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Properties

Label 32.0.535...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $5.350\times 10^{55}$
Root discriminant \(55.15\)
Ramified primes $2,5,17$
Class number not computed
Class group not computed
Galois group $C_2^3\times C_4$ (as 32T34)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^32 - 49*x^28 + 2145*x^24 - 92561*x^20 + 3986369*x^16 - 23695616*x^12 + 140574720*x^8 - 822083584*x^4 + 4294967296)
 
Copy content gp:K = bnfinit(y^32 - 49*y^28 + 2145*y^24 - 92561*y^20 + 3986369*y^16 - 23695616*y^12 + 140574720*y^8 - 822083584*y^4 + 4294967296, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 49*x^28 + 2145*x^24 - 92561*x^20 + 3986369*x^16 - 23695616*x^12 + 140574720*x^8 - 822083584*x^4 + 4294967296);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 - 49*x^28 + 2145*x^24 - 92561*x^20 + 3986369*x^16 - 23695616*x^12 + 140574720*x^8 - 822083584*x^4 + 4294967296)
 

\( x^{32} - 49 x^{28} + 2145 x^{24} - 92561 x^{20} + 3986369 x^{16} - 23695616 x^{12} + 140574720 x^{8} + \cdots + 4294967296 \) 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:  $32$
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:  $[0, 16]$
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:   \(53503546288734745169056818528256000000000000000000000000\) \(\medspace = 2^{64}\cdot 5^{24}\cdot 17^{16}\) 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:  \(55.15\)
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:  $2^{2}5^{3/4}17^{1/2}\approx 55.14573827051111$
Ramified primes:   \(2\), \(5\), \(17\) 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:  \(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_2^3\times C_4$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(680=2^{3}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{680}(1,·)$, $\chi_{680}(647,·)$, $\chi_{680}(137,·)$, $\chi_{680}(271,·)$, $\chi_{680}(273,·)$, $\chi_{680}(67,·)$, $\chi_{680}(407,·)$, $\chi_{680}(409,·)$, $\chi_{680}(543,·)$, $\chi_{680}(33,·)$, $\chi_{680}(679,·)$, $\chi_{680}(169,·)$, $\chi_{680}(171,·)$, $\chi_{680}(307,·)$, $\chi_{680}(441,·)$, $\chi_{680}(443,·)$, $\chi_{680}(577,·)$, $\chi_{680}(579,·)$, $\chi_{680}(69,·)$, $\chi_{680}(203,·)$, $\chi_{680}(339,·)$, $\chi_{680}(341,·)$, $\chi_{680}(477,·)$, $\chi_{680}(101,·)$, $\chi_{680}(611,·)$, $\chi_{680}(613,·)$, $\chi_{680}(103,·)$, $\chi_{680}(237,·)$, $\chi_{680}(239,·)$, $\chi_{680}(373,·)$, $\chi_{680}(509,·)$, $\chi_{680}(511,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{9}a^{16}-\frac{2}{9}a^{12}+\frac{4}{9}a^{8}+\frac{1}{9}a^{4}-\frac{2}{9}$, $\frac{1}{36}a^{17}+\frac{7}{36}a^{13}+\frac{13}{36}a^{9}-\frac{17}{36}a^{5}-\frac{11}{36}a$, $\frac{1}{144}a^{18}-\frac{65}{144}a^{14}+\frac{49}{144}a^{10}-\frac{17}{144}a^{6}-\frac{47}{144}a^{2}$, $\frac{1}{576}a^{19}+\frac{79}{576}a^{15}-\