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Properties

Label 16.8.438...048.1
Degree $16$
Signature $[8, 4]$
Discriminant $4.384\times 10^{43}$
Root discriminant \(534.10\)
Ramified primes $2,3,13,1327$
Class number $96$ (GRH)
Class group [96] (GRH)
Galois group $C_2^7.(C_2\times S_4)$ (as 16T1675)

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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 - 137*x^14 + 2120*x^12 + 225613*x^10 - 2409767*x^8 - 86396905*x^6 + 63138950*x^4 + 3934885019*x^2 + 3379364677)
 
Copy content gp:K = bnfinit(y^16 - 137*y^14 + 2120*y^12 + 225613*y^10 - 2409767*y^8 - 86396905*y^6 + 63138950*y^4 + 3934885019*y^2 + 3379364677, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 137*x^14 + 2120*x^12 + 225613*x^10 - 2409767*x^8 - 86396905*x^6 + 63138950*x^4 + 3934885019*x^2 + 3379364677);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 137*x^14 + 2120*x^12 + 225613*x^10 - 2409767*x^8 - 86396905*x^6 + 63138950*x^4 + 3934885019*x^2 + 3379364677)
 

\( x^{16} - 137 x^{14} + 2120 x^{12} + 225613 x^{10} - 2409767 x^{8} - 86396905 x^{6} + 63138950 x^{4} + \cdots + 3379364677 \) 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:  $[8, 4]$
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:   \(43843732772164778616036288947079464176386048\) \(\medspace = 2^{16}\cdot 3^{8}\cdot 13^{9}\cdot 1327^{8}\) 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:  \(534.10\)
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:  not computed
Ramified primes:   \(2\), \(3\), \(13\), \(1327\) 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:  \(\Q(\sqrt{13}) \)
$\Aut(K/\Q)$:   $C_2$
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.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{6}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{7}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{9}a^{12}+\frac{1}{9}a^{6}+\frac{1}{3}a^{4}+\frac{2}{9}$, $\frac{1}{9}a^{13}+\frac{1}{9}a^{7}+\frac{1}{3}a^{5}+\frac{2}{9}a$, $\frac{1}{18\cdots 47}a^{14}-\frac{68\cdots 05}{18\cdots 47}a^{12}+\frac{93\cdots 58}{61\cdots 49}a^{10}-\frac{84\cdots 31}{18\cdots 47}a^{8}-\frac{30\cdots 57}{18\cdots 47}a^{6}-\frac{15\cdots 80}{20\cdots 83}a^{4}+\frac{20\cdots 24}{18\cdots 47}a^{2}+\frac{50\cdots 22}{18\cdots 47}$, $\frac{1}{29\cdots 81}a^{15}+\frac{97\cdots 95}{99\cdots 27}a^{13}-\frac{60\cdots 09}{99\cdots 27}a^{11}-\frac{79\cdots 88}{29\cdots 81}a^{9}+\frac{47\cdots 56}{99\cdots 27}a^{7}-\frac{49\cdots 72}{99\cdots 27}a^{5}-\frac{60\cdots 35}{29\cdots 81}a^{3}-\frac{37\cdots 91}{99\cdots 27}a$ 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:  $C_{96}$, which has order $96$ (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:  $C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{96}$, which has order $1536$ (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:  $11$
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:   $\frac{67\cdots 44}{20\cdots 83}a^{14}-\frac{25\cdots 26}{61\cdots 49}a^{12}+\frac{10\cdots 93}{61\cdots 49}a^{10}+\frac{54\cdots 12}{61\cdots 49}a^{8}-\frac{47\cdots 21}{20\cdots 83}a^{6}-\frac{25\cdots 49}{61\cdots 49}a^{4}-\frac{37\cdots 07}{61\cdots 49}a^{2}+\frac{10\cdots 71}{61\cdots 49}$, $\frac{67\cdots 44}{20\cdots 83}a^{14}-\frac{25\cdots 26}{61\cdots 49}a^{12}+\frac{10\cdots 93}{61\cdots 49}a^{10}+\frac{54\cdots 12}{61\cdots 49}a^{8}-\frac{47\cdots 21}{20\cdots 83}a^{6}-\frac{25\cdots 49}{61\cdots 49}a^{4}-\frac{37\cdots 07}{61\cdots 49}a^{2}+\frac{16\cdots 20}{61\cdots 49}$, $\frac{10\cdots 32}{61\cdots 49}a^{14}-\frac{14\cdots 39}{61\cdots 49}a^{12}+\frac{22\cdots 79}{61\cdots 49}a^{10}+\frac{24\cdots 31}{61\cdots 49}a^{8}-\frac{86\cdots 20}{20\cdots 83}a^{6}-\frac{31\cdots 67}{20\cdots 83}a^{4}+\frac{21\cdots 66}{20\cdots 83}a^{2}+\frac{35\cdots 63}{61\cdots 49}$, $\frac{26\cdots 63}{61\cdots 49}a^{14}-\frac{10\cdots 18}{18\cdots 47}a^{12}+\frac{15\cdots 88}{20\cdots 83}a^{10}+\frac{17\cdots 01}{20\cdots 83}a^{8}-\frac{38\cdots 01}{18\cdots 47}a^{6}-\frac{17\cdots 29}{61\cdots 49}a^{4}-\frac{11\cdots 58}{61\cdots 49}a^{2}-\frac{27\cdots 75}{18\cdots 47}$, $\frac{22\cdots 37}{18\cdots 47}a^{14}-\frac{11\cdots 62}{61\cdots 