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Properties

Label 16.0.148...625.2
Degree $16$
Signature $[0, 8]$
Discriminant $1.482\times 10^{28}$
Root discriminant \(57.63\)
Ramified primes $5,41$
Class number $54$ (GRH)
Class group [3, 3, 6] (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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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 - 6*x^15 - 8*x^14 + 160*x^13 - 441*x^12 - 134*x^11 + 4342*x^10 - 16579*x^9 + 43548*x^8 - 89962*x^7 + 151269*x^6 - 210738*x^5 + 244446*x^4 - 223845*x^3 + 154504*x^2 - 70325*x + 18419)
 
Copy content gp:K = bnfinit(y^16 - 6*y^15 - 8*y^14 + 160*y^13 - 441*y^12 - 134*y^11 + 4342*y^10 - 16579*y^9 + 43548*y^8 - 89962*y^7 + 151269*y^6 - 210738*y^5 + 244446*y^4 - 223845*y^3 + 154504*y^2 - 70325*y + 18419, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 - 8*x^14 + 160*x^13 - 441*x^12 - 134*x^11 + 4342*x^10 - 16579*x^9 + 43548*x^8 - 89962*x^7 + 151269*x^6 - 210738*x^5 + 244446*x^4 - 223845*x^3 + 154504*x^2 - 70325*x + 18419);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 6*x^15 - 8*x^14 + 160*x^13 - 441*x^12 - 134*x^11 + 4342*x^10 - 16579*x^9 + 43548*x^8 - 89962*x^7 + 151269*x^6 - 210738*x^5 + 244446*x^4 - 223845*x^3 + 154504*x^2 - 70325*x + 18419)
 

\( x^{16} - 6 x^{15} - 8 x^{14} + 160 x^{13} - 441 x^{12} - 134 x^{11} + 4342 x^{10} - 16579 x^{9} + \cdots + 18419 \) 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:   \(14816104373013890157094140625\) \(\medspace = 5^{8}\cdot 41^{14}\) 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:  \(57.63\)
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:  $5^{1/2}41^{7/8}\approx 57.632953563318466$
Ramified primes:   \(5\), \(41\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2\times C_4$
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 a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{118}a^{14}-\frac{1}{118}a^{13}+\frac{15}{118}a^{12}-\frac{29}{118}a^{11}+\frac{11}{118}a^{10}+\frac{53}{118}a^{9}-\frac{41}{118}a^{8}-\frac{19}{118}a^{7}-\frac{57}{118}a^{6}-\frac{37}{118}a^{5}+\frac{41}{118}a^{4}+\frac{5}{118}a^{3}-\frac{57}{118}a^{2}+\frac{33}{118}a+\frac{27}{118}$, $\frac{1}{31\cdots 66}a^{15}+\frac{31\cdots 10}{15\cdots 83}a^{14}+\frac{21\cdots 82}{15\cdots 83}a^{13}+\frac{31\cdots 41}{15\cdots 83}a^{12}+\frac{40\cdots 07}{15\cdots 83}a^{11}-\frac{68\cdots 51}{15\cdots 83}a^{10}-\frac{41\cdots 21}{15\cdots 83}a^{9}+\frac{70\cdots 77}{15\cdots 83}a^{8}-\frac{32\cdots 36}{15\cdots 83}a^{7}-\frac{15\cdots 58}{15\cdots 83}a^{6}-\frac{22\cdots 53}{15\cdots 83}a^{5}-\frac{72\cdots 67}{15\cdots 83}a^{4}+\frac{40\cdots 08}{15\cdots 83}a^{3}+\frac{37\cdots 73}{15\cdots 83}a^{2}+\frac{39\cdots 69}{15\cdots 83}a+\frac{83\cdots 49}{31\cdots 66}$ 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:  $C_{3}\times C_{3}\times C_{6}$, which has order $54$ (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:  $C_{3}\times C_{3}\times C_{6}$, which has order $54$ (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);
 
Relative class number:   $54$ (assuming GRH)

