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Properties

Label 16.0.193...009.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.935\times 10^{24}$
Root discriminant \(32.95\)
Ramified primes $3,61$
Class number $3$ (GRH)
Class group [3] (GRH)
Galois group $D_4:C_4$ (as 16T26)

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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^15 + 24*x^13 + 68*x^12 - 27*x^11 + 120*x^10 + 1216*x^9 + 1120*x^8 - 486*x^7 + 1699*x^6 + 4434*x^5 + 2142*x^4 - 72*x^3 + 468*x^2 + 189*x + 81)
 
Copy content gp:K = bnfinit(y^16 - 2*y^15 + 24*y^13 + 68*y^12 - 27*y^11 + 120*y^10 + 1216*y^9 + 1120*y^8 - 486*y^7 + 1699*y^6 + 4434*y^5 + 2142*y^4 - 72*y^3 + 468*y^2 + 189*y + 81, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 24*x^13 + 68*x^12 - 27*x^11 + 120*x^10 + 1216*x^9 + 1120*x^8 - 486*x^7 + 1699*x^6 + 4434*x^5 + 2142*x^4 - 72*x^3 + 468*x^2 + 189*x + 81);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 + 24*x^13 + 68*x^12 - 27*x^11 + 120*x^10 + 1216*x^9 + 1120*x^8 - 486*x^7 + 1699*x^6 + 4434*x^5 + 2142*x^4 - 72*x^3 + 468*x^2 + 189*x + 81)
 

\( x^{16} - 2 x^{15} + 24 x^{13} + 68 x^{12} - 27 x^{11} + 120 x^{10} + 1216 x^{9} + 1120 x^{8} - 486 x^{7} + \cdots + 81 \) 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:   \(1935020402262940309416009\) \(\medspace = 3^{6}\cdot 61^{12}\) 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:  \(32.95\)
Copy content comment:Root discriminant
 
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:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $3^{1/2}61^{3/4}\approx 37.80576266511387$
Ramified primes:   \(3\), \(61\) 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:  4.0.226981.1

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}$, $\frac{1}{15}a^{13}-\frac{1}{15}a^{12}-\frac{2}{15}a^{11}-\frac{2}{5}a^{10}-\frac{7}{15}a^{9}+\frac{1}{3}a^{8}+\frac{2}{5}a^{7}-\frac{1}{3}a^{6}-\frac{1}{15}a^{5}+\frac{4}{15}a^{4}+\frac{4}{15}a^{3}-\frac{2}{15}a^{2}-\frac{2}{5}$, $\frac{1}{45}a^{14}+\frac{1}{45}a^{13}+\frac{2}{15}a^{12}-\frac{4}{45}a^{10}+\frac{2}{15}a^{9}-\frac{1}{5}a^{8}-\frac{8}{45}a^{7}-\frac{11}{45}a^{6}-\frac{1}{15}a^{5}-\frac{8}{45}a^{4}-\frac{1}{5}a^{3}+\frac{7}{15}a^{2}+\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{99\cdots 25}a^{15}+\frac{51\cdots 29}{99\cdots 25}a^{14}-\frac{26\cdots 97}{33\cdots 75}a^{13}-\frac{16\cdots 89}{33\cdots 75}a^{12}-\frac{65\cdots 44}{99\cdots 25}a^{11}+\frac{14\cdots 73}{33\cdots 75}a^{10}-\frac{33\cdots 67}{33\cdots 75}a^{9}-\frac{66\cdots 79}{19\cdots 65}a^{8}-\frac{39\cdots 08}{19\cdots 65}a^{7}-\frac{87\cdots 87}{33\cdots 75}a^{6}-\frac{27\cdots 27}{99\cdots 25}a^{5}-\frac{83\cdots 92}{11\cdots 25}a^{4}-\frac{23\cdots 39}{11\cdots 25}a^{3}+\frac{26\cdots 93}{11\cdots 25}a^{2}-\frac{23\cdots 37}{22\cdots 85}a-\frac{50\cdots 43}{36\cdots 75}$ 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:  $3$

