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Properties

Label 20.4.267...000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2.677\times 10^{29}$
Root discriminant \(29.61\)
Ramified primes $2,5,3803$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^8.(C_4\times S_5)$ (as 20T802)

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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^20 + 8*x^18 + 29*x^16 + 97*x^14 + 296*x^12 + 350*x^10 - 1255*x^8 - 3750*x^6 - 2925*x^4 - 375*x^2 + 125)
 
Copy content gp:K = bnfinit(y^20 + 8*y^18 + 29*y^16 + 97*y^14 + 296*y^12 + 350*y^10 - 1255*y^8 - 3750*y^6 - 2925*y^4 - 375*y^2 + 125, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 8*x^18 + 29*x^16 + 97*x^14 + 296*x^12 + 350*x^10 - 1255*x^8 - 3750*x^6 - 2925*x^4 - 375*x^2 + 125);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 + 8*x^18 + 29*x^16 + 97*x^14 + 296*x^12 + 350*x^10 - 1255*x^8 - 3750*x^6 - 2925*x^4 - 375*x^2 + 125)
 

\( x^{20} + 8 x^{18} + 29 x^{16} + 97 x^{14} + 296 x^{12} + 350 x^{10} - 1255 x^{8} - 3750 x^{6} + \cdots + 125 \) 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:  $20$
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:  $[4, 8]$
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:   \(267741240538215680000000000000\) \(\medspace = 2^{20}\cdot 5^{13}\cdot 3803^{4}\) 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:  \(29.61\)
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:   \(2\), \(5\), \(3803\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5}a^{8}-\frac{1}{5}a^{6}-\frac{1}{5}a^{4}$, $\frac{1}{5}a^{9}-\frac{1}{5}a^{7}-\frac{1}{5}a^{5}$, $\frac{1}{5}a^{10}-\frac{2}{5}a^{6}-\frac{1}{5}a^{4}$, $\frac{1}{5}a^{11}-\frac{2}{5}a^{7}-\frac{1}{5}a^{5}$, $\frac{1}{5}a^{12}+\frac{2}{5}a^{6}-\frac{2}{5}a^{4}$, $\frac{1}{5}a^{13}+\frac{2}{5}a^{7}-\frac{2}{5}a^{5}$, $\frac{1}{35}a^{14}+\frac{2}{35}a^{12}+\frac{3}{35}a^{8}+\frac{11}{35}a^{6}-\frac{1}{7}a^{4}-\frac{2}{7}a^{2}+\frac{1}{7}$, $\frac{1}{35}a^{15}+\frac{2}{35}a^{13}+\frac{3}{35}a^{9}+\frac{11}{35}a^{7}-\frac{1}{7}a^{5}-\frac{2}{7}a^{3}+\frac{1}{7}a$, $\frac{1}{175}a^{16}-\frac{2}{175}a^{14}-\frac{1}{175}a^{12}+\frac{17}{175}a^{10}+\frac{6}{175}a^{8}-\frac{1}{5}a^{6}+\frac{16}{35}a^{4}-\frac{1}{7}a^{2}+\frac{2}{7}$, $\frac{1}{175}a^{17}-\frac{2}{175}a^{15}-\frac{1}{175}a^{13}+\frac{17}{175}a^{11}+\frac{6}{175}a^{9}-\frac{1}{5}a^{7}+\frac{16}{35}a^{5}-\frac{1}{7}a^{3}+\frac{2}{7}a$, $\frac{1}{11790626575}a^{18}-\frac{5614242}{11790626575}a^{16}+\frac{44563104}{11790626575}a^{14}-\frac{743211663}{11790626575}a^{12}-\frac{125362}{1684375225}a^{10}-\frac{314687}{36278851}a^{8}+\frac{1021243557}{2358125315}a^{6}+\frac{55949826}{471625063}a^{4}+\frac{130089273}{471625063}a^{2}+\frac{87995896}{471625063}$, $\frac{1}{11790626575}a^{19}-\frac{5614242}{11790626575}a^{17}+\frac{44563104}{11790626575}a^{15}-\frac{743211663}{11790626575}a^{13}-\frac{125362}{1684375225}a^{11}-\frac{314687}{36278851}a^{9}+\frac{1021243557}{2358125315}a^{7}+\frac{55949826}{471625063}a^{5}+\frac{130089273}{471625063}a^{3}+\frac{87995896}{471625063}a$ 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:  Trivial group, which has order $1$ (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:   $\frac{48808973}{11790626575}a^{18}+\frac{363911082}{11790626575}a^{16}+\frac{1211758761}{11790626575}a^{14}+\frac{4049541033}{11790626575}a^{12}+\frac{12167535509}{11790626575}a^{10}+\frac{112277143}{129567325}a^{8}-\frac{1920201089}{336875045}a^{6}-\frac{4172429554}{336875045}a^{4}-\frac{324925347}{67375009}a^{2}+\frac{790997290}{471625063}$, $\frac{193637331}{11790626575}a^{18}+\frac{191589461}{1684375225}a^{16}+\frac{4210332481}{11790626575}a^{14}+\frac{14479052748}{11790626575}a^{12}+\frac{42380832384}{11790626575}a^{10}+\frac{1881330463}{906971275}a^{8}-\frac{52692653666}{2358125315}a^{6}-\frac{17763782789}{471625063}a^{4}-\frac{805691650}{67375009}a^{2}+\frac{822273560}{471625063}$, $\frac{85576}{9593675}a^{18}+\frac{566193}{9593675}a^{16}+\frac{242327}{1370525}a^{14}+\frac{5956207}{9593675}a^{12}+\frac{17143566}{9593675}a^{10}+\frac{97872}{147595}a^{8}-\frac{3297881}{274105}a^{6}-\frac{6303665}{383747}a^{4}-\frac{1127522}{383747}a^{2}-\frac{317216}{383747}$, $\frac{8713638}{2358125315}a^{18}+\frac{49977441}{2358125315}a^{16}+\frac{3497631}{67375009}a^{14}+\frac{13105541}{67375009}a^{12}+\frac{1223662898}{2358125315}a^{10}-\frac{63692781}{181394255}a^{8}-\frac{2429295653}{471625063}a^{6}-\frac{5592304269}{2358125315}a^{4}+\frac{2326548685}{471625063}a^{2}+\frac{307148715}{471625063}$, $\frac{33886503}{1684375225}a^{18}+\frac{1591013432}{11790626575}a^{16}+\frac{4822417906}{11790626575}a