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Properties

Label 16.16.150...000.1
Degree $16$
Signature $[16, 0]$
Discriminant $1.504\times 10^{26}$
Root discriminant \(43.26\)
Ramified primes $2,3,5,29$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $\OD_{16}:C_2^2$ (as 16T99)

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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 - 41*x^14 + 677*x^12 - 5843*x^10 + 28585*x^8 - 79537*x^6 + 116702*x^4 - 69544*x^2 + 1)
 
Copy content gp:K = bnfinit(y^16 - 41*y^14 + 677*y^12 - 5843*y^10 + 28585*y^8 - 79537*y^6 + 116702*y^4 - 69544*y^2 + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 41*x^14 + 677*x^12 - 5843*x^10 + 28585*x^8 - 79537*x^6 + 116702*x^4 - 69544*x^2 + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 41*x^14 + 677*x^12 - 5843*x^10 + 28585*x^8 - 79537*x^6 + 116702*x^4 - 69544*x^2 + 1)
 

\( x^{16} - 41x^{14} + 677x^{12} - 5843x^{10} + 28585x^{8} - 79537x^{6} + 116702x^{4} - 69544x^{2} + 1 \) 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:  $[16, 0]$
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:   \(150351248768400000000000000\) \(\medspace = 2^{16}\cdot 3^{12}\cdot 5^{14}\cdot 29^{4}\) 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:  \(43.26\)
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:  $2\cdot 3^{3/4}5^{7/8}29^{1/2}\approx 100.38496905417634$
Ramified primes:   \(2\), \(3\), \(5\), \(29\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2^2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1509309031}a^{14}-\frac{224586322}{1509309031}a^{12}-\frac{36399689}{1509309031}a^{10}+\frac{277455466}{1509309031}a^{8}-\frac{54505536}{1509309031}a^{6}+\frac{572580726}{1509309031}a^{4}-\frac{399038036}{1509309031}a^{2}-\frac{247327078}{1509309031}$, $\frac{1}{1509309031}a^{15}-\frac{224586322}{1509309031}a^{13}-\frac{36399689}{1509309031}a^{11}+\frac{277455466}{1509309031}a^{9}-\frac{54505536}{1509309031}a^{7}+\frac{572580726}{1509309031}a^{5}-\frac{399038036}{1509309031}a^{3}-\frac{247327078}{1509309031}a$ 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:  Trivial group, which has order $1$ (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:  $C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$ (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:  $15$
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:   \( -1 \)  (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:   $\frac{44065}{882121}a^{14}-\frac{1630507}{882121}a^{12}+\frac{23314572}{882121}a^{10}-\frac{164292518}{882121}a^{8}+\frac{603058464}{882121}a^{6}-\frac{1094772282}{882121}a^{4}+\frac{765381860}{882121}a^{2}+\frac{534175}{882121}$, $\frac{51123103}{1509309031}a^{15}-\frac{1930280361}{1509309031}a^{13}+\frac{28464002297}{1509309031}a^{11}-\frac{210577357234}{1509309031}a^{9}+\frac{837905782321}{1509309031}a^{7}-\frac{1766289245086}{1509309031}a^{5}+\frac{1764119352491}{1509309031}a^{3}-\frac{591595470918}{1509309031}a$, $\frac{16114034}{1509309031}a^{15}-\frac{620291768}{1509309031}a^{13}+\frac{9396036949}{1509309031}a^{11}-\frac{72162358898}{1509309031}a^{9}+\frac{302459378651}{1509309031}a^{7}-\frac{687163378110}{1509309031}a^{5}+\frac{772570214010}{1509309031}a^{3}-\frac{323788157616}{1509309031}a$, $\frac{13542064}{1509309031}a^{15}-\frac{506260469}{1509309031}a^{13}+\frac{7374270380}{1509309031}a^{11}-\frac{53838445622}{1509309031}a^{9}+\frac{212372154400}{1509309031}a^{7}-\frac{452171910236}{1509309031}a^{5}+\frac{481592341288}{1509309031}a^{3}-\frac{200112145767}{1509309031}a-1$, $\frac{51123103}{1509309031}a^{15}-\frac{1930280361}{1509309031}a^{13}+\frac{28464002297}{1509309031}a^{11}-\frac{210577357234}{1509309031}a^{9}+\frac{837905782321}{1509309031}a^{7}-\frac{1766289245086}{1509309031}a^{5}+\frac{1764119352491}{1509309031}a^{3}-\frac{591595470918}{1509309031}a-1$, $\frac{51123103}{1509309031}a^{15}-\frac{36696023}{1509309031}a^{14}-\frac{1930280361}{1509309031}a^{13}+\frac{1351270161}{1509309031}a^{12}+\frac{28464002297}{1509309031}a^{11}-\frac{19195625246}{1509309031}a^{10}-\frac{210577357234}{1509309031}a^{9}+\frac{134121102151}{1509309031}a^{8}+\frac{837905782321}{1509309031}a^{7}-\frac{487432014885}{1509309031}a^{6}-\frac{1766289245086}{1509309031}a^{5}+\frac{876560904381}{1509309031}a^{4}+\frac{1764119352491}{1509309031}a^{3}-\frac{610257342449}{1509309031}a^{2}-\frac{591595470918}{1509309031}a-\frac{1816065382}{1509309031}$, $\frac{51123103}{1509309031}a^{15}+\frac{44065}{882121}a^{14}-\frac{1930280361}{1509309031}a^{13}-\frac{1630507}{882121}a^{12}+\frac{28464002297}{1509309031}a^{11}+\frac{23314572}{882121}a^{10}-\frac{210577357234}{1509309031}a^{9}-\frac{164292518}{882121}a^{8}+\frac{837905782321}{1509309031}a^{7}+\frac{603058464}{882121}a^{6}-\frac{1766289245086}{1509309