Class group and class number

Ideal class group:  $C_{3}$, which has order $3$ (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:  $C_{3}$, which has order $3$ (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:  $7$
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:   $\frac{26\cdots 89}{99\cdots 25}a^{15}-\frac{76\cdots 64}{99\cdots 25}a^{14}+\frac{17\cdots 42}{33\cdots 75}a^{13}+\frac{20\cdots 74}{33\cdots 75}a^{12}+\frac{12\cdots 69}{99\cdots 25}a^{11}-\frac{71\cdots 33}{33\cdots 75}a^{10}+\frac{13\cdots 67}{33\cdots 75}a^{9}+\frac{11\cdots 07}{39\cdots 13}a^{8}+\frac{73\cdots 59}{19\cdots 65}a^{7}-\frac{10\cdots 03}{33\cdots 75}a^{6}+\frac{59\cdots 12}{99\cdots 25}a^{5}+\frac{87\cdots 72}{11\cdots 25}a^{4}-\frac{31\cdots 81}{11\cdots 25}a^{3}-\frac{35\cdots 88}{11\cdots 25}a^{2}+\frac{42\cdots 63}{22\cdots 85}a+\frac{12\cdots 83}{36\cdots 75}$, $\frac{20\cdots 89}{99\cdots 25}a^{15}-\frac{57\cdots 49}{99\cdots 25}a^{14}+\frac{12\cdots 32}{33\cdots 75}a^{13}+\frac{16\cdots 94}{33\cdots 75}a^{12}+\frac{10\cdots 59}{99\cdots 25}a^{11}-\frac{52\cdots 13}{33\cdots 75}a^{10}+\frac{10\cdots 02}{33\cdots 75}a^{9}+\frac{45\cdots 53}{19\cdots 65}a^{8}+\frac{92\cdots 56}{19\cdots 65}a^{7}-\frac{81\cdots 83}{33\cdots 75}a^{6}+\frac{43\cdots 62}{99\cdots 25}a^{5}+\frac{74\cdots 58}{12\cdots 25}a^{4}-\frac{24\cdots 91}{11\cdots 25}a^{3}-\frac{29\cdots 18}{11\cdots 25}a^{2}+\frac{31\cdots 16}{22\cdots 85}a-\frac{13\cdots 47}{36\cdots 75}$, $\frac{57\cdots 34}{99\cdots 25}a^{15}+\frac{95\cdots 46}{99\cdots 25}a^{14}-\frac{16\cdots 68}{33\cdots 75}a^{13}+\frac{56\cdots 84}{33\cdots 75}a^{12}+\frac{84\cdots 19}{99\cdots 25}a^{11}+\frac{36\cdots 42}{33\cdots 75}a^{10}-\frac{61\cdots 03}{33\cdots 75}a^{9}+\frac{20\cdots 76}{19\cdots 65}a^{8}+\frac{58\cdots 93}{19\cdots 65}a^{7}+\frac{39\cdots 22}{33\cdots 75}a^{6}+\frac{35\cdots 72}{99\cdots 25}a^{5}+\frac{22\cdots 19}{36\cdots 75}a^{4}+\frac{77\cdots 69}{11\cdots 25}a^{3}+\frac{41\cdots 12}{11\cdots 25}a^{2}+\frac{38\cdots 09}{22\cdots 85}a-\frac{20\cdots 87}{36\cdots 75}$, $\frac{26\cdots 98}{99\cdots 25}a^{15}-\frac{39\cdots 23}{99\cdots 25}a^{14}-\frac{11\cdots 06}{33\cdots 75}a^{13}+\frac{21\cdots 93}{33\cdots 75}a^{12}+\frac{21\cdots 58}{99\cdots 25}a^{11}+\frac{19\cdots 94}{33\cdots 75}a^{10}+\frac{74\cdots 69}{33\cdots 75}a^{9}+\frac{13\cdots 94}{39\cdots 13}a^{8}+\frac{90\cdots 98}{19\cdots 65}a^{7}-\frac{20\cdots 46}{33\cdots 75}a^{6}+\frac{26\cdots 84}{99\cdots 25}a^{5}+\frac{15\cdots 04}{11\cdots 25}a^{4}+\frac{11\cdots 58}{11\cdots 25}a^{3}-\frac{13\cdots 91}{11\cdots 25}a^{2}-\frac{12\cdots 64}{22\cdots 85}a-\frac{10\cdots 69}{36\cdots 75}$, $\frac{25\cdots 38}{33\cdots 75}a^{15}-\frac{55\cdots 48}{33\cdots 75}a^{14}-\frac{43\cdots 96}{11\cdots 25}a^{13}+\frac{21\cdots 53}{11\cdots 25}a^{12}+\frac{15\cdots 88}{33\cdots 75}a^{11}-\frac{42\cdots 91}{11\cdots 25}a^{10}+\frac{92\cdots 49}{11\cdots 25}a^{9}+\frac{62\cdots 18}{66\cdots 55}a^{8}+\frac{76\cdots 31}{13\cdots 71}a^{7}-\frac{96\cdots 81}{11\cdots 25}a^{6}+\frac{51\cdots 79}{33\cdots 75}a^{5}+\frac{39\cdots 58}{12\cdots 25}a^{4}-\frac{10\cdots 37}{36\cdots 75}a^{3}-\frac{21\cdots 01}{36\cdots 75}a^{2}+\frac{46\cdots 19}{73\cdots 95}a-\frac{56\cdots 19}{12\cdots 25}$, $\frac{48\cdots 34}{99\cdots 25}a^{15}-\frac{15\cdots 84}{99\cdots 25}a^{14}+\frac{51\cdots 22}{33\cdots 75}a^{13}+\frac{35\cdots 