^{14}+\frac{16772522423}{11790626575}a^{12}+\frac{48499146874}{11790626575}a^{10}+\frac{1562866558}{906971275}a^{8}-\frac{9262733133}{336875045}a^{6}-\frac{94411218214}{2358125315}a^{4}-\frac{3313292865}{471625063}a^{2}+\frac{657797212}{471625063}$, $\frac{5909246}{1684375225}a^{18}+\frac{259700962}{11790626575}a^{16}+\frac{721932506}{11790626575}a^{14}+\frac{2569359503}{11790626575}a^{12}+\frac{7286394264}{11790626575}a^{10}+\frac{10044947}{906971275}a^{8}-\frac{1634152176}{336875045}a^{6}-\frac{11036321262}{2358125315}a^{4}+\frac{1078390331}{471625063}a^{2}+\frac{144332145}{471625063}$, $\frac{11931186}{906971275}a^{18}+\frac{88194921}{906971275}a^{16}+\frac{296316623}{906971275}a^{14}+\frac{1002102284}{906971275}a^{12}+\frac{427658601}{129567325}a^{10}+\frac{2615271893}{906971275}a^{8}-\frac{3169371087}{181394255}a^{6}-\frac{1000442283}{25913465}a^{4}-\frac{754536641}{36278851}a^{2}+\frac{33388065}{36278851}$, $\frac{101211583}{11790626575}a^{19}-\frac{103928029}{11790626575}a^{18}+\frac{125297812}{2358125315}a^{17}-\frac{4077442}{67375009}a^{16}+\frac{342048916}{2358125315}a^{15}-\frac{445961531}{2358125315}a^{14}+\frac{1218223033}{2358125315}a^{13}-\frac{1545413196}{2358125315}a^{12}+\frac{3374493023}{2358125315}a^{11}-\frac{4524162029}{2358125315}a^{10}-\frac{27098834}{129567325}a^{9}-\frac{1018660686}{906971275}a^{8}-\frac{28929481978}{2358125315}a^{7}+\frac{5506273326}{471625063}a^{6}-\frac{27592139741}{2358125315}a^{5}+\frac{44228836074}{2358125315}a^{4}+\frac{2051103805}{471625063}a^{3}+\frac{2215567815}{471625063}a^{2}+\frac{1274341697}{471625063}a-\frac{880991654}{471625063}$, $\frac{4902566}{1684375225}a^{19}+\frac{10543688}{11790626575}a^{18}+\frac{9771347}{336875045}a^{17}-\frac{8972213}{11790626575}a^{16}+\frac{289752461}{2358125315}a^{15}-\frac{305501639}{11790626575}a^{14}+\frac{188935133}{471625063}a^{13}-\frac{791869352}{11790626575}a^{12}+\frac{430294546}{336875045}a^{11}-\frac{3270844881}{11790626575}a^{10}+\frac{2008935063}{906971275}a^{9}-\frac{1141090009}{906971275}a^{8}-\frac{7260982156}{2358125315}a^{7}-\frac{3901085522}{2358125315}a^{6}-\frac{43573688463}{2358125315}a^{5}+\frac{17046735786}{2358125315}a^{4}-\frac{9643687112}{471625063}a^{3}+\frac{5057062463}{471625063}a^{2}-\frac{2884933711}{471625063}a+\frac{842643229}{471625063}$, $\frac{96798042}{11790626575}a^{19}+\frac{21302829}{11790626575}a^{18}+\frac{739305043}{11790626575}a^{17}+\frac{162435552}{11790626575}a^{16}+\frac{2558426224}{11790626575}a^{15}+\frac{580535451}{11790626575}a^{14}+\frac{8566648457}{11790626575}a^{13}+\frac{2024098338}{11790626575}a^{12}+\frac{3687605263}{1684375225}a^{11}+\frac{853857752}{1684375225}a^{10}+\frac{2007596544}{906971275}a^{9}+\frac{82258682}{181394255}a^{8}-\frac{24884323502}{2358125315}a^{7}-\frac{5997976239}{2358125315}a^{6}-\frac{12435911744}{471625063}a^{5}-\frac{14299655283}{2358125315}a^{4}-\frac{7929997290}{471625063}a^{3}-\frac{1825610016}{471625063}a^{2}-\frac{704357105}{471625063}a+\frac{57409575}{471625063}$, $\frac{678291052}{11790626575}a^{19}+\frac{308788936}{11790626575}a^{18}+\frac{5562200453}{11790626575}a^{17}+\frac{2457951408}{11790626575}a^{16}+\frac{20688712509}{11790626575}a^{15}+\frac{1268784857}{1684375225}a^{14}+\frac{69335940372}{11790626575}a^{13}+\frac{29921865182}{11790626575}a^{12}+\frac{213018763046}{11790626575}a^{11}+\frac{13050952873}{1684375225}a^{10}+\frac{21050629494}{906971275}a^{9}+\frac{235362904}{25913465}a^{8}-\frac{32572607153}{471625063}a^{7}-\frac{75867247343}{2358125315}a^{6}-\frac{541448900844}{2358125315}a^{5}-\frac{226318188319}{2358125315}a^{4}-\frac{96774372480}{471625063}a^{3}-\frac{37499649809}{471625063}a^{2}-\frac{24051988236}{471625063}a-\frac{8877720214}{471625063}$ 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:  \( 6003782.69041 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 6003782.69041 \cdot 1}{2\cdot\sqrt{267741240538215680000000000000}}\cr\approx \mathstrut & 0.225473892594 \end{aligned}\] (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^20 + 8*x^18 + 29*x^16 + 97*x^14 + 296*x^12 + 350*x^10 - 1255*x^8 - 3750*x^6 - 2925*x^4 - 375*x^2 + 125) 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^20 + 8*x^18 + 29*x^16 + 97*x^14 + 296*x^12 + 350*x^10 - 1255*x^8 - 3750*x^6 - 2925*x^4 - 375*x^2 + 125, 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^20 + 8*x^18 + 29*x^16 + 97*x^14 + 296*x^12 + 350*x^10 - 1255*x^8 - 3750*x^6 - 2925*x^4 - 375*x^2 + 125); 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^20 + 8*x^18 + 29*x^16 + 97*x^14 + 296*x^12 + 350*x^10 - 1255*x^8 - 3750*x^6 - 2925*x^4 - 375*x^2 + 125); 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