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:   $\frac{22\cdots 10}{42\cdots 71}a^{15}-\frac{13\cdots 31}{42\cdots 71}a^{14}-\frac{21\cdots 57}{42\cdots 71}a^{13}+\frac{36\cdots 59}{42\cdots 71}a^{12}-\frac{92\cdots 09}{42\cdots 71}a^{11}-\frac{64\cdots 38}{42\cdots 71}a^{10}+\frac{10\cdots 90}{42\cdots 71}a^{9}-\frac{35\cdots 79}{42\cdots 71}a^{8}+\frac{88\cdots 49}{42\cdots 71}a^{7}-\frac{17\cdots 76}{42\cdots 71}a^{6}+\frac{28\cdots 12}{42\cdots 71}a^{5}-\frac{37\cdots 76}{42\cdots 71}a^{4}+\frac{40\cdots 05}{42\cdots 71}a^{3}-\frac{34\cdots 69}{42\cdots 71}a^{2}+\frac{18\cdots 94}{42\cdots 71}a-\frac{57\cdots 46}{42\cdots 71}$, $\frac{68\cdots 50}{42\cdots 71}a^{15}-\frac{40\cdots 54}{42\cdots 71}a^{14}-\frac{60\cdots 66}{42\cdots 71}a^{13}+\frac{11\cdots 87}{42\cdots 71}a^{12}-\frac{29\cdots 64}{42\cdots 71}a^{11}-\frac{17\cdots 32}{42\cdots 71}a^{10}+\frac{31\cdots 31}{42\cdots 71}a^{9}-\frac{11\cdots 56}{42\cdots 71}a^{8}+\frac{27\cdots 36}{42\cdots 71}a^{7}-\frac{54\cdots 93}{42\cdots 71}a^{6}+\frac{87\cdots 50}{42\cdots 71}a^{5}-\frac{11\cdots 37}{42\cdots 71}a^{4}+\frac{12\cdots 67}{42\cdots 71}a^{3}-\frac{10\cdots 68}{42\cdots 71}a^{2}+\frac{56\cdots 45}{42\cdots 71}a-\frac{17\cdots 01}{42\cdots 71}$, $\frac{65\cdots 28}{15\cdots 83}a^{15}-\frac{34\cdots 98}{15\cdots 83}a^{14}-\frac{73\cdots 97}{15\cdots 83}a^{13}+\frac{98\cdots 05}{15\cdots 83}a^{12}-\frac{22\cdots 58}{15\cdots 83}a^{11}-\frac{20\cdots 14}{15\cdots 83}a^{10}+\frac{26\cdots 47}{15\cdots 83}a^{9}-\frac{92\cdots 30}{15\cdots 83}a^{8}+\frac{22\cdots 79}{15\cdots 83}a^{7}-\frac{44\cdots 03}{15\cdots 83}a^{6}+\frac{71\cdots 85}{15\cdots 83}a^{5}-\frac{92\cdots 95}{15\cdots 83}a^{4}+\frac{97\cdots 87}{15\cdots 83}a^{3}-\frac{77\cdots 25}{15\cdots 83}a^{2}+\frac{41\cdots 03}{15\cdots 83}a-\frac{10\cdots 71}{15\cdots 83}$, $\frac{64\cdots 05}{15\cdots 83}a^{15}-\frac{40\cdots 72}{15\cdots 83}a^{14}-\frac{42\cdots 25}{15\cdots 83}a^{13}+\frac{10\cdots 56}{15\cdots 83}a^{12}-\frac{31\cdots 09}{15\cdots 83}a^{11}-\frac{83\cdots 77}{15\cdots 83}a^{10}+\frac{31\cdots 30}{15\cdots 83}a^{9}-\frac{19\cdots 77}{26\cdots 37}a^{8}+\frac{28\cdots 04}{15\cdots 83}a^{7}-\frac{55\cdots 03}{15\cdots 83}a^{6}+\frac{89\cdots 82}{15\cdots 83}a^{5}-\frac{11\cdots 84}{15\cdots 83}a^{4}+\frac{12\cdots 08}{15\cdots 83}a^{3}-\frac{92\cdots 56}{15\cdots 83}a^{2}+\frac{47\cdots 05}{15\cdots 83}a-\frac{13\cdots 42}{15\cdots 83}$, $\frac{31\cdots 72}{15\cdots 83}a^{15}-\frac{18\cdots 24}{15\cdots 83}a^{14}-\frac{28\cdots 22}{15\cdots 83}a^{13}+\frac{50\cdots 43}{15\cdots 83}a^{12}-\frac{13\cdots 08}{15\cdots 83}a^{11}-\frac{80\cdots 62}{15\cdots 83}a^{10}+\frac{14\cdots 61}{15\cdots 83}a^{9}-\frac{49\cdots 79}{15\cdots 83}a^{8}+\frac{12\cdots 14}{15\cdots 83}a^{7}-\frac{24\cdots 63}{15\cdots 83}a^{6}+\frac{39\cdots 46}{15\cdots 83}a^{5}-\frac{51\cdots 64}{15\cdots 83}a^{4}+\frac{94\cdots 61}{26\cdots 37}a^{3}-\frac{45\cdots 68}{15\cdots 83}a^{2}+\frac{24\cdots 65}{15\cdots 83}a-\frac{66\cdots 65}{15\cdots 83}$, $\frac{49\cdots 32}{15\cdots 83}a^{15}-\frac{25\cdots 18}{15\cdots 83}a^{14}-\frac{58\cdots 32}{15\cdots 83}a^{13}+\frac{73\cdots 17}{15\cdots 83}a^{12}-\frac{16\cdots 70}{15\cdots 83}a^{11}-\frac{17\cdots 58}{15\cdots 83}a^{10}+\frac{20\cdots 29}{15\cdots 83}a^{9}-\frac{67\cdots 30}{15\cdots 83}a^{8}+\frac{16\cdots 36}{15\cdots 83}a^{7}-\frac{32\cdots 23}{15\cdots 83}a^{6}+\frac{51\cdots 52}{15\cdots 83}a^{5}-\frac{66\cdots 96}{15\cdots 83}a^{4}+\frac{71\cdots 89}{15\cdots 83}a^{3}-\frac{57\cdots 55}{15\cdots 83}a^{2}+\frac{30\cdots 49}{15\cdots 83}a-\frac{95\cdots 88}{15\cdots 83}$, $\frac{22\cdots 50}{42\cdots 71}a^{15}-\frac{24\cdots 29}{42\cdots 71}a^{14}+\frac{38\cdots 41}{42\cdots 71}a^{13}+\frac{53\cdots 85}{42\cdots 71}a^{12}-\frac{28\cdots 79}{42\cdots 71}a^{11}+\frac{25\cdots 66}{42\cdots 71}a^{10}+\frac{19\cdots 14}{42\cdots 71}a^{9}-\frac{90\cdots 76}{42\cdots 71}a^{8}+\frac{21\cdots 39}{42\cdots 71}a^{7}-\frac{40\cdots 52}{42\cdots 71}a^{6}+\frac{65\cdots 42}{42\cdots 71}a^{5}-\frac{82\cdots 88}{42\cdots 71}a^{4}+\frac{72\cdots 55}{42\cdots 71}a^{3}-\frac{40\cdots 74}{42\cdots 71}a^{2}+\frac{12\cdots 10}{42\cdots 71}a-\frac{13\cdots 66}{42\cdots 71}$ 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:  \( 645179.367444 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 645179.367444 \cdot 54}{2\cdot\sqrt{14816104373013890157094140625}}\cr\approx \mathstrut & 0.347629134405 \end{aligned}\] (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 - 8*x^14 + 160*x^13 - 441*x^12 - 134*x^11 + 4342*x^10 - 16579*x^9 + 43548*x^8 - 89962*x^7 + 151269*x^6 - 210738*x^5 + 244446*x^4 - 223845*x^3 + 154504*x^2 - 70325*x + 18419) 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 - 8*x^14 + 160*x^13 - 441*x^12 - 134*x^11 + 4342*x^10 - 16579*x^9 + 43548*x^8 - 89962*x^7 + 151269*x^6 - 210738*x^5 + 244446*x^4 - 223845*x^3 + 154504*x^2 - 70325*x + 18419, 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 - 8*x^14 + 160*x^13 - 441*x^12 - 134*x^11 + 4342*x^10 - 16579*x^9 + 43548*x^8 - 89962*x^7 + 151269*x^6 - 210738*x^5 + 244446*x^4 - 223845*x^3 + 154504*x^2 - 70325*x + 18419); 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 - 6*x^15 - 8*x^14 + 160*x^13 - 441*x^12 - 134*x^11 + 4342*x^10 - 16579*x^9 + 43548*x^8 - 89962*x^7 + 151269*x^6 - 210738*x^5 + 244446*x^4 - 223845*x^3 + 154504*x^2 - 70325*x + 18419); 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