49}a^{12}+\frac{34\cdots 67}{61\cdots 49}a^{10}+\frac{35\cdots 15}{18\cdots 47}a^{8}-\frac{36\cdots 90}{61\cdots 49}a^{6}-\frac{93\cdots 58}{61\cdots 49}a^{4}+\frac{54\cdots 48}{18\cdots 47}a^{2}+\frac{20\cdots 63}{61\cdots 49}$, $\frac{76\cdots 36}{18\cdots 47}a^{14}-\frac{11\cdots 47}{18\cdots 47}a^{12}+\frac{10\cdots 24}{61\cdots 49}a^{10}+\frac{14\cdots 41}{18\cdots 47}a^{8}-\frac{41\cdots 74}{18\cdots 47}a^{6}-\frac{35\cdots 73}{61\cdots 49}a^{4}+\frac{20\cdots 18}{18\cdots 47}a^{2}+\frac{18\cdots 03}{18\cdots 47}$, $\frac{16\cdots 38}{18\cdots 47}a^{14}-\frac{22\cdots 33}{18\cdots 47}a^{12}+\frac{12\cdots 51}{61\cdots 49}a^{10}+\frac{36\cdots 72}{18\cdots 47}a^{8}-\frac{42\cdots 09}{18\cdots 47}a^{6}-\frac{46\cdots 51}{61\cdots 49}a^{4}+\frac{22\cdots 13}{18\cdots 47}a^{2}+\frac{62\cdots 78}{18\cdots 47}$, $\frac{98\cdots 14}{99\cdots 27}a^{15}-\frac{49\cdots 42}{18\cdots 47}a^{14}-\frac{42\cdots 70}{33\cdots 09}a^{13}+\frac{63\cdots 20}{18\cdots 47}a^{12}+\frac{11\cdots 86}{99\cdots 27}a^{11}-\frac{64\cdots 31}{20\cdots 83}a^{10}+\frac{22\cdots 88}{99\cdots 27}a^{9}-\frac{11\cdots 68}{18\cdots 47}a^{8}-\frac{68\cdots 54}{99\cdots 27}a^{7}+\frac{34\cdots 80}{18\cdots 47}a^{6}-\frac{29\cdots 70}{33\cdots 09}a^{5}+\frac{50\cdots 61}{20\cdots 83}a^{4}-\frac{59\cdots 62}{99\cdots 27}a^{3}+\frac{29\cdots 44}{18\cdots 47}a^{2}-\frac{45\cdots 72}{99\cdots 27}a+\frac{22\cdots 79}{18\cdots 47}$, $\frac{52\cdots 39}{33\cdots 09}a^{15}+\frac{27\cdots 45}{18\cdots 47}a^{14}-\frac{24\cdots 90}{33\cdots 09}a^{13}-\frac{42\cdots 94}{61\cdots 49}a^{12}-\frac{32\cdots 55}{99\cdots 27}a^{11}-\frac{19\cdots 66}{61\cdots 49}a^{10}+\frac{57\cdots 17}{99\cdots 27}a^{9}+\frac{10\cdots 82}{18\cdots 47}a^{8}+\frac{49\cdots 94}{33\cdots 09}a^{7}+\frac{28\cdots 75}{20\cdots 83}a^{6}-\frac{18\cdots 34}{99\cdots 27}a^{5}-\frac{10\cdots 53}{61\cdots 49}a^{4}-\frac{22\cdots 70}{33\cdots 09}a^{3}-\frac{12\cdots 47}{18\cdots 47}a^{2}-\frac{58\cdots 41}{99\cdots 27}a-\frac{11\cdots 80}{20\cdots 83}$, $\frac{40\cdots 97}{29\cdots 81}a^{15}-\frac{75\cdots 20}{61\cdots 49}a^{14}-\frac{92\cdots 31}{99\cdots 27}a^{13}+\frac{16\cdots 56}{18\cdots 47}a^{12}-\frac{36\cdots 18}{99\cdots 27}a^{11}+\frac{18\cdots 14}{61\cdots 49}a^{10}+\frac{18\cdots 06}{29\cdots 81}a^{9}-\frac{46\cdots 55}{61\cdots 49}a^{8}+\frac{16\cdots 58}{99\cdots 27}a^{7}-\frac{30\cdots 82}{18\cdots 47}a^{6}-\frac{12\cdots 62}{99\cdots 27}a^{5}+\frac{17\cdots 40}{61\cdots 49}a^{4}-\frac{20\cdots 83}{29\cdots 81}a^{3}+\frac{48\cdots 95}{61\cdots 49}a^{2}-\frac{59\cdots 50}{99\cdots 27}a+\frac{12\cdots 18}{18\cdots 47}$, $\frac{18\cdots 14}{12\cdots 47}a^{15}-\frac{13\cdots 18}{18\cdots 47}a^{14}-\frac{20\cdots 06}{12\cdots 47}a^{13}+\frac{14\cdots 23}{18\cdots 47}a^{12}-\frac{15\cdots 94}{14\cdots 83}a^{11}+\frac{11\cdots 60}{20\cdots 83}a^{10}+\frac{37\cdots 05}{12\cdots 47}a^{9}-\frac{27\cdots 81}{18\cdots 47}a^{8}+\frac{54\cdots 39}{12\cdots 47}a^{7}-\frac{39\cdots 71}{18\cdots 47}a^{6}-\frac{50\cdots 10}{43\cdots 49}a^{5}+\frac{36\cdots 90}{61\cdots 49}a^{4}-\frac{28\cdots 34}{12\cdots 47}a^{3}+\frac{20\cdots 66}{18\cdots 47}a^{2}-\frac{23\cdots 75}{12\cdots 47}a+\frac{17\cdots 98}{18\cdots 47}$ 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:  \( 67240256894900 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 67240256894900 \cdot 96}{2\cdot\sqrt{43843732772164778616036288947079464176386048}}\cr\approx \mathstrut & 0.194480597048614 \end{aligned}\] (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 - 137*x^14 + 2120*x^12 + 225613*x^10 - 2409767*x^8 - 86396905*x^6 + 63138950*x^4 + 3934885019*x^2 + 3379364677) 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 - 137*x^14 + 2120*x^12 + 225613*x^10 - 2409767*x^8 - 86396905*x^6 + 63138950*x^4 + 3934885019*x^2 + 3379364677, 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 - 137*x^14 + 2120*x^12 + 225613*x^10 - 2409767*x^8 - 86396905*x^6 + 63138950*x^4 + 3934885019*x^2 + 3379364677); 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 - 137*x^14 + 2120*x^12 + 225613*x^10 - 2409767*x^8 - 86396905*x^6 + 63138950*x^4 + 3934885019*x^2 + 3379364677); 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

