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Properties

Label 16.0.141...176.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.419\times 10^{19}$
Root discriminant \(15.74\)
Ramified primes $2,3,7,13$
Class number $1$
Class group trivial
Galois group $D_4^2:D_4$ (as 16T877)

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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 + 2*x^14 + 16*x^12 + 65*x^10 - 126*x^9 + 172*x^8 - 144*x^7 + 137*x^6 - 186*x^5 + 64*x^4 - 30*x^3 + 80*x^2 - 42*x + 7)
 
Copy content gp:K = bnfinit(y^16 + 2*y^14 + 16*y^12 + 65*y^10 - 126*y^9 + 172*y^8 - 144*y^7 + 137*y^6 - 186*y^5 + 64*y^4 - 30*y^3 + 80*y^2 - 42*y + 7, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 2*x^14 + 16*x^12 + 65*x^10 - 126*x^9 + 172*x^8 - 144*x^7 + 137*x^6 - 186*x^5 + 64*x^4 - 30*x^3 + 80*x^2 - 42*x + 7);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 + 2*x^14 + 16*x^12 + 65*x^10 - 126*x^9 + 172*x^8 - 144*x^7 + 137*x^6 - 186*x^5 + 64*x^4 - 30*x^3 + 80*x^2 - 42*x + 7)
 

