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Properties

Label 16.0.148...761.11
Degree $16$
Signature $[0, 8]$
Discriminant $1.485\times 10^{38}$
Root discriminant \(243.07\)
Ramified primes $37,41$
Class number $192$ (GRH)
Class group [2, 96] (GRH)
Galois group $C_2^3.D_4$ (as 16T153)

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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 - x^15 - 21*x^14 + 279*x^13 - 1802*x^12 - 13059*x^11 + 276033*x^10 + 1587075*x^9 - 11621499*x^8 - 96153882*x^7 + 330208021*x^6 + 4122671339*x^5 - 1269953021*x^4 - 86907862279*x^3 - 115905519180*x^2 + 675228426700*x + 1550616680000)
 
Copy content gp:K = bnfinit(y^16 - y^15 - 21*y^14 + 279*y^13 - 1802*y^12 - 13059*y^11 + 276033*y^10 + 1587075*y^9 - 11621499*y^8 - 96153882*y^7 + 330208021*y^6 + 4122671339*y^5 - 1269953021*y^4 - 86907862279*y^3 - 115905519180*y^2 + 675228426700*y + 1550616680000, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 21*x^14 + 279*x^13 - 1802*x^12 - 13059*x^11 + 276033*x^10 + 1587075*x^9 - 11621499*x^8 - 96153882*x^7 + 330208021*x^6 + 4122671339*x^5 - 1269953021*x^4 - 86907862279*x^3 - 115905519180*x^2 + 675228426700*x + 1550616680000);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - x^15 - 21*x^14 + 279*x^13 - 1802*x^12 - 13059*x^11 + 276033*x^10 + 1587075*x^9 - 11621499*x^8 - 96153882*x^7 + 330208021*x^6 + 4122671339*x^5 - 1269953021*x^4 - 86907862279*x^3 - 115905519180*x^2 + 675228426700*x + 1550616680000)
 

\( x^{16} - x^{15} - 21 x^{14} + 279 x^{13} - 1802 x^{12} - 13059 x^{11} + 276033 x^{10} + \cdots + 1550616680000 \) 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:   \(148534373731547764810930925932451123761\) \(\medspace = 37^{12}\cdot 41^{12}\) 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:  \(243.07\)
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:  $37^{3/4}41^{3/4}\approx 243.0743793855133$
Ramified primes:   \(37\), \(41\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_4$
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:  8.0.8902460114416801.2

