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Properties

Label 16.0.162...625.2
Degree $16$
Signature $[0, 8]$
Discriminant $1.620\times 10^{31}$
Root discriminant \(89.25\)
Ramified primes $5,61$
Class number $32$ (GRH)
Class group [2, 2, 2, 4] (GRH)
Galois group $C_2^3.D_4$ (as 16T153)

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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 - 4*x^15 + 43*x^14 - 153*x^13 + 1031*x^12 - 5509*x^11 + 31605*x^10 - 104029*x^9 + 435089*x^8 - 841163*x^7 + 5233845*x^6 - 6798197*x^5 + 30254141*x^4 + 28820519*x^3 + 45021737*x^2 + 24440568*x + 9101501)
 
Copy content gp:K = bnfinit(y^16 - 4*y^15 + 43*y^14 - 153*y^13 + 1031*y^12 - 5509*y^11 + 31605*y^10 - 104029*y^9 + 435089*y^8 - 841163*y^7 + 5233845*y^6 - 6798197*y^5 + 30254141*y^4 + 28820519*y^3 + 45021737*y^2 + 24440568*y + 9101501, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 43*x^14 - 153*x^13 + 1031*x^12 - 5509*x^11 + 31605*x^10 - 104029*x^9 + 435089*x^8 - 841163*x^7 + 5233845*x^6 - 6798197*x^5 + 30254141*x^4 + 28820519*x^3 + 45021737*x^2 + 24440568*x + 9101501);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 43*x^14 - 153*x^13 + 1031*x^12 - 5509*x^11 + 31605*x^10 - 104029*x^9 + 435089*x^8 - 841163*x^7 + 5233845*x^6 - 6798197*x^5 + 30254141*x^4 + 28820519*x^3 + 45021737*x^2 + 24440568*x + 9101501)
 

\( x^{16} - 4 x^{15} + 43 x^{14} - 153 x^{13} + 1031 x^{12} - 5509 x^{11} + 31605 x^{10} - 104029 x^{9} + \cdots + 9101501 \) 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:   \(16200860438828040517095947265625\) \(\medspace = 5^{14}\cdot 61^{12}\) 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:  \(89.25\)
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:  $5^{7/8}61^{3/4}\approx 89.24751449797871$
Ramified primes:   \(5\), \(61\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_4$
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.
Maximal CM subfield:  8.0.216341265625.1

