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Properties

Label 16.8.233...000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2.330\times 10^{23}$
Root discriminant \(28.87\)
Ramified primes $2,5,17$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $(C_2^3\times C_4):C_4$ (as 16T292)

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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 - 6*x^15 + 2*x^14 + 56*x^13 - 144*x^12 + 98*x^11 + 39*x^10 + 113*x^9 - 451*x^8 + 364*x^7 + 49*x^6 - 213*x^5 + 106*x^4 - 2*x^3 - 17*x^2 + 5*x + 1)
 
Copy content gp:K = bnfinit(y^16 - 6*y^15 + 2*y^14 + 56*y^13 - 144*y^12 + 98*y^11 + 39*y^10 + 113*y^9 - 451*y^8 + 364*y^7 + 49*y^6 - 213*y^5 + 106*y^4 - 2*y^3 - 17*y^2 + 5*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 2*x^14 + 56*x^13 - 144*x^12 + 98*x^11 + 39*x^10 + 113*x^9 - 451*x^8 + 364*x^7 + 49*x^6 - 213*x^5 + 106*x^4 - 2*x^3 - 17*x^2 + 5*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 6*x^15 + 2*x^14 + 56*x^13 - 144*x^12 + 98*x^11 + 39*x^10 + 113*x^9 - 451*x^8 + 364*x^7 + 49*x^6 - 213*x^5 + 106*x^4 - 2*x^3 - 17*x^2 + 5*x + 1)
 

\( x^{16} - 6 x^{15} + 2 x^{14} + 56 x^{13} - 144 x^{12} + 98 x^{11} + 39 x^{10} + 113 x^{9} - 451 x^{8} + \cdots + 1 \) 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:  $[8, 4]$
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:   \(233048894891904400000000\) \(\medspace = 2^{10}\cdot 5^{8}\cdot 17^{12}\) 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:  \(28.87\)
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:  $2^{2}5^{1/2}17^{3/4}\approx 74.88273266140878$
Ramified primes:   \(2\), \(5\), \(17\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_4$
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 not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{13}+\frac{1}{4}a^{11}+\frac{1}{4}a^{10}-\frac{1}{2}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{3212390396}a^{15}-\frac{59823005}{3212390396}a^{14}-\frac{15196942}{803097599}a^{13}-\frac{456853211}{3212390396}a^{12}-\frac{852823611}{3212390396}a^{11}-\frac{537543303}{1606195198}a^{10}+\frac{160136945}{803097599}a^{9}-\frac{474170717}{3212390396}a^{8}+\frac{309537040}{803097599}a^{7}-\frac{411393531}{3212390396}a^{6}+\frac{290135377}{1606195198}a^{5}+\frac{302066796}{803097599}a^{4}-\frac{397318230}{803097599}a^{3}+\frac{23078665}{1606195198}a^{2}+\frac{1293866637}{3212390396}a-\frac{165216425}{803097599}$ 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:  No
Index:  Not computed
Inessential primes:  $2$

