Properties

Label 20.0.975...000.1
Degree $20$
Signature $(0, 10)$
Discriminant $9.753\times 10^{32}$
Root discriminant \(44.61\)
Ramified primes $2,5,47$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_2^9.A_5^2.C_4$ (as 20T1025)

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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 + 30*x^18 + 395*x^16 + 2930*x^14 + 13305*x^12 + 39200*x^10 + 79900*x^8 + 109000*x^6 + 88000*x^4 + 50000*x^2 + 10000)
 
Copy content gp:K = bnfinit(y^20 + 30*y^18 + 395*y^16 + 2930*y^14 + 13305*y^12 + 39200*y^10 + 79900*y^8 + 109000*y^6 + 88000*y^4 + 50000*y^2 + 10000, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 30*x^18 + 395*x^16 + 2930*x^14 + 13305*x^12 + 39200*x^10 + 79900*x^8 + 109000*x^6 + 88000*x^4 + 50000*x^2 + 10000);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 + 30*x^18 + 395*x^16 + 2930*x^14 + 13305*x^12 + 39200*x^10 + 79900*x^8 + 109000*x^6 + 88000*x^4 + 50000*x^2 + 10000)
 

\( x^{20} + 30 x^{18} + 395 x^{16} + 2930 x^{14} + 13305 x^{12} + 39200 x^{10} + 79900 x^{8} + 109000 x^{6} + \cdots + 10000 \) 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:  $(0, 10)$
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:   \(975310301665730560000000000000000\) \(\medspace = 2^{28}\cdot 5^{16}\cdot 47^{8}\) 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:  \(44.61\)
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:   \(2\), \(5\), \(47\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{10}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{10}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{10}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{20}a^{11}-\frac{1}{20}a^{10}-\frac{1}{20}a^{8}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{100}a^{12}-\frac{1}{20}a^{10}-\frac{1}{20}a^{9}-\frac{1}{20}a^{8}-\frac{1}{4}a^{7}-\frac{1}{5}a^{6}+\frac{1}{4}a^{5}-\frac{1}{5}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{100}a^{13}-\frac{1}{20}a^{9}-\frac{1}{5}a^{7}-\frac{1}{2}a^{6}+\frac{1}{20}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{100}a^{14}-\frac{1}{20}a^{10}+\frac{1}{20}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{200}a^{15}-\frac{1}{40}a^{11}+\frac{1}{40}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{1000}a^{16}-\frac{1}{200}a^{12}-\frac{1}{50}a^{10}+\frac{1}{200}a^{8}+\frac{1}{20}a^{6}-\frac{1}{10}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2000}a^{17}-\frac{1}{200}a^{14}-\frac{1}{400}a^{13}-\frac{1}{100}a^{11}-\frac{1}{40}a^{10}+\frac{1}{400}a^{9}+\frac{1}{40}a^{7}+\frac{19}{40}a^{6}-\frac{1}{20}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{1277931578000}a^{18}+\frac{1953803}{9128082700}a^{16}-\frac{724563593}{255586315600}a^{14}+\frac{254362663}{63896578900}a^{12}+\frac{5872042441}{255586315600}a^{10}-\frac{1}{20}a^{9}-\frac{165108501}{5111726312}a^{8}-\frac{1}{4}a^{7}-\frac{92178028}{638965789}a^{6}+\frac{1}{4}a^{5}+\frac{438858779}{12779315780}a^{4}-\frac{1}{2}a^{3}+\frac{162034277}{638965789}a^{2}+\frac{68952141}{638965789}$, $\frac{1}{1277931578000}a^{19}+\frac{1953803}{9128082700}a^{17}+\frac{110673597}{51117263120}a^{15}+\frac{254362663}{63896578900}a^{13}-\frac{517615449}{255586315600}a^{11}-\frac{1}{20}a^{10}-\frac{165108501}{5111726312}a^{9}-\frac{1}{20}a^{8}-\frac{3048155331}{25558631560}a^{7}-\frac{1}{4}a^{6}+\frac{908421931}{3194828945}a^{5}+\frac{162034277}{638965789}a^{3}-\frac{1}{2}a^{2}-\frac{501061507}{1277931578}a$ 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:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