$C_2^2:C_8$ (as 16T24):

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 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.0.8405.1, 4.4.68921.1, 4.0.344605.1, 8.8.4868856847025.1, 8.0.121721421175625.1, 8.0.118752606025.1

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: 16.0.23705766996822224251350625.1
Minimal sibling: 16.0.23705766996822224251350625.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{4}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ R ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{8}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ R ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.1.0.1}{1} }^{16}$

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
\(5\) Copy content Toggle raw display 5.2.2.2a1.1$x^{4} + 8 x^{3} + 20 x^{2} + 21 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
5.2.2.2a1.1$x^{4} + 8 x^{3} + 20 x^{2} + 21 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(41\) Copy content Toggle raw display 41.2.8.14a1.2$x^{16} + 304 x^{15} + 40480 x^{14} + 3085600 x^{13} + 147416080 x^{12} + 4529584192 x^{11} + 87831092608 x^{10} + 996302227840 x^{9} + 5391168776800 x^{8} + 5977813367040 x^{7} + 3161919333888 x^{6} + 978390185472 x^{5} + 191051239680 x^{4} + 23993625600 x^{3} + 1888634880 x^{2} + 85100544 x + 1679657$$8$$2$$14$$C_8\times C_2$$$[\ ]_{8}^{2}$$

Spectrum of ring of integers

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