031}a^{5}-\frac{1094772282}{882121}a^{4}+\frac{1764119352491}{1509309031}a^{3}+\frac{765381860}{882121}a^{2}-\frac{591595470918}{1509309031}a+\frac{534175}{882121}$, $\frac{81877905}{1509309031}a^{14}-\frac{3032299158}{1509309031}a^{12}+\frac{43402491946}{1509309031}a^{10}-\frac{306143892264}{1509309031}a^{8}+\frac{1124235649334}{1509309031}a^{6}-\frac{2039153690675}{1509309031}a^{4}+\frac{1422343281558}{1509309031}a^{2}-\frac{349748002}{1509309031}$, $\frac{24427891}{1509309031}a^{14}-\frac{908833374}{1509309031}a^{12}+\frac{13103416162}{1509309031}a^{10}-\frac{93514459262}{1509309031}a^{8}+\frac{349940795638}{1509309031}a^{6}-\frac{654328197122}{1509309031}a^{4}+\frac{480366840880}{1509309031}a^{2}-\frac{3900872327}{1509309031}$, $\frac{96157089}{1509309031}a^{14}-\frac{3556476063}{1509309031}a^{12}+\frac{50824650733}{1509309031}a^{10}-\frac{357910867559}{1509309031}a^{8}+\frac{1313144568138}{1509309031}a^{6}-\frac{2385128030399}{1509309031}a^{4}+\frac{1671256824125}{1509309031}a^{2}+\frac{7524871639}{1509309031}$, $\frac{62059662}{1509309031}a^{15}-\frac{112091238}{1509309031}a^{14}-\frac{2272265951}{1509309031}a^{13}+\frac{4141067638}{1509309031}a^{12}+\frac{31939761613}{1509309031}a^{11}-\frac{59086857938}{1509309031}a^{10}-\frac{218503203217}{1509309031}a^{9}+\frac{415225600449}{1509309031}a^{8}+\frac{758013215535}{1509309031}a^{7}-\frac{1519265046789}{1509309031}a^{6}-\frac{1204731381480}{1509309031}a^{5}+\frac{2749716278883}{1509309031}a^{4}+\frac{462233857812}{1509309031}a^{3}-\frac{1919825704909}{1509309031}a^{2}+\frac{435733732210}{1509309031}a-\frac{2730038807}{1509309031}$, $\frac{29113655}{1509309031}a^{15}-\frac{67606604}{1509309031}a^{14}-\frac{1236033787}{1509309031}a^{13}+\frac{2502605216}{1509309031}a^{12}+\frac{21273991107}{1509309031}a^{11}-\frac{35810300662}{1509309031}a^{10}-\frac{192417101095}{1509309031}a^{9}+\frac{252674955379}{1509309031}a^{8}+\frac{989046424974}{1509309031}a^{7}-\frac{929775427953}{1509309031}a^{6}-\frac{2890890882620}{1509309031}a^{5}+\frac{1696887417676}{1509309031}a^{4}+\frac{4445695145977}{1509309031}a^{3}-\frac{1203399102427}{1509309031}a^{2}-\frac{2766861808454}{1509309031}a+\frac{10895073527}{1509309031}$, $\frac{35009069}{1509309031}a^{15}+\frac{46147482}{1509309031}a^{14}-\frac{1309988593}{1509309031}a^{13}-\frac{1693360055}{1509309031}a^{12}+\frac{19067965348}{1509309031}a^{11}+\frac{23946573197}{1509309031}a^{10}-\frac{138414998336}{1509309031}a^{9}-\frac{166396441075}{1509309031}a^{8}+\frac{535446403670}{1509309031}a^{7}+\frac{601149876106}{1509309031}a^{6}-\frac{1079125866976}{1509309031}a^{5}-\frac{1075150836692}{1509309031}a^{4}+\frac{991549138481}{1509309031}a^{3}+\frac{746051304135}{1509309031}a^{2}-\frac{269316622333}{1509309031}a-\frac{1715046023}{1509309031}$, $\frac{17239096}{1509309031}a^{15}-\frac{9451459}{1509309031}a^{14}-\frac{732014022}{1509309031}a^{13}+\frac{342089894}{1509309031}a^{12}+\frac{12589687416}{1509309031}a^{11}-\frac{4750947951}{1509309031}a^{10}-\frac{113606005263}{1509309031}a^{9}+\frac{32275338924}{1509309031}a^{8}+\frac{581290826653}{1509309031}a^{7}-\frac{113717861221}{1509309031}a^{6}-\frac{1687206031761}{1509309031}a^{5}+\frac{198589932311}{1509309031}a^{4}+\frac{2571597682649}{1509309031}a^{3}-\frac{134284652655}{1509309031}a^{2}-\frac{1586145272177}{1509309031}a-\frac{5524742781}{1509309031}$, $\frac{2689680}{52045139}a^{15}-\frac{218397034}{1509309031}a^{14}-\frac{101864564}{52045139}a^{13}+\frac{8082200170}{1509309031}a^{12}+\frac{1509991164}{52045139}a^{11}-\frac{115592766046}{1509309031}a^{10}-\frac{11276451653}{52045139}a^{9}+\frac{814883951975}{1509309031}a^{8}+\frac{45658543210}{52045139}a^{7}-\frac{2993441491761}{1509309031}a^{6}-\frac{99547763736}{52045139}a^{5}+\frac{5443198166680}{1509309031}a^{4}+\frac{106656765463}{52045139}a^{3}-\frac{3822535827347}{1509309031}a^{2}-\frac{42242002291}{52045139}a+\frac{4539199584}{1509309031}$ 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:  \( 61330889.5255 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 61330889.5255 \cdot 1}{2\cdot\sqrt{150351248768400000000000000}}\cr\approx \mathstrut & 0.163898764125 \end{aligned}\] (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 - 41*x^14 + 677*x^12 - 5843*x^10 + 28585*x^8 - 79537*x^6 + 116702*x^4 - 69544*x^2 + 1) 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 - 41*x^14 + 677*x^12 - 5843*x^10 + 28585*x^8 - 79537*x^6 + 116702*x^4 - 69544*x^2 + 1, 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 - 41*x^14 + 677*x^12 - 5843*x^10 + 28585*x^8 - 79537*x^6 + 116702*x^4 - 69544*x^2 + 1); 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 - 41*x^14 + 677*x^12 - 5843*x^10 + 28585*x^8 - 79537*x^6 + 116702*x^4 - 69544*x^2 + 1); 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