$C_2^7.(C_2\times S_4)$ (as 16T1675):

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 6144
The 54 conjugacy class representatives for $C_2^7.(C_2\times S_4)$
Character table for $C_2^7.(C_2\times S_4)$

Intermediate fields

\(\Q(\sqrt{51753}) \), 4.4.3981.1, 8.8.7173681975339714081.1

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

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.8.43843732772164778616036288947079464176386048.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }$ ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ R ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}$ ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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
\(2\) Copy content Toggle raw display 2.4.2.8a5.2$x^{8} + 2 x^{7} + 4 x^{5} + 6 x^{4} + 2 x^{3} + 7 x^{2} + 6 x + 5$$2$$4$$8$$C_2^3: C_4$$$[2, 2, 2]^{4}$$
2.4.2.8a5.1$x^{8} + 2 x^{7} + 4 x^{5} + 6 x^{4} + 2 x^{3} + 3 x^{2} + 6 x + 5$$2$$4$$8$$(C_8:C_2):C_2$$$[2, 2, 2]^{4}$$
\(3\) Copy content Toggle raw display 3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(13\) Copy content Toggle raw display 13.1.4.3a1.3$x^{4} + 52$$4$$1$$3$$C_4$$$[\ ]_{4}$$
13.6.2.6a1.2$x^{12} + 20 x^{9} + 22 x^{8} + 22 x^{7} + 104 x^{6} + 220 x^{5} + 341 x^{4} + 282 x^{3} + 165 x^{2} + 44 x + 17$$2$$6$$6$$C_6\times C_2$$$[\ ]_{2}^{6}$$
\(1327\) Copy content Toggle raw display Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$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)