frac{95}{576}a^{11}-\frac{17}{576}a^{7}-\frac{191}{576}a^{3}$, $\frac{1}{9184594176}a^{20}-\frac{113}{2304}a^{16}-\frac{863}{2304}a^{12}-\frac{209}{2304}a^{8}+\frac{769}{2304}a^{4}+\frac{11866546}{35877321}$, $\frac{1}{36738376704}a^{21}-\frac{113}{9216}a^{17}-\frac{863}{9216}a^{13}-\frac{2513}{9216}a^{9}+\frac{3073}{9216}a^{5}+\frac{47743867}{143509284}a$, $\frac{1}{146953506816}a^{22}-\frac{113}{36864}a^{18}+\frac{17569}{36864}a^{14}+\frac{15919}{36864}a^{10}-\frac{6143}{36864}a^{6}-\frac{95765417}{574037136}a^{2}$, $\frac{1}{587814027264}a^{23}-\frac{113}{147456}a^{19}+\frac{54433}{147456}a^{15}+\frac{15919}{147456}a^{11}+\frac{30721}{147456}a^{7}+\frac{1052308855}{2296148544}a^{3}$, $\frac{1}{2351256109056}a^{24}-\frac{49}{2351256109056}a^{20}-\frac{9055}{589824}a^{16}+\frac{13871}{589824}a^{12}-\frac{131071}{589824}a^{8}+\frac{1360649765}{3061531392}a^{4}+\frac{3988514}{35877321}$, $\frac{1}{9405024436224}a^{25}-\frac{49}{9405024436224}a^{21}-\frac{9055}{2359296}a^{17}-\frac{575953}{2359296}a^{13}-\frac{131071}{2359296}a^{9}-\frac{4762413019}{12246125568}a^{5}+\frac{39865835}{143509284}a$, $\frac{1}{37620097744896}a^{26}-\frac{49}{37620097744896}a^{22}-\frac{9055}{9437184}a^{18}+\frac{4142639}{9437184}a^{14}-\frac{2490367}{9437184}a^{10}+\frac{7483712549}{48984502272}a^{6}+\frac{39865835}{574037136}a^{2}$, $\frac{1}{150480390979584}a^{27}-\frac{49}{150480390979584}a^{23}-\frac{9055}{37748736}a^{19}+\frac{13579823}{37748736}a^{15}+\frac{16384001}{37748736}a^{11}+\frac{56468214821}{195938009088}a^{7}-\frac{534171301}{2296148544}a^{3}$, $\frac{1}{601921563918336}a^{28}-\frac{49}{601921563918336}a^{24}+\frac{715}{200640521306112}a^{20}+\frac{6895151}{150994944}a^{16}+\frac{16777217}{150994944}a^{12}-\frac{58055716681}{261250678784}a^{8}+\frac{4082044001}{9184594176}a^{4}+\frac{3986320}{35877321}$, $\frac{1}{24\cdots 44}a^{29}-\frac{49}{24\cdots 44}a^{25}+\frac{715}{802562085224448}a^{21}+\frac{6895151}{603979776}a^{17}+\frac{167772161}{603979776}a^{13}-\frac{58055716681}{1045002715136}a^{9}-\frac{14287144351}{36738376704}a^{5}+\frac{39863641}{143509284}a$, $\frac{1}{96\cdots 76}a^{30}-\frac{49}{96\cdots 76}a^{26}+\frac{715}{32\cdots 92}a^{22}+\frac{6895151}{2415919104}a^{18}-\frac{436207615}{2415919104}a^{14}-\frac{1103058431817}{4180010860544}a^{10}+\frac{22451232353}{146953506816}a^{6}+\frac{39863641}{574037136}a^{2}$, $\frac{1}{38\cdots 04}a^{31}-\frac{49}{38\cdots 04}a^{27}+\frac{715}{12\cdots 68}a^{23}+\frac{6895151}{9663676416}a^{19}-\frac{436207615}{9663676416}a^{15}+\frac{7256963289271}{16720043442176}a^{11}+\frac{169404739169}{587814027264}a^{7}-\frac{534173495}{2296148544}a^{3}$ 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:  not computed
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:  not computed
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:   data not computed