$C_2^8.(C_4\times S_5)$ (as 20T802):

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 non-solvable group of order 122880
The 138 conjugacy class representatives for $C_2^8.(C_4\times S_5)$
Character table for $C_2^8.(C_4\times S_5)$

Intermediate fields

\(\Q(\sqrt{5}) \), 5.3.19015.1, 10.6.45196278125.1

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 20 siblings: data not computed
Degree 40 siblings: data not computed
Minimal sibling: 20.4.274167030311132856320000000000000.4

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ R ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{8}$ ${\href{/padicField/11.10.0.1}{10} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/padicField/29.10.0.1}{10} }^{2}$ ${\href{/padicField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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 2.10.2.20a22.1$x^{20} + 2 x^{17} + 2 x^{16} + 2 x^{15} + 2 x^{14} + 4 x^{13} + 7 x^{12} + 6 x^{11} + 7 x^{10} + 6 x^{9} + 8 x^{8} + 12 x^{7} + 9 x^{6} + 8 x^{5} + 9 x^{4} + 8 x^{3} + 7 x^{2} + 4 x + 3$$2$$10$$20$20T344not computed
\(5\) Copy content Toggle raw display 5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.3.4.9a1.3$x^{12} + 12 x^{10} + 12 x^{9} + 54 x^{8} + 108 x^{7} + 162 x^{6} + 324 x^{5} + 405 x^{4} + 432 x^{3} + 486 x^{2} + 324 x + 86$$4$$3$$9$$C_{12}$$$[\ ]_{4}^{3}$$
\(3803\) Copy content Toggle raw display Deg $8$$2$$4$$4$
Deg $12$$1$$12$$0$$C_{12}$$$[\ ]^{12}$$

Spectrum of ring of integers

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