\( x^{16} + 2 x^{14} + 16 x^{12} + 65 x^{10} - 126 x^{9} + 172 x^{8} - 144 x^{7} + 137 x^{6} - 186 x^{5} + \cdots + 7 \) 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:   \(14190242108422738176\) \(\medspace = 2^{8}\cdot 3^{14}\cdot 7^{4}\cdot 13^{6}\) 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:  \(15.74\)
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:  $2^{3/2}3^{7/8}7^{1/2}13^{1/2}\approx 70.55808537927624$
Ramified primes:   \(2\), \(3\), \(7\), \(13\) 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.
Maximal CM subfield:  \(\Q(\sqrt{-3}) \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{28\cdots 68}a^{15}+\frac{171635900812509}{28\cdots 68}a^{14}-\frac{12\cdots 81}{28\cdots 68}a^{13}+\frac{48\cdots 35}{28\cdots 68}a^{12}-\frac{25\cdots 65}{28\cdots 68}a^{11}+\frac{73\cdots 19}{28\cdots 68}a^{10}-\frac{14\cdots 71}{35\cdots 21}a^{9}-\frac{41\cdots 83}{14\cdots 84}a^{8}-\frac{47\cdots 41}{14\cdots 84}a^{7}+\frac{833567199264603}{14\cdots 84}a^{6}+\frac{22264898447749}{655519955292376}a^{5}-\frac{25\cdots 11}{28\cdots 68}a^{4}+\frac{89\cdots 05}{28\cdots 68}a^{3}-\frac{73\cdots 17}{28\cdots 68}a^{2}+\frac{95\cdots 07}{28\cdots 68}a-\frac{37\cdots 03}{28\cdots 68}$ 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:   \( -\frac{4076193189}{12850056952} a^{15} - \frac{2671797213}{12850056952} a^{14} - \frac{10111722155}{12850056952} a^{13} - \frac{6898073547}{12850056952} a^{12} - \frac{70510333935}{12850056952} a^{11} - \frac{46916645247}{12850056952} a^{10} - \frac{37486766943}{1606257119} a^{9} + \frac{156062310935}{6425028476} a^{8} - \frac{258112940495}{6425028476} a^{7} + \frac{127018761617}{6425028476} a^{6} - \frac{419732636451}{12850056952} a^{5} + \frac{496011729783}{12850056952} a^{4} + \frac{21497035083}{12850056952} a^{3} + \frac{161538632161}{12850056952} a^{2} - \frac{217968203579}{12850056952} a + \frac{59905736031}{12850056952} \)  (order $6$) 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:   $\frac{18964671625713}{81939994411547}a^{15}+\frac{37790542651971}{163879988823094}a^{14}+\frac{54347340163424}{81939994411547}a^{13}+\frac{52436926556142}{81939994411547}a^{12}+\frac{349093163851566}{81939994411547}a^{11}+\frac{342161318051601}{81939994411547}a^{10}+\frac{30\cdots 03}{163879988823094}a^{9}-\frac{17\cdots 09}{163879988823094}a^{8}+\frac{43\cdots 35}{163879988823094}a^{7}-\frac{817036253785667}{163879988823094}a^{6}+\frac{37\cdots 51}{163879988823094}a^{5}-\frac{15\cdots 96}{81939994411547}a^{4}-\frac{11\cdots 67}{163879988823094}a^{3}-\frac{19\cdots 69}{163879988823094}a^{2}+\frac{10\cdots 23}{163879988823094}a-\frac{274304604972033}{163879988823094}$, $\frac{6666191185530}{25348343594939}a^{15}+\frac{12319298452015}{50696687189878}a^{14}+\frac{18246530439955}{25348343594939}a^{13}+\frac{16401982836546}{25348343594939}a^{12}+\frac{119903983432235}{25348343594939}a^{11}+\frac{109364314135018}{25348343594939}a^{10}+\frac{10\cdots 43}{50696687189878}a^{9}-\frac{734438124751833}{50696687189878}a^{8}+\frac{15\cdots 07}{50696687189878}a^{7}-\frac{413941645876441}{50696687189878}a^{6}+\frac{28802927313339}{1178992725346}a^{5}-\frac{627283592058569}{25348343594939}a^{4}-\frac{415628715440253}{50696687189878}a^{3}-\frac{691689668052829}{50696687189878}a^{2}+\frac{527207962226073}{50696687189878}a-\frac{71793919104659}{50696687189878}$, $\frac{34\cdots 07}{70\cdots 42}a^{15}+\frac{33\cdots 63}{14\cdots 84}a^{14}+\frac{81\cdots 01}{70\cdots 42}a^{13}+\frac{86\cdots 89}{14\cdots 84}a^{12}+\frac{29\cdots 57}{35\cdots 21}a^{11}+\frac{59\cdots 53}{14\cdots 84}a^{10}+\frac{24\cdots 69}{70\cdots 42}a^{9}-\frac{31\cdots 89}{70\cdots 42}a^{8}+\frac{23\cdots 46}{35\cdots 21}a^{7}-\frac{28\cdots 25}{70\cdots 42}a^{6}+\frac{86\cdots 35}{163879988823094}a^{5}-\frac{92\cdots 43}{14\cdots 84}a^{4}+\frac{14\cdots 56}{35\cdots 21}a^{3}-\frac{21\cdots 95}{14\cdots 84}a^{2}+\frac{10\cdots 43}{35\cdots 21}a-\frac{12\cdots 87}{14\cdots 84}$, $\frac{10\cdots 01}{28\cdots 68}a^{15}+\frac{41\cdots 43}{28\cdots 68}a^{14}+\frac{23\cdots 59}{28\cdots 68}a^{13}+\frac{10\cdots 97}{28\cdots 68}a^{12}+\frac{17\cdots 35}{28\cdots 68}a^{11}+\frac{71\cdots 45}{28\cdots 68}a^{10}+\frac{18\cdots 81}{70\cdots 42}a^{9}-\frac{52\cdots 13}{14\cdots 84}a^{8}+\frac{75\cdots 47}{14\cdots 84}a^{7}-\frac{48\cdots 47}{14\cdots 84}a^{6}+\frac{26\cdots 41}{655519955292376}a^{5}-\frac{15\cdots 45}{28\cdots 68}a^{4}+\frac{18\cdots 29}{28\cdots 68}a^{3}-\frac{29\cdots 51}{28\cdots 68}a^{2}+\frac{68\cdots 11}{28\cdots 68}a-\frac{24\cdots 21}{28\cdots 68}$, $\frac{53\cdots 93}{28\cdots 68}a^{15}+\frac{45\cdots 23}{28\cdots 68}a^{14}+\frac{14\cdots 19}{28\cdots 68}a^{13}+\frac{12\cdots 09}{28\cdots 68}a^{12}+\frac{95\cdots 07}{28\cdots 68}a^{11}+\frac{82\cdots 57}{28\cdots 68}a^{10}+\frac{51\cdots 73}{35\cdots 21}a^{9}-\frac{15\cdots 67}{14\cdots 84}a^{8}+\frac{30\cdots 73}{14\cdots 84}a^{7}-\frac{84\cdots 97}{14\cdots 84}a^{6}+\frac{11\cdots 25}{655519955292376}a^{5}-\frac{50\cdots 53}{28\cdots 68}a^{4}-\frac{12\cdots 63}{28\cdots 68}a^{3}-\frac{24\cdots 11}{28\cdots 68}a^{2}+\frac{20\cdots 59}{28\cdots 68}a-\frac{28\cdots 09}{28\cdots 68}$, $\frac{44\cdots 61}{28\cdots 68}a^{15}+\frac{78\cdots 67}{28\cdots 68}a^{14}+\frac{16\cdots 15}{28\cdots 68}a^{13}+\frac{22\cdots 17}{28\cdots 68}a^{12}+\frac{92\cdots 35}{28\cdots 68}a^{11}+\frac{14\cdots 49}{28\cdots 68}a^{10}+\frac{10\cdots 67}{70\cdots 42}a^{9}+\frac{29\cdots 95}{14\cdots 84}a^{8}+\frac{19\cdots 83}{14\cdots 84}a^{7}+\frac{11\cdots 05}{14\cdots 84}a^{6}+\frac{10\cdots 65}{655519955292376}a^{5}-\frac{13\cdots 93}{28\cdots 68}a^{4}-\frac{31\cdots 31}{28\cdots 68}a^{3}-\frac{40\cdots 99}{28\cdots 68}a^{2}-\frac{38\cdots 25}{28\cdots 68}a+\frac{66\cdots 03}{28\cdots 68}$, $\frac{121091134839045}{70\cdots 42}a^{15}+\frac{336701232084069}{70\cdots 42}a^{14}+\frac{276565925272664}{35\cdots 21}a^{13}+\frac{975084281314225}{70\cdots 42}a^{12}+\frac{14\cdots 39}{35\cdots 21}a^{11}+\frac{31\cdots 85}{35\cdots 21}a^{10}+\frac{13\cdots 71}{70\cdots 42}a^{9}+\frac{11\cdots 57}{70\cdots 42}a^{8}+\frac{34\cdots 37}{70\cdots 42}a^{7}+\frac{24\cdots 27}{70\cdots 42}a^{6}+\frac{50520348995566}{81939994411547}a^{5}+\frac{51\cdots 40}{35\cdots 21}a^{4}-\frac{28\cdots 75}{70\cdots 42}a^{3}-\frac{47\cdots 64}{35\cdots 21}a^{2}-\frac{36\cdots 81}{70\cdots 42}a+\frac{25\cdots 97}{70\cdots 42}$ 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:  \( 2827.57534373 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2827.57534373 \cdot 1}{6\cdot\sqrt{14190242108422738176}}\cr\approx \mathstrut & 0.303883402072 \end{aligned}\]