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{10}a^{12}-\frac{3}{10}a^{11}+\frac{1}{5}a^{10}+\frac{2}{5}a^{9}+\frac{2}{5}a^{8}+\frac{1}{10}a^{7}-\frac{1}{10}a^{6}+\frac{2}{5}a^{5}-\frac{2}{5}a^{4}+\frac{2}{5}a^{3}-\frac{1}{2}a^{2}-\frac{3}{10}a$, $\frac{1}{10}a^{13}+\frac{3}{10}a^{11}-\frac{2}{5}a^{9}+\frac{3}{10}a^{8}+\frac{1}{5}a^{7}+\frac{1}{10}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{4}-\frac{3}{10}a^{3}+\frac{1}{5}a^{2}+\frac{1}{10}a$, $\frac{1}{180}a^{14}-\frac{2}{45}a^{13}+\frac{1}{60}a^{12}+\frac{23}{90}a^{11}-\frac{2}{15}a^{10}-\frac{7}{36}a^{9}-\frac{41}{90}a^{8}-\frac{11}{36}a^{7}-\frac{1}{6}a^{6}-\frac{13}{45}a^{5}-\frac{79}{180}a^{4}-\frac{1}{45}a^{3}-\frac{13}{36}a^{2}-\frac{4}{15}a+\frac{2}{9}$, $\frac{1}{21\cdots 00}a^{15}+\frac{19\cdots 23}{72\cdots 00}a^{14}-\frac{47\cdots 31}{21\cdots 00}a^{13}-\frac{39\cdots 51}{21\cdots 00}a^{12}-\frac{20\cdots 03}{54\cdots 50}a^{11}-\frac{82\cdots 19}{21\cdots 00}a^{10}-\frac{30\cdots 77}{21\cdots 00}a^{9}-\frac{58\cdots 85}{29\cdots 84}a^{8}-\frac{10\cdots 89}{21\cdots 00}a^{7}-\frac{59\cdots 34}{44\cdots 75}a^{6}+\frac{33\cdots 07}{72\cdots 00}a^{5}-\frac{64\cdots 59}{24\cdots 00}a^{4}-\frac{92\cdots 11}{21\cdots 00}a^{3}+\frac{65\cdots 71}{21\cdots 00}a^{2}+\frac{10\cdots 63}{21\cdots 80}a+\frac{19\cdots 17}{10\cdots 19}$ 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:  $C_{2}\times C_{96}$, which has order $192$ (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:  $C_{2}\times C_{96}$, which has order $192$ (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:   $\frac{34\cdots 60}{21\cdots 71}a^{15}-\frac{19\cdots 60}{21\cdots 71}a^{14}-\frac{43\cdots 00}{21\cdots 71}a^{13}+\frac{15\cdots 50}{21\cdots 71}a^{12}-\frac{13\cdots 80}{21\cdots 71}a^{11}+\frac{72\cdots 60}{21\cdots 71}a^{10}+\frac{11\cdots 90}{21\cdots 71}a^{9}-\frac{99\cdots 60}{21\cdots 71}a^{8}-\frac{54\cdots 40}{21\cdots 71}a^{7}-\frac{87\cdots 50}{21\cdots 71}a^{6}+\frac{22\cdots 00}{21\cdots 71}a^{5}+\frac{55\cdots 60}{21\cdots 71}a^{4}-\frac{53\cdots 30}{21\cdots 71}a^{3}-\frac{12\cdots 00}{21\cdots 71}a^{2}+\frac{60\cdots 00}{21\cdots 71}a+\frac{17\cdots 17}{21\cdots 71}$, $\frac{72\cdots 67}{90\cdots 50}a^{15}-\frac{45\cdots 77}{15\cdots 75}a^{14}-\frac{79\cdots 07}{90\cdots 50}a^{13}+\frac{11\cdots 34}{45\cdots 25}a^{12}-\frac{98\cdots 32}{45\cdots 25}a^{11}-\frac{39\cdots 53}{90\cdots 50}a^{10}+\frac{10\cdots 08}{45\cdots 25}a^{9}+\frac{70\cdots 53}{12\cdots 78}a^{8}-\frac{50\cdots 34}{45\cdots 25}a^{7}-\frac{32\cdots 12}{73\cdots 25}a^{6}+\frac{12\cdots 69}{30\cdots 50}a^{5}+\frac{32\cdots 93}{15\cdots 75}a^{4}-\frac{69\cdots 07}{90\cdots 50}a^{3}-\frac{21\cdots 69}{45\cdots 25}a^{2}+\frac{52\cdots 36}{90\cdots 85}a+\frac{70\cdots 73}{18\cdots 17}$, $\frac{34\cdots 49}{54\cdots 50}a^{15}+\frac{18\cdots 09}{36\cdots 00}a^{14}+\frac{11\cdots 81}{54\cdots 50}a^{13}+\frac{23\cdots 27}{10\cdots 00}a^{12}+\frac{40\cdots 21}{27\cdots 75}a^{11}-\frac{75\cdots 31}{54\cdots 50}a^{10}+\frac{22\cdots 19}{10\cdots 00}a^{9}+\frac{53\cdots 62}{36\cdots 73}a^{8}+\frac{66\cdots 73}{10\cdots 00}a^{7}-\frac{16\cdots 54}{44\cdots 75}a^{6}-\frac{58\cdots 07}{18\cdots 50}a^{5}-\frac{83\cdots 17}{12\cdots 00}a^{4}+\frac{27\cdots 61}{54\cdots 50}a^{3}+\frac{10\cdots 23}{10\cdots 00}a^{2}-\frac{51\cdots 63}{21\cdots 38}a-\frac{71\cdots 79}{10\cdots 19}$, $\frac{85\cdots 93}{60\cdots 50}a^{15}-\frac{17\cdots 97}{54\cdots 50}a^{14}-\frac{18\cdots 71}{27\cdots 75}a^{13}+\frac{14\cdots 38}{91\cdots 25}a^{12}-\frac{10\cdots 37}{27\cdots 75}a^{11}-\frac{52\cdots 41}{18\cdots 50}a^{10}+\frac{21\cdots 91}{54\cdots 50}a^{9}+\frac{25\cdots 61}{10\cdots 19}a^{8}-\frac{70\cdots 84}{27\cdots 75}a^{7}-\frac{35\cdots 54}{14\cdots 25}a^{6}+\frac{13\cdots 77}{54\cdots 50}a^{5}+\frac{42\cdots 53}{54\cdots 50}a^{4}+\frac{69\cdots 44}{27\cdots 75}a^{3}-\frac{61\cdots 19}{27\cdots 75}a^{2}-\frac{16\cdots 21}{18\cdots 65}a-\frac{12\cdots 71}{10\cdots 19}$, $\frac{63\cdots 19}{21\cdots 80}a^{15}-\frac{12\cdots 09}{72\cdots 46}a^{14}+\frac{42\cdots 31}{21\cdots 80}a^{13}+\frac{82\cdots 91}{10\cdots 90}a^{12}-\frac{10\cdots 47}{10\cdots 19}a^{11}+\frac{17\cdots 