 07}{11\cdots 11}a^{15}-\frac{37\cdots 41}{11\cdots 11}a^{14}+\frac{13\cdots 25}{11\cdots 11}a^{13}-\frac{38\cdots 17}{91\cdots 47}a^{12}-\frac{11\cdots 77}{11\cdots 11}a^{11}+\frac{36\cdots 80}{91\cdots 47}a^{10}+\frac{24\cdots 00}{11\cdots 11}a^{9}-\frac{10\cdots 07}{91\cdots 47}a^{8}+\frac{63\cdots 33}{11\cdots 11}a^{7}+\frac{10\cdots 46}{11\cdots 11}a^{6}-\frac{46\cdots 53}{11\cdots 11}a^{5}+\frac{94\cdots 51}{11\cdots 11}a^{4}-\frac{80\cdots 20}{11\cdots 11}a^{3}+\frac{14\cdots 39}{11\cdots 11}a^{2}+\frac{15\cdots 42}{11\cdots 11}a-\frac{66\cdots 43}{11\cdots 11}$, $\frac{24\cdots 33}{11\cdots 11}a^{15}-\frac{35\cdots 88}{11\cdots 11}a^{14}+\frac{58\cdots 46}{11\cdots 11}a^{13}-\frac{21\cdots 21}{91\cdots 47}a^{12}+\frac{51\cdots 24}{11\cdots 11}a^{11}+\frac{25\cdots 30}{91\cdots 47}a^{10}-\frac{55\cdots 11}{11\cdots 11}a^{9}-\frac{85\cdots 36}{91\cdots 47}a^{8}+\frac{16\cdots 41}{11\cdots 11}a^{7}+\frac{31\cdots 99}{11\cdots 11}a^{6}-\frac{23\cdots 98}{11\cdots 11}a^{5}+\frac{80\cdots 05}{11\cdots 11}a^{4}-\frac{12\cdots 97}{11\cdots 11}a^{3}+\frac{54\cdots 45}{11\cdots 11}a^{2}+\frac{17\cdots 68}{11\cdots 11}a-\frac{13\cdots 51}{11\cdots 11}$, $\frac{14\cdots 08}{11\cdots 11}a^{15}-\frac{49\cdots 94}{11\cdots 11}a^{14}+\frac{20\cdots 89}{11\cdots 11}a^{13}-\frac{51\cdots 57}{91\cdots 47}a^{12}-\frac{13\cdots 17}{11\cdots 11}a^{11}+\frac{30\cdots 73}{91\cdots 47}a^{10}-\frac{61\cdots 11}{11\cdots 11}a^{9}-\frac{20\cdots 76}{91\cdots 47}a^{8}+\frac{27\cdots 06}{11\cdots 11}a^{7}-\frac{29\cdots 10}{11\cdots 11}a^{6}-\frac{36\cdots 08}{11\cdots 11}a^{5}+\frac{15\cdots 92}{11\cdots 11}a^{4}-\frac{28\cdots 47}{11\cdots 11}a^{3}+\frac{33\cdots 15}{11\cdots 11}a^{2}-\frac{21\cdots 15}{11\cdots 11}a+\frac{53\cdots 63}{11\cdots 11}$ 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:  \( 3083861.57137 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 3083861.57137 \cdot 1}{2\cdot\sqrt{52703064995487500398531609}}\cr\approx \mathstrut & 0.209096197920 \end{aligned}\] (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^16 - 3*x^15 + 10*x^14 - 36*x^13 - 45*x^12 + 334*x^11 - 90*x^10 - 918*x^9 + 1057*x^8 + 305*x^7 - 3257*x^6 + 7824*x^5 - 9426*x^4 + 4926*x^3 + 102*x^2 - 1076*x + 293) 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 + 10*x^14 - 36*x^13 - 45*x^12 + 334*x^11 - 90*x^10 - 918*x^9 + 1057*x^8 + 305*x^7 - 3257*x^6 + 7824*x^5 - 9426*x^4 + 4926*x^3 + 102*x^2 - 1076*x + 293, 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 + 10*x^14 - 36*x^13 - 45*x^12 + 334*x^11 - 90*x^10 - 918*x^9 + 1057*x^8 + 305*x^7 - 3257*x^6 + 7824*x^5 - 9426*x^4 + 4926*x^3 + 102*x^2 - 1076*x + 293); 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 + 10*x^14 - 36*x^13 - 45*x^12 + 334*x^11 - 90*x^10 - 918*x^9 + 1057*x^8 + 305*x^7 - 3257*x^6 + 7824*x^5 - 9426*x^4 + 4926*x^3 + 102*x^2 - 1076*x + 293); 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

$C_4^2.(C_2\times C_4)$ (as 16T362):

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 14 conjugacy class representatives for $C_4^2.(C_2\times C_4)$
Character table for $C_4^2.(C_2\times C_4)$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.4.108353547241.2

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 32 siblings: data not computed
Arithmetically equivalent siblings: data not computed
Minimal sibling: 16.4.52703064995487500398531609.6

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ R ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(17\) Copy content Toggle raw display 17.2.4.6a1.2$x^{8} + 64 x^{7} + 1548 x^{6} + 16960 x^{5} + 74806 x^{4} + 50880 x^{3} + 13932 x^{2} + 1728 x + 98$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
17.2.4.6a1.2$x^{8} + 64 x^{7} + 1548 x^{6} + 16960 x^{5} + 74806 x^{4} + 50880 x^{3} + 13932 x^{2} + 1728 x + 98$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
\(67\) Copy content Toggle raw display $\Q_{67}$$x + 65$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{67}$$x + 65$$1$$1$$0$Trivial$$[\ ]$$
67.1.2.1a1.2$x^{2} + 134$$2$$1$$1$$C_2$$$[\ ]_{2}$$
67.1.2.1a1.1$x^{2} + 67$$2$$1$$1$$C_2$$$[\ ]_{2}$$
67.1.2.1a1.1$x^{2} + 67$$2$$1$$1$$C_2$$$[\ ]_{2}$$
67.2.1.0a1.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
67.1.2.1a1.1$x^{2} + 67$$2$$1$$1$$C_2$$$[\ ]_{2}$$
67.2.2.2a1.2$x^{4} + 126 x^{3} + 3973 x^{2} + 252 x + 71$$2$$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)