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:  $15$
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:   \( -\frac{181}{587814027264} a^{27} - \frac{25963647845}{587814027264} a^{7} \)  (order $40$) 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:  not computed
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:  not computed
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot R \cdot h}{40\cdot\sqrt{53503546288734745169056818528256000000000000000000000000}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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^32 - 49*x^28 + 2145*x^24 - 92561*x^20 + 3986369*x^16 - 23695616*x^12 + 140574720*x^8 - 822083584*x^4 + 4294967296) 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^32 - 49*x^28 + 2145*x^24 - 92561*x^20 + 3986369*x^16 - 23695616*x^12 + 140574720*x^8 - 822083584*x^4 + 4294967296, 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^32 - 49*x^28 + 2145*x^24 - 92561*x^20 + 3986369*x^16 - 23695616*x^12 + 140574720*x^8 - 822083584*x^4 + 4294967296); 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^32 - 49*x^28 + 2145*x^24 - 92561*x^20 + 3986369*x^16 - 23695616*x^12 + 140574720*x^8 - 822083584*x^4 + 4294967296); 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

$C_2^3\times C_4$ (as 32T34):

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)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2^3\times C_4$
Character table for $C_2^3\times C_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-170}) \), \(\Q(\sqrt{170}) \), \(\Q(\sqrt{-85}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{17}) \), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{170})\), \(\Q(i, \sqrt{85})\), \(\Q(\sqrt{2}, \sqrt{-85})\), \(\Q(\sqrt{2}, \sqrt{85})\), \(\Q(\sqrt{-2}, \sqrt{85})\), \(\Q(\sqrt{-2}, \sqrt{-85})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{34})\), \(\Q(i, \sqrt{17})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{-17})\), \(\Q(\sqrt{2}, \sqrt{17})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{17})\), \(\Q(\sqrt{-2}, \sqrt{-17})\), \(\Q(\sqrt{5}, \sqrt{-34})\), \(\Q(\sqrt{-5}, \sqrt{34})\), \(\Q(\sqrt{10}, \sqrt{-17})\), \(\Q(\sqrt{-10}, \sqrt{17})\), \(\Q(\sqrt{5}, \sqrt{34})\), \(\Q(\sqrt{-5}, \sqrt{-34})\), \(\Q(\sqrt{10}, \sqrt{17})\), \(\Q(\sqrt{-10}, \sqrt{-17})\), \(\Q(\sqrt{5}, \sqrt{-17})\), \(\Q(\sqrt{-5}, \sqrt{17})\), \(\Q(\sqrt{10}, \sqrt{-34})\), \(\Q(\sqrt{-10}, \sqrt{34})\), \(\Q(\sqrt{5}, \sqrt{17})\), \(\Q(\sqrt{-5}, \sqrt{-17})\), \(\Q(\sqrt{10}, \sqrt{34})\), \(\Q(\sqrt{-10}, \sqrt{-34})\), 4.4.2312000.1, 4.0.2312000.1, 4.4.578000.1, 4.0.36125.1, \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), 4.0.8000.2, 4.4.8000.1, 8.0.3421020160000.10, 8.0.40960000.1, 8.0.5473632256.1, 8.0.3421020160000.9, 8.0.3421020160000.6, 8.0.13363360000.1, 8.0.3421020160000.1, 8.0.3421020160000.8, 8.0.3421020160000.5, 8.8.213813760000.1, 8.0.3421020160000.4, 8.0.213813760000.1, 8.0.3421020160000.2, 8.0.3421020160000.7, 8.0.3421020160000.3, 8.0.85525504000000.31, 8.0.334084000000.2, \(\Q(\zeta_{20})\), 8.0.1024000000.2, 8.8.85525504000000.3, 8.0.5345344000000.5, 8.0.64000000.2, \(\Q(\zeta_{40})^+\), 8.0.5345344000000.1, 8.0.85525504000000.26, 8.0.64000000.1, 8.0.1024000000.1, 8.0.5345344000000.6, 8.0.85525504000000.22, 8.0.85525504000000.5, 8.0.5345344000000.4, 8.8.85525504000000.2, 8.0.5345344000000.7, 8.8.85525504000000.1, 8.0.5345344000000.3, 8.0.85525504000000.17, 8.0.85525504000000.30, 8.0.334084000000.3, 8.0.334084000000.1, 8.8.5345344000000.2, 8.0.5345344000000.2, 8.8.334084000000.1, 8.0.1305015625.1, 16.0.11703378935126425600000000.1, 16.0.7314611834454016000000000000.9, \(\Q(\zeta_{40})\), 16.0.7314611834454016000000000000.10, 16.0.7314611834454016000000000000.4, 16.0.7314611834454016000000000000.3, 16.0.111612119056000000000000.1, 16.0.7314611834454016000000000000.7, 16.0.7314611834454016000000000000.6, 16.16.7314611834454016000000000000.1, 16.0.28572702478336000000000000.1, 16.0.28572702478336000000000000.2, 16.0.7314611834454016000000000000.5, 16.0.7314611834454016000000000000.11, 16.0.7314611834454016000000000000.8

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{8}$ R ${\href{/padicField/7.4.0.1}{4} }^{8}$ ${\href{/padicField/11.2.0.1}{2} }^{16}$ ${\href{/padicField/13.4.0.1}{4} }^{8}$ R ${\href{/padicField/19.2.0.1}{2} }^{16}$ ${\href{/padicField/23.4.0.1}{4} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{16}$ ${\href{/padicField/31.2.0.1}{2} }^{16}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{16}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{16}$

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
\(2\) Copy content Toggle raw display 2.4.4.32b1.1$x^{16} + 4 x^{13} + 4 x^{12} + 6 x^{10} + 12 x^{9} + 8 x^{8} + 4 x^{7} + 12 x^{6} + 16 x^{5} + 13 x^{4} + 4 x^{3} + 8 x^{2} + 12 x + 9$$4$$4$$32$$C_4\times C_2^2$$$[2, 3]^{4}$$
2.4.4.32b1.1$x^{16} + 4 x^{13} + 4 x^{12} + 6 x^{10} + 12 x^{9} + 8 x^{8} + 4 x^{7} + 12 x^{6} + 16 x^{5} + 13 x^{4} + 4 x^{3} + 8 x^{2} + 12 x + 9$$4$$4$$32$$C_4\times C_2^2$$$[2, 3]^{4}$$
\(5\) Copy content Toggle raw display 5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
\(17\) Copy content Toggle raw display 17.4.2.4a1.2$x^{8} + 14 x^{6} + 20 x^{5} + 55 x^{4} + 140 x^{3} + 142 x^{2} + 60 x + 26$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
17.4.2.4a1.2$x^{8} + 14 x^{6} + 20 x^{5} + 55 x^{4} + 140 x^{3} + 142 x^{2} + 60 x + 26$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
17.4.2.4a1.2$x^{8} + 14 x^{6} + 20 x^{5} + 55 x^{4} + 140 x^{3} + 142 x^{2} + 60 x + 26$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
17.4.2.4a1.2$x^{8} + 14 x^{6} + 20 x^{5} + 55 x^{4} + 140 x^{3} + 142 x^{2} + 60 x + 26$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$

Spectrum of ring of integers

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