74}{33\cdots 75}a^{12}+\frac{20\cdots 69}{99\cdots 25}a^{11}-\frac{13\cdots 18}{33\cdots 75}a^{10}+\frac{32\cdots 37}{33\cdots 75}a^{9}+\frac{19\cdots 03}{39\cdots 13}a^{8}-\frac{13\cdots 69}{19\cdots 65}a^{7}-\frac{10\cdots 43}{33\cdots 75}a^{6}+\frac{12\cdots 12}{99\cdots 25}a^{5}+\frac{77\cdots 67}{11\cdots 25}a^{4}-\frac{20\cdots 51}{11\cdots 25}a^{3}+\frac{16\cdots 67}{11\cdots 25}a^{2}+\frac{10\cdots 83}{22\cdots 85}a+\frac{85\cdots 43}{36\cdots 75}$, $\frac{75\cdots 11}{99\cdots 25}a^{15}-\frac{10\cdots 11}{99\cdots 25}a^{14}-\frac{24\cdots 62}{33\cdots 75}a^{13}+\frac{59\cdots 46}{33\cdots 75}a^{12}+\frac{62\cdots 26}{99\cdots 25}a^{11}+\frac{41\cdots 53}{33\cdots 75}a^{10}+\frac{29\cdots 73}{33\cdots 75}a^{9}+\frac{38\cdots 05}{39\cdots 13}a^{8}+\frac{27\cdots 24}{19\cdots 65}a^{7}+\frac{95\cdots 78}{33\cdots 75}a^{6}+\frac{11\cdots 48}{99\cdots 25}a^{5}+\frac{48\cdots 27}{12\cdots 25}a^{4}+\frac{41\cdots 71}{11\cdots 25}a^{3}+\frac{15\cdots 43}{11\cdots 25}a^{2}+\frac{72\cdots 42}{22\cdots 85}a+\frac{31\cdots 72}{36\cdots 75}$ 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:  \( 135710.355692 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 135710.355692 \cdot 3}{2\cdot\sqrt{1935020402262940309416009}}\cr\approx \mathstrut & 0.355467926147 \end{aligned}\] (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 - 2*x^15 + 24*x^13 + 68*x^12 - 27*x^11 + 120*x^10 + 1216*x^9 + 1120*x^8 - 486*x^7 + 1699*x^6 + 4434*x^5 + 2142*x^4 - 72*x^3 + 468*x^2 + 189*x + 81) 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^15 + 24*x^13 + 68*x^12 - 27*x^11 + 120*x^10 + 1216*x^9 + 1120*x^8 - 486*x^7 + 1699*x^6 + 4434*x^5 + 2142*x^4 - 72*x^3 + 468*x^2 + 189*x + 81, 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^15 + 24*x^13 + 68*x^12 - 27*x^11 + 120*x^10 + 1216*x^9 + 1120*x^8 - 486*x^7 + 1699*x^6 + 4434*x^5 + 2142*x^4 - 72*x^3 + 468*x^2 + 189*x + 81); 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 = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 + 24*x^13 + 68*x^12 - 27*x^11 + 120*x^10 + 1216*x^9 + 1120*x^8 - 486*x^7 + 1699*x^6 + 4434*x^5 + 2142*x^4 - 72*x^3 + 468*x^2 + 189*x + 81); 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

$D_4:C_4$ (as 16T26):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{61}) \), 4.0.226981.1, 4.2.11163.1, 4.2.680943.1, 8.2.373837707.1, 8.2.1391050107747.2, 8.0.463683369249.1

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

Sibling fields

Galois closure: deg 32
Degree 16 sibling: 16.4.17415183620366462784744081.2
Minimal sibling: This field is its own minimal sibling

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 ${\href{/padicField/5.2.0.1}{2} }^{8}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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 3.2.1.0a1.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
3.2.1.0a1.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(61\) Copy content Toggle raw display 61.2.4.6a1.2$x^{8} + 240 x^{7} + 21608 x^{6} + 865440 x^{5} + 13046424 x^{4} + 1730880 x^{3} + 86432 x^{2} + 1920 x + 77$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
61.2.4.6a1.2$x^{8} + 240 x^{7} + 21608 x^{6} + 865440 x^{5} + 13046424 x^{4} + 1730880 x^{3} + 86432 x^{2} + 1920 x + 77$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$

Spectrum of ring of integers

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