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}+\frac{2}{5}a^{10}-\frac{1}{5}a^{9}-\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{4}+\frac{2}{5}a^{3}-\frac{1}{5}a^{2}+\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{13}+\frac{2}{5}a^{11}-\frac{1}{5}a^{10}-\frac{1}{5}a^{9}+\frac{1}{5}a^{8}-\frac{1}{5}a^{7}-\frac{2}{5}a^{6}+\frac{2}{5}a^{5}+\frac{2}{5}a^{4}-\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{10}a^{14}-\frac{1}{10}a^{13}-\frac{1}{10}a^{12}-\frac{3}{10}a^{11}-\frac{1}{10}a^{10}-\frac{1}{2}a^{9}+\frac{1}{10}a^{8}+\frac{1}{10}a^{7}-\frac{3}{10}a^{6}+\frac{1}{10}a^{5}+\frac{1}{10}a^{4}-\frac{3}{10}a^{3}-\frac{3}{10}a^{2}+\frac{3}{10}a-\frac{3}{10}$, $\frac{1}{36\cdots 90}a^{15}-\frac{45\cdots 73}{36\cdots 90}a^{14}+\frac{19\cdots 73}{36\cdots 90}a^{13}+\frac{13\cdots 89}{73\cdots 78}a^{12}-\frac{13\cdots 01}{36\cdots 90}a^{11}+\frac{13\cdots 67}{36\cdots 90}a^{10}-\frac{17\cdots 27}{36\cdots 90}a^{9}-\frac{22\cdots 19}{73\cdots 78}a^{8}-\frac{17\cdots 51}{36\cdots 90}a^{7}+\frac{12\cdots 27}{36\cdots 90}a^{6}+\frac{10\cdots 11}{36\cdots 90}a^{5}-\frac{11\cdots 79}{36\cdots 90}a^{4}-\frac{60\cdots 67}{36\cdots 90}a^{3}+\frac{11\cdots 57}{36\cdots 90}a^{2}+\frac{26\cdots 61}{73\cdots 78}a+\frac{76\cdots 66}{18\cdots 95}$ 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:  $C_{2}\times C_{2}\times C_{2}\times C_{4}$, which has order $32$ (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:  $C_{2}\times C_{2}\times C_{2}\times C_{4}$, which has order $32$ (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:   $\frac{85\cdots 82}{56\cdots 81}a^{15}+\frac{67\cdots 67}{28\cdots 05}a^{14}+\frac{25\cdots 16}{28\cdots 05}a^{13}+\frac{96\cdots 21}{28\cdots 05}a^{12}+\frac{12\cdots 73}{56\cdots 81}a^{11}-\frac{14\cdots 29}{28\cdots 05}a^{10}+\frac{26\cdots 74}{28\cdots 05}a^{9}+\frac{87\cdots 42}{28\cdots 05}a^{8}+\frac{67\cdots 62}{28\cdots 05}a^{7}-\frac{11\cdots 10}{56\cdots 81}a^{6}+\frac{32\cdots 02}{28\cdots 05}a^{5}+\frac{20\cdots 39}{28\cdots 05}a^{4}+\frac{24\cdots 87}{28\cdots 05}a^{3}+\frac{27\cdots 72}{28\cdots 05}a^{2}+\frac{28\cdots 99}{56\cdots 81}a+\frac{46\cdots 92}{28\cdots 05}$, $\frac{31\cdots 39}{59\cdots 41}a^{15}-\frac{99\cdots 87}{59\cdots 41}a^{14}+\frac{46\cdots 44}{29\cdots 05}a^{13}-\frac{16\cdots 44}{29\cdots 05}a^{12}+\frac{87\cdots 13}{29\cdots 05}a^{11}-\frac{66\cdots 72}{29\cdots 05}a^{10}+\frac{64\cdots 47}{59\cdots 41}a^{9}-\frac{73\cdots 17}{29\cdots 05}a^{8}+\frac{26\cdots 02}{29\cdots 05}a^{7}-\frac{35\cdots 29}{29\cdots 05}a^{6}+\frac{41\cdots 21}{29\cdots 05}a^{5}-\frac{17\cdots 38}{59\cdots 41}a^{4}-\frac{93\cdots 72}{29\cdots 05}a^{3}-\frac{16\cdots 93}{29\cdots 05}a^{2}-\frac{89\cdots 14}{29\cdots 05}a-\frac{19\cdots 04}{29\cdots 05}$, $\frac{27\cdots 91}{29\cdots 05}a^{15}-\frac{12\cdots 43}{29\cdots 05}a^{14}+\frac{59\cdots 92}{29\cdots 05}a^{13}-\frac{47\cdots 92}{59\cdots 41}a^{12}+\frac{10\cdots 57}{29\cdots 05}a^{11}-\frac{95\cdots 72}{29\cdots 05}a^{10}+\frac{65\cdots 09}{29\cdots 05}a^{9}-\frac{81\cdots 01}{29\cdots 05}a^{8}+\frac{63\cdots 98}{59\cdots 41}a^{7}+\frac{47\cdots 29}{29\cdots 05}a^{6}+\frac{85\cdots 99}{29\cdots 05}a^{5}-\frac{23\cdots 76}{29\cdots 05}a^{4}-\frac{26\cdots 76}{29\cdots 05}a^{3}-\frac{82\cdots 29}{59\cdots 41}a^{2}-\frac{22\cdots 96}{29\cdots 05}a-\frac{60\cdots 03}{29\cdots 05}$, $\frac{14\cdots 25}{36\cdots 39}a^{15}-\frac{21\cdots 69}{18\cdots 95}a^{14}+\frac{12\cdots 76}{18\cdots 95}a^{13}-\frac{16\cdots 65}{36\cdots 39}a^{12}+\frac{26\cdots 11}{18\cdots 95}a^{11}-\frac{13\cdots 96}{18\cdots 95}a^{10}+\frac{76\cdots 57}{18\cdots 95}a^{9}-\frac{43\cdots 98}{18\cdots 95}a^{8}+\frac{22\cdots 36}{36\cdots 39}a^{7}-\frac{13\cdots 03}{18\cdots 95}a^{6}-\frac{14\cdots 57}{18\cdots 95}a^{5}+\frac{28\cdots 02}{18\cdots 95}a^{4}-\frac{84\cdots 23}{18\cdots 95}a^{3}-\frac{35\cdots 52}{36\cdots 39}a^{2}-\frac{14\cdots 38}{18\cdots 