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 + 2*x^14 + 16*x^12 + 65*x^10 - 126*x^9 + 172*x^8 - 144*x^7 + 137*x^6 - 186*x^5 + 64*x^4 - 30*x^3 + 80*x^2 - 42*x + 7) 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 + 2*x^14 + 16*x^12 + 65*x^10 - 126*x^9 + 172*x^8 - 144*x^7 + 137*x^6 - 186*x^5 + 64*x^4 - 30*x^3 + 80*x^2 - 42*x + 7, 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 + 2*x^14 + 16*x^12 + 65*x^10 - 126*x^9 + 172*x^8 - 144*x^7 + 137*x^6 - 186*x^5 + 64*x^4 - 30*x^3 + 80*x^2 - 42*x + 7); 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 + 2*x^14 + 16*x^12 + 65*x^10 - 126*x^9 + 172*x^8 - 144*x^7 + 137*x^6 - 186*x^5 + 64*x^4 - 30*x^3 + 80*x^2 - 42*x + 7); 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

$D_4^2:D_4$ (as 16T877):

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 512
The 53 conjugacy class representatives for $D_4^2:D_4$
Character table for $D_4^2:D_4$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.117.1, 4.0.2457.1, 4.0.189.1, 8.0.6036849.2

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: 16.0.21495277986723201024.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.8.0.1}{8} }^{2}$ R ${\href{/padicField/11.4.0.1}{4} }^{4}$ R ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(2\) Copy content Toggle raw display 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$$$[\ ]^{4}$$
2.4.2.8a3.1$x^{8} + 2 x^{6} + 4 x^{5} + 2 x^{4} + 2 x^{3} + 5 x^{2} + 4 x + 3$$2$$4$$8$$C_2^2:C_4$$$[2, 2]^{4}$$
\(3\) Copy content Toggle raw display 3.2.8.14a1.2$x^{16} + 16 x^{15} + 128 x^{14} + 672 x^{13} + 2576 x^{12} + 7616 x^{11} + 17920 x^{10} + 34176 x^{9} + 53344 x^{8} + 68352 x^{7} + 71680 x^{6} + 60928 x^{5} + 41216 x^{4} + 21504 x^{3} + 8192 x^{2} + 2048 x + 259$$8$$2$$14$$QD_{16}$$$[\ ]_{8}^{2}$$
\(7\) Copy content Toggle raw display 7.1.2.1a1.2$x^{2} + 21$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.1.2.1a1.2$x^{2} + 21$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.4.1.0a1.1$x^{4} + 5 x^{2} + 4 x + 3$$1$$4$$0$$C_4$$$[\ ]^{4}$$
7.4.1.0a1.1$x^{4} + 5 x^{2} + 4 x + 3$$1$$4$$0$$C_4$$$[\ ]^{4}$$
7.2.2.2a1.2$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(13\) Copy content Toggle raw display 13.4.1.0a1.1$x^{4} + 3 x^{2} + 12 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
13.2.2.2a1.2$x^{4} + 24 x^{3} + 148 x^{2} + 48 x + 17$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
13.2.2.2a1.2$x^{4} + 24 x^{3} + 148 x^{2} + 48 x + 17$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
13.2.2.2a1.1$x^{4} + 24 x^{3} + 148 x^{2} + 61 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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