31}{21\cdots 80}a^{10}+\frac{84\cdots 87}{10\cdots 19}a^{9}+\frac{56\cdots 43}{72\cdots 60}a^{8}-\frac{41\cdots 09}{10\cdots 90}a^{7}-\frac{15\cdots 80}{17\cdots 79}a^{6}+\frac{10\cdots 61}{72\cdots 60}a^{5}+\frac{59\cdots 81}{12\cdots 10}a^{4}-\frac{12\cdots 01}{43\cdots 76}a^{3}-\frac{58\cdots 63}{54\cdots 95}a^{2}+\frac{11\cdots 22}{54\cdots 95}a+\frac{96\cdots 49}{10\cdots 19}$, $\frac{33\cdots 03}{36\cdots 00}a^{15}+\frac{48\cdots 49}{27\cdots 75}a^{14}-\frac{19\cdots 59}{10\cdots 00}a^{13}+\frac{19\cdots 16}{30\cdots 75}a^{12}+\frac{12\cdots 93}{27\cdots 75}a^{11}-\frac{12\cdots 39}{12\cdots 00}a^{10}+\frac{32\cdots 21}{54\cdots 50}a^{9}+\frac{26\cdots 87}{43\cdots 76}a^{8}-\frac{66\cdots 19}{27\cdots 75}a^{7}-\frac{15\cdots 58}{49\cdots 75}a^{6}+\frac{51\cdots 89}{10\cdots 00}a^{5}+\frac{38\cdots 74}{27\cdots 75}a^{4}-\frac{64\cdots 19}{10\cdots 00}a^{3}-\frac{17\cdots 73}{54\cdots 50}a^{2}+\frac{12\cdots 01}{18\cdots 65}a+\frac{25\cdots 53}{10\cdots 19}$, $\frac{15\cdots 23}{60\cdots 50}a^{15}+\frac{11\cdots 51}{10\cdots 00}a^{14}-\frac{19\cdots 67}{54\cdots 50}a^{13}+\frac{24\cdots 57}{36\cdots 00}a^{12}-\frac{34\cdots 52}{27\cdots 75}a^{11}-\frac{74\cdots 41}{18\cdots 50}a^{10}+\frac{54\cdots 47}{10\cdots 00}a^{9}+\frac{36\cdots 92}{54\cdots 95}a^{8}+\frac{37\cdots 69}{10\cdots 00}a^{7}-\frac{35\cdots 84}{14\cdots 25}a^{6}-\frac{21\cdots 33}{54\cdots 50}a^{5}+\frac{95\cdots 91}{10\cdots 00}a^{4}+\frac{22\cdots 93}{54\cdots 50}a^{3}-\frac{17\cdots 21}{10\cdots 00}a^{2}-\frac{16\cdots 09}{36\cdots 30}a-\frac{80\cdots 63}{10\cdots 19}$ 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:  \( 592324580865 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 592324580865 \cdot 192}{2\cdot\sqrt{148534373731547764810930925932451123761}}\cr\approx \mathstrut & 11.3333017221292 \end{aligned}\] (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 - x^15 - 21*x^14 + 279*x^13 - 1802*x^12 - 13059*x^11 + 276033*x^10 + 1587075*x^9 - 11621499*x^8 - 96153882*x^7 + 330208021*x^6 + 4122671339*x^5 - 1269953021*x^4 - 86907862279*x^3 - 115905519180*x^2 + 675228426700*x + 1550616680000) 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 - x^15 - 21*x^14 + 279*x^13 - 1802*x^12 - 13059*x^11 + 276033*x^10 + 1587075*x^9 - 11621499*x^8 - 96153882*x^7 + 330208021*x^6 + 4122671339*x^5 - 1269953021*x^4 - 86907862279*x^3 - 115905519180*x^2 + 675228426700*x + 1550616680000, 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 - x^15 - 21*x^14 + 279*x^13 - 1802*x^12 - 13059*x^11 + 276033*x^10 + 1587075*x^9 - 11621499*x^8 - 96153882*x^7 + 330208021*x^6 + 4122671339*x^5 - 1269953021*x^4 - 86907862279*x^3 - 115905519180*x^2 + 675228426700*x + 1550616680000); 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 - x^15 - 21*x^14 + 279*x^13 - 1802*x^12 - 13059*x^11 + 276033*x^10 + 1587075*x^9 - 11621499*x^8 - 96153882*x^7 + 330208021*x^6 + 4122671339*x^5 - 1269953021*x^4 - 86907862279*x^3 - 115905519180*x^2 + 675228426700*x + 1550616680000); 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

\(\Q(\sqrt{41}) \), 4.4.94352849.1, 4.0.62197.1, 4.0.2550077.1, 8.0.8902460114416801.2

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: 16.8.148534373731547764810930925932451123761.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }^{4}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ R R ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(37\) Copy content Toggle raw display 37.2.4.6a1.2$x^{8} + 132 x^{7} + 6542 x^{6} + 144540 x^{5} + 1212081 x^{4} + 289080 x^{3} + 26168 x^{2} + 1056 x + 53$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
37.2.4.6a1.2$x^{8} + 132 x^{7} + 6542 x^{6} + 144540 x^{5} + 1212081 x^{4} + 289080 x^{3} + 26168 x^{2} + 1056 x + 53$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
\(41\) Copy content Toggle raw display 41.2.4.6a1.2$x^{8} + 152 x^{7} + 8688 x^{6} + 222224 x^{5} + 2189320 x^{4} + 1333344 x^{3} + 312768 x^{2} + 32832 x + 1337$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
41.2.4.6a1.2$x^{8} + 152 x^{7} + 8688 x^{6} + 222224 x^{5} + 2189320 x^{4} + 1333344 x^{3} + 312768 x^{2} + 32832 x + 1337$$4$$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)