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:  $11$
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:   $\frac{51626212}{803097599}a^{15}-\frac{447719913}{803097599}a^{14}+\frac{606202311}{803097599}a^{13}+\frac{4067964176}{803097599}a^{12}-\frac{13567074307}{803097599}a^{11}+\frac{9076030402}{803097599}a^{10}+\frac{10858171829}{803097599}a^{9}+\frac{3990617268}{803097599}a^{8}-\frac{46998539481}{803097599}a^{7}+\frac{29456319428}{803097599}a^{6}+\frac{22279041863}{803097599}a^{5}-\frac{24847144085}{803097599}a^{4}+\frac{5072113959}{803097599}a^{3}+\frac{1804206729}{803097599}a^{2}-\frac{2296155414}{803097599}a+\frac{960393402}{803097599}$, $\frac{66054379}{803097599}a^{15}-\frac{381898107}{803097599}a^{14}+\frac{197930564}{803097599}a^{13}+\frac{3290773477}{803097599}a^{12}-\frac{10289021275}{803097599}a^{11}+\frac{9751382174}{803097599}a^{10}+\frac{3251472553}{803097599}a^{9}-\frac{142554449}{803097599}a^{8}-\frac{33923696646}{803097599}a^{7}+\frac{37118636791}{803097599}a^{6}+\frac{10898981934}{803097599}a^{5}-\frac{25449642656}{803097599}a^{4}+\frac{6399265603}{803097599}a^{3}+\frac{1195042886}{803097599}a^{2}-\frac{1732088286}{803097599}a+\frac{91241424}{803097599}$, $\frac{137962641}{803097599}a^{15}-\frac{502949887}{803097599}a^{14}-\frac{1176896304}{803097599}a^{13}+\frac{6132899779}{803097599}a^{12}-\frac{4016661626}{803097599}a^{11}-\frac{8844749561}{803097599}a^{10}+\frac{1843144688}{803097599}a^{9}+\frac{23715117869}{803097599}a^{8}-\frac{10664378260}{803097599}a^{7}-\frac{19749357475}{803097599}a^{6}+\frac{13850760929}{803097599}a^{5}+\frac{400264513}{803097599}a^{4}-\frac{1907459153}{803097599}a^{3}+\frac{1418509810}{803097599}a^{2}-\frac{702262342}{803097599}a+\frac{51626212}{803097599}$, $\frac{433463602}{803097599}a^{15}-\frac{2138878454}{803097599}a^{14}-\frac{1195988302}{803097599}a^{13}+\frac{21945845767}{803097599}a^{12}-\frac{39660174274}{803097599}a^{11}+\frac{10952749526}{803097599}a^{10}+\frac{9333158971}{803097599}a^{9}+\frac{65069086957}{803097599}a^{8}-\frac{124158363494}{803097599}a^{7}+\frac{58023579873}{803097599}a^{6}+\frac{24760311800}{803097599}a^{5}-\frac{34990266304}{803097599}a^{4}+\frac{13724015763}{803097599}a^{3}-\frac{2186821197}{803097599}a^{2}+\frac{509539975}{803097599}a-\frac{218028409}{803097599}$, $\frac{418836388}{803097599}a^{15}-\frac{5089346395}{1606195198}a^{14}+\frac{972909205}{803097599}a^{13}+\frac{47289080825}{1606195198}a^{12}-\frac{61773869194}{803097599}a^{11}+\frac{42581834255}{803097599}a^{10}+\frac{16797612228}{803097599}a^{9}+\frac{49747495720}{803097599}a^{8}-\frac{393936370897}{1606195198}a^{7}+\frac{309540159401}{1606195198}a^{6}+\frac{43253965779}{1606195198}a^{5}-\frac{80090727192}{803097599}a^{4}+\frac{78326589651}{1606195198}a^{3}-\frac{10466256999}{1606195198}a^{2}-\frac{4595169970}{803097599}a+\frac{614883264}{803097599}$, $\frac{66253426}{803097599}a^{15}-\frac{83850339}{1606195198}a^{14}-\frac{1562695196}{803097599}a^{13}+\frac{4738539061}{1606195198}a^{12}+\frac{8546620613}{803097599}a^{11}-\frac{22553054327}{803097599}a^{10}+\frac{3393718572}{803097599}a^{9}+\frac{19312208505}{803097599}a^{8}+\frac{51622564947}{1606195198}a^{7}-\frac{134580360799}{1606195198}a^{6}+\frac{50824741547}{1606195198}a^{5}+\frac{20253316803}{803097599}a^{4}-\frac{40734330207}{1606195198}a^{3}+\frac{9701028063}{1606195198}a^{2}+\frac{2005456932}{803097599}a-\frac{1478713469}{803097599}$, $\frac{150660907}{3212390396}a^{15}+\frac{1085534883}{3212390396}a^{14}-\frac{4796004267}{1606195198}a^{13}+\frac{2290852073}{3212390396}a^{12}+\frac{83274500025}{3212390396}a^{11}-\frac{81958925107}{1606195198}a^{10}+\frac{4408398760}{803097599}a^{9}+\frac{103675938845}{3212390396}a^{8}+\frac{128699182941}{1606195198}a^{7}-\frac{522532382089}{3212390396}a^{6}+\frac{68698407377}{1606195198}a^{5}+\frac{96751776733}{1606195198}a^{4}-\frac{66221683537}{1606195198}a^{3}+\frac{12634865091}{1606195198}a^{2}+\frac{16654470393}{3212390396}a-\frac{1329931233}{803097599}$, $\frac{2334778411}{1606195198}a^{15}-\frac{22989544509}{3212390396}a^{14}-\frac{12864166057}{3212390396}a^{13}+\frac{58728366121}{803097599}a^{12}-\frac{425521513077}{3212390396}a^{11}+\frac{127638634351}{3212390396}a^{10}+\frac{38989007663}{1606195198}a^{9}+\frac{351718458449}{1606195198}a^{8}-\frac{1321406755469}{3212390396}a^{7}+\frac{329953218379}{1606195198}a^{6}+\frac{194340634307}{3212390396}a^{5}-\frac{88727277766}{803097599}a^{4}+\frac{79529132081}{1606195198}a^{3}-\frac{6813874406}{803097599}a^{2}-\frac{1596120845}{1606195198}a-\frac{1230353237}{3212390396}$, $\frac{2103258549}{3212390396}a^{15}-\frac{2397369639}{803097599}a^{14}-\frac{10674669291}{3212390396}a^{13}+\frac{108220625221}{3212390396}a^{12}-\frac{36537163971}{803097599}a^{11}-\frac{62340864465}{3212390396}a^{10}+\frac{56834710097}{1606195198}a^{9}+\frac{350404971143}{3212390396}a^{8}-\frac{479768183281}{3212390396}a^{7}-\frac{87997507973}{3212390396}a^{6}+\frac{355903003145}{3212390396}a^{5}-\frac{34336215549}{803097599}a^{4}-\frac{8651146757}{803097599}a^{3}+\frac{14869168859}{803097599}a^{2}-\frac{17652575043}{3212390396}a-\frac{4224792075}{3212390396}$, $\frac{40240256}{803097599}a^{15}+\frac{48089210}{803097599}a^{14}-\frac{1327172057}{803097599}a^{13}+\frac{2569091909}{1606195198}a^{12}+\frac{8918734130}{803097599}a^{11}-\frac{41080117863}{1606195198}a^{10}+\frac{5326874358}{803097599}a^{9}+\frac{8675465493}{803097599}a^{8}+\frac{30995455955}{803097599}a^{7}-\frac{59552106073}{803097599}a^{6}+\frac{56018510829}{1606195198}a^{5}-\frac{2096012573}{1606195198}a^{4}-\frac{17056325017}{1606195198}a^{3}+\frac{9969315641}{803097599}a^{2}-\frac{7566709685}{1606195198}a-\frac{1765555079}{1606195198}$, $\frac{655572915}{803097599}a^{15}-\frac{6645211627}{1606195198}a^{14}-\frac{1520841902}{803097599}a^{13}+\frac{33877392145}{803097599}a^{12}-\frac{63143038114}{803097599}a^{11}+\frac{39456168945}{1606195198}a^{10}+\frac{14712501300}{803097599}a^{9}+\frac{98254216101}{803097599}a^{8}-\frac{383616566239}{1606195198}a^{7}+\frac{193408978099}{1606195198}a^{6}+\frac{29621291629}{803097599}a^{5}-\frac{130434448639}{1606195198}a^{4}+\frac{35404301273}{803097599}a^{3}-\frac{11724578729}{1606195198}a^{2}-\frac{2587825615}{1606195198}a+\frac{1862944505}{1606195198}$ 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:  \( 628294.842196 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 628294.842196 \cdot 1}{2\cdot\sqrt{233048894891904400000000}}\cr\approx \mathstrut & 0.259638692642 \end{aligned}\] (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 - 6*x^15 + 2*x^14 + 56*x^13 - 144*x^12 + 98*x^11 + 39*x^10 + 113*x^9 - 451*x^8 + 364*x^7 + 49*x^6 - 213*x^5 + 106*x^4 - 2*x^3 - 17*x^2 + 5*x + 1) 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 - 6*x^15 + 2*x^14 + 56*x^13 - 144*x^12 + 98*x^11 + 39*x^10 + 113*x^9 - 451*x^8 + 364*x^7 + 49*x^6 - 213*x^5 + 106*x^4 - 2*x^3 - 17*x^2 + 5*x + 1, 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 - 6*x^15 + 2*x^14 + 56*x^13 - 144*x^12 + 98*x^11 + 39*x^10 + 113*x^9 - 451*x^8 + 364*x^7 + 49*x^6 - 213*x^5 + 106*x^4 - 2*x^3 - 17*x^2 + 5*x + 1); 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 - 6*x^15 + 2*x^14 + 56*x^13 - 144*x^12 + 98*x^11 + 39*x^10 + 113*x^9 - 451*x^8 + 364*x^7 + 49*x^6 - 213*x^5 + 106*x^4 - 2*x^3 - 17*x^2 + 5*x + 1); 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