95}a-\frac{14\cdots 42}{18\cdots 95}$, $\frac{18\cdots 82}{18\cdots 95}a^{15}-\frac{10\cdots 71}{18\cdots 95}a^{14}+\frac{84\cdots 41}{18\cdots 95}a^{13}-\frac{34\cdots 87}{18\cdots 95}a^{12}+\frac{18\cdots 93}{18\cdots 95}a^{11}-\frac{22\cdots 79}{36\cdots 39}a^{10}+\frac{67\cdots 03}{18\cdots 95}a^{9}-\frac{26\cdots 13}{18\cdots 95}a^{8}+\frac{10\cdots 21}{18\cdots 95}a^{7}-\frac{27\cdots 89}{18\cdots 95}a^{6}+\frac{11\cdots 46}{18\cdots 95}a^{5}-\frac{29\cdots 07}{18\cdots 95}a^{4}+\frac{81\cdots 57}{18\cdots 95}a^{3}-\frac{46\cdots 19}{18\cdots 95}a^{2}+\frac{75\cdots 81}{18\cdots 95}a-\frac{27\cdots 23}{18\cdots 95}$, $\frac{83\cdots 67}{73\cdots 78}a^{15}-\frac{88\cdots 22}{18\cdots 95}a^{14}+\frac{88\cdots 69}{18\cdots 95}a^{13}-\frac{32\cdots 58}{18\cdots 95}a^{12}+\frac{42\cdots 79}{36\cdots 39}a^{11}-\frac{23\cdots 20}{36\cdots 39}a^{10}+\frac{65\cdots 18}{18\cdots 95}a^{9}-\frac{43\cdots 56}{36\cdots 39}a^{8}+\frac{86\cdots 36}{18\cdots 95}a^{7}-\frac{16\cdots 48}{18\cdots 95}a^{6}+\frac{10\cdots 07}{18\cdots 95}a^{5}-\frac{14\cdots 08}{18\cdots 95}a^{4}+\frac{53\cdots 39}{18\cdots 95}a^{3}+\frac{13\cdots 31}{36\cdots 39}a^{2}+\frac{38\cdots 66}{18\cdots 95}a+\frac{14\cdots 87}{36\cdots 90}$, $\frac{23\cdots 89}{36\cdots 90}a^{15}-\frac{11\cdots 11}{36\cdots 90}a^{14}+\frac{10\cdots 69}{36\cdots 90}a^{13}-\frac{43\cdots 87}{36\cdots 90}a^{12}+\frac{27\cdots 09}{36\cdots 90}a^{11}-\frac{14\cdots 63}{36\cdots 90}a^{10}+\frac{16\cdots 45}{73\cdots 78}a^{9}-\frac{59\cdots 89}{73\cdots 78}a^{8}+\frac{12\cdots 43}{36\cdots 90}a^{7}-\frac{54\cdots 85}{73\cdots 78}a^{6}+\frac{13\cdots 03}{36\cdots 90}a^{5}-\frac{24\cdots 71}{36\cdots 90}a^{4}+\frac{84\cdots 09}{36\cdots 90}a^{3}+\frac{21\cdots 97}{36\cdots 90}a^{2}+\frac{74\cdots 99}{36\cdots 90}a+\frac{45\cdots 97}{18\cdots 95}$ 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:  \( 28292261.4648 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 28292261.4648 \cdot 32}{2\cdot\sqrt{16200860438828040517095947265625}}\cr\approx \mathstrut & 0.273185441961 \end{aligned}\] (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 - 4*x^15 + 43*x^14 - 153*x^13 + 1031*x^12 - 5509*x^11 + 31605*x^10 - 104029*x^9 + 435089*x^8 - 841163*x^7 + 5233845*x^6 - 6798197*x^5 + 30254141*x^4 + 28820519*x^3 + 45021737*x^2 + 24440568*x + 9101501) 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 - 4*x^15 + 43*x^14 - 153*x^13 + 1031*x^12 - 5509*x^11 + 31605*x^10 - 104029*x^9 + 435089*x^8 - 841163*x^7 + 5233845*x^6 - 6798197*x^5 + 30254141*x^4 + 28820519*x^3 + 45021737*x^2 + 24440568*x + 9101501, 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 - 4*x^15 + 43*x^14 - 153*x^13 + 1031*x^12 - 5509*x^11 + 31605*x^10 - 104029*x^9 + 435089*x^8 - 841163*x^7 + 5233845*x^6 - 6798197*x^5 + 30254141*x^4 + 28820519*x^3 + 45021737*x^2 + 24440568*x + 9101501); 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 - 4*x^15 + 43*x^14 - 153*x^13 + 1031*x^12 - 5509*x^11 + 31605*x^10 - 104029*x^9 + 435089*x^8 - 841163*x^7 + 5233845*x^6 - 6798197*x^5 + 30254141*x^4 + 28820519*x^3 + 45021737*x^2 + 24440568*x + 9101501); 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

$C_2^3.D_4$ (as 16T153):

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 64
The 13 conjugacy class representatives for $C_2^3.D_4$
Character table for $C_2^3.D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.465125.1, 4.4.7625.1, 4.0.1525.1, 8.0.216341265625.1

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: 16.0.16200860438828040517095947265625.3

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 ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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
\(5\) Copy content Toggle raw display 5.1.8.7a1.1$x^{8} + 5$$8$$1$$7$$C_8:C_2$$$[\ ]_{8}^{2}$$
5.1.8.7a1.1$x^{8} + 5$$8$$1$$7$$C_8:C_2$$$[\ ]_{8}^{2}$$
\(61\) 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)