Ideal class group:  $C_{2}$, which has order $2$ (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:  $C_{2}$, which has order $2$ (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:  $9$
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:   $\frac{204037}{4189939600}a^{18}-\frac{80973}{59856280}a^{16}-\frac{67509711}{4189939600}a^{14}-\frac{43425043}{418993960}a^{12}-\frac{317819753}{837987920}a^{10}-\frac{172024899}{209496980}a^{8}-\frac{250070419}{209496980}a^{6}-\frac{40951941}{41899396}a^{4}-\frac{6031573}{10474849}a^{2}-\frac{7707793}{10474849}$, $\frac{6229431}{58087799000}a^{18}+\frac{25893669}{8298257000}a^{16}+\frac{456935419}{11617559800}a^{14}+\frac{3168057037}{11617559800}a^{12}+\frac{12827755929}{11617559800}a^{10}+\frac{30868028173}{11617559800}a^{8}+\frac{2188239939}{580877990}a^{6}+\frac{507750234}{290438995}a^{4}-\frac{388168157}{116175598}a^{2}-\frac{187550373}{58087799}$, $\frac{4553863}{255586315600}a^{18}-\frac{5393834}{11410103375}a^{16}-\frac{1396347201}{255586315600}a^{14}-\frac{2288626743}{63896578900}a^{12}-\frac{39945468043}{255586315600}a^{10}-\frac{71388608209}{127793157800}a^{8}-\frac{20841215083}{12779315780}a^{6}-\frac{33231042527}{12779315780}a^{4}-\frac{1801403803}{1277931578}a^{2}-\frac{401654139}{638965789}$, $\frac{155136501}{638965789000}a^{18}+\frac{163800173}{22820206750}a^{16}+\frac{11904414449}{127793157800}a^{14}+\frac{4339947351}{6389657890}a^{12}+\frac{77101356403}{25558631560}a^{10}+\frac{551352423407}{63896578900}a^{8}+\frac{216080145091}{12779315780}a^{6}+\frac{68882841274}{3194828945}a^{4}+\frac{9649975576}{638965789}a^{2}+\frac{4886078326}{638965789}$, $\frac{12480133}{182561654000}a^{19}-\frac{596399}{5280709000}a^{18}-\frac{339423379}{182561654000}a^{17}-\frac{2341991}{754387000}a^{16}-\frac{775672983}{36512330800}a^{15}-\frac{3856619}{105614180}a^{14}-\frac{4542024717}{36512330800}a^{13}-\frac{48649437}{211228360}a^{12}-\frac{12755578457}{36512330800}a^{11}-\frac{430834023}{528070900}a^{10}-\frac{7137999449}{36512330800}a^{9}-\frac{1796451417}{1056141800}a^{8}+\frac{979782781}{730246616}a^{7}-\frac{111402847}{42245672}a^{6}+\frac{3881199741}{912808270}a^{5}-\frac{33022221}{10561418}a^{4}+\frac{2129549569}{365123308}a^{3}-\frac{11470104}{5280709}a^{2}+\frac{339090483}{182561654}a-\frac{36782435}{10561418}$, $\frac{6861109}{255586315600}a^{19}+\frac{30214887}{255586315600}a^{18}+\frac{34453911}{45640413500}a^{17}+\frac{62846261}{18256165400}a^{16}+\frac{2341202551}{255586315600}a^{15}+\frac{11098248961}{255586315600}a^{14}+\frac{38999010}{638965789}a^{13}+\frac{38501039637}{127793157800}a^{12}+\frac{61620824033}{255586315600}a^{11}+\frac{62457653367}{51117263120}a^{10}+\frac{78447816119}{127793157800}a^{9}+\frac{9506194239}{3194828945}a^{8}+\frac{2880811977}{2555863156}a^{7}+\frac{11767814801}{2555863156}a^{6}+\frac{13453295739}{12779315780}a^{5}+\frac{53115754701}{12779315780}a^{4}-\frac{393684278}{638965789}a^{3}+\frac{611279360}{638965789}a^{2}-\frac{351503531}{638965789}a-\frac{162662305}{638965789}$, $\frac{50594823}{638965789000}a^{19}-\frac{7513511}{638965789000}a^{18}-\frac{100766173}{45640413500}a^{17}-\frac{325181}{1825616540}a^{16}-\frac{1709339323}{63896578900}a^{15}-\frac{110569693}{127793157800}a^{14}-\frac{459641573}{2555863156}a^{13}-\frac{175142127}{