$(C_2^3\times C_4):C_4$ (as 16T292):

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 26 conjugacy class representatives for $(C_2^3\times C_4):C_4$
Character table for $(C_2^3\times C_4):C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{85}) \), 4.4.4913.1, 4.4.122825.1, \(\Q(\sqrt{5}, \sqrt{17})\), 8.8.15085980625.1

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: 16.0.23864206836931010560000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ R ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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
\(2\) Copy content Toggle raw display 2.2.2.4a1.2$x^{4} + 2 x^{3} + 5 x^{2} + 8 x + 5$$2$$2$$4$$C_4$$$[2]^{2}$$
2.2.2.6a1.2$x^{4} + 2 x^{3} + 3 x^{2} + 10 x + 3$$2$$2$$6$$C_4$$$[3]^{2}$$
2.4.1.0a1.1$x^{4} + x + 1$$1$$4$$0$$C_4$$$[\ ]^{4}$$
2.4.1.0a1.1$x^{4} + x + 1$$1$$4$$0$$C_4$$$[\ ]^{4}$$
\(5\) Copy content Toggle raw display 5.8.2.8a1.2$x^{16} + 2 x^{12} + 6 x^{10} + 8 x^{9} + 5 x^{8} + 6 x^{6} + 8 x^{5} + 13 x^{4} + 24 x^{3} + 28 x^{2} + 16 x + 9$$2$$8$$8$$C_8\times C_2$$$[\ ]_{2}^{8}$$
\(17\) Copy content Toggle raw display 17.4.4.12a1.4$x^{16} + 28 x^{14} + 40 x^{13} + 306 x^{12} + 840 x^{11} + 2224 x^{10} + 6240 x^{9} + 12619 x^{8} + 22760 x^{7} + 37872 x^{6} + 46720 x^{5} + 37954 x^{4} + 19560 x^{3} + 6156 x^{2} + 1080 x + 98$$4$$4$$12$$C_4^2$$$[\ ]_{4}^{4}$$

Spectrum of ring of integers

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