$\OD_{16}:C_2^2$ (as 16T99):

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 64
The 22 conjugacy class representatives for $\OD_{16}:C_2^2$
Character table for $\OD_{16}:C_2^2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{15}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{3}, \sqrt{5})\), 8.8.47897578125.1, 8.8.12261780000000.3, \(\Q(\zeta_{60})^+\)

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: 16.16.126445400214224400000000000000.3

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 R ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ R ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$

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 2.8.2.16a1.1$x^{16} + 2 x^{12} + 2 x^{11} + 2 x^{10} + 5 x^{8} + 2 x^{7} + 3 x^{6} + 2 x^{5} + 5 x^{4} + 4 x^{3} + 4 x^{2} + 5$$2$$8$$16$$C_8\times C_2$$$[2]^{8}$$
\(3\) Copy content Toggle raw display 3.4.4.12a1.1$x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 24 x^{12} + 48 x^{11} + 96 x^{10} + 64 x^{9} + 24 x^{8} + 96 x^{7} + 96 x^{6} + 32 x^{4} + 64 x^{3} + 3 x^{2} + 16$$4$$4$$12$$C_8: C_2$$$[\ ]_{4}^{4}$$
\(5\) Copy content Toggle raw display 5.2.8.14a1.5$x^{16} + 32 x^{15} + 464 x^{14} + 4032 x^{13} + 23408 x^{12} + 95872 x^{11} + 285376 x^{10} + 627456 x^{9} + 1027168 x^{8} + 1254912 x^{7} + 1141504 x^{6} + 766976 x^{5} + 374528 x^{4} + 129024 x^{3} + 29696 x^{2} + 4106 x + 266$$8$$2$$14$$C_8: C_2$$$[\ ]_{8}^{2}$$
\(29\) Copy content Toggle raw display 29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.2.2a1.2$x^{4} + 48 x^{3} + 580 x^{2} + 96 x + 33$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
29.2.2.2a1.2$x^{4} + 48 x^{3} + 580 x^{2} + 96 x + 33$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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