63896578900}a^{12}-\frac{2348878927}{3194828945}a^{11}-\frac{7022180181}{127793157800}a^{10}-\frac{65456780693}{31948289450}a^{9}-\frac{7034074573}{12779315780}a^{8}-\frac{22566758021}{5111726312}a^{7}-\frac{25490055179}{12779315780}a^{6}-\frac{19004245252}{3194828945}a^{5}-\frac{14145891707}{6389657890}a^{4}-\frac{1833271974}{638965789}a^{3}+\frac{229661848}{638965789}a^{2}-\frac{1070411675}{1277931578}a+\frac{305756720}{638965789}$, $\frac{45100849}{127793157800}a^{18}+\frac{957312047}{91280827000}a^{16}+\frac{17403225527}{127793157800}a^{14}+\frac{5053880631}{5111726312}a^{12}+\frac{553547603069}{127793157800}a^{10}+\frac{1535922333059}{127793157800}a^{8}+\frac{284701671607}{12779315780}a^{6}+\frac{81349047606}{3194828945}a^{4}+\frac{15462313919}{1277931578}a^{2}+\frac{1211141482}{638965789}$, $\frac{547730221}{182561654000}a^{19}-\frac{5046563689}{638965789000}a^{18}+\frac{7177643199}{91280827000}a^{17}-\frac{552306001}{2282020675}a^{16}+\frac{31014048651}{36512330800}a^{15}-\frac{416941296133}{127793157800}a^{14}+\frac{81602628773}{18256165400}a^{13}-\frac{1588596825847}{63896578900}a^{12}+\frac{326339195649}{36512330800}a^{11}-\frac{14852090014049}{127793157800}a^{10}-\frac{142478922613}{9128082700}a^{9}-\frac{1117217705507}{3194828945}a^{8}-\frac{109675355321}{912808270}a^{7}-\frac{9051029817883}{12779315780}a^{6}-\frac{581492221121}{1825616540}a^{5}-\frac{1181726276725}{1277931578}a^{4}-\frac{78759950575}{182561654}a^{3}-\frac{392033464111}{638965789}a^{2}-\frac{10044854927}{91280827}a-\frac{79130457569}{638965789}$ 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:  \( 133777304.43188025 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 133777304.43188025 \cdot 2}{2\cdot\sqrt{975310301665730560000000000000000}}\cr\approx \mathstrut & 0.410780404921031 \end{aligned}\] (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 + 30*x^18 + 395*x^16 + 2930*x^14 + 13305*x^12 + 39200*x^10 + 79900*x^8 + 109000*x^6 + 88000*x^4 + 50000*x^2 + 10000) 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 + 30*x^18 + 395*x^16 + 2930*x^14 + 13305*x^12 + 39200*x^10 + 79900*x^8 + 109000*x^6 + 88000*x^4 + 50000*x^2 + 10000, 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 + 30*x^18 + 395*x^16 + 2930*x^14 + 13305*x^12 + 39200*x^10 + 79900*x^8 + 109000*x^6 + 88000*x^4 + 50000*x^2 + 10000); 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 + 30*x^18 + 395*x^16 + 2930*x^14 + 13305*x^12 + 39200*x^10 + 79900*x^8 + 109000*x^6 + 88000*x^4 + 50000*x^2 + 10000); 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

$C_2^9.A_5^2.C_4$ (as 20T1025):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma: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 7372800
The 216 conjugacy class representatives for $C_2^9.A_5^2.C_4$
Character table for $C_2^9.A_5^2.C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 10.2.19518724000000.1

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 sibling: data not computed
Degree 40 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $16{,}\,{\href{/padicField/3.4.0.1}{4} }$ R ${\href{/padicField/7.8.0.1}{8} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ $16{,}\,{\href{/padicField/13.4.0.1}{4} }$ ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ $16{,}\,{\href{/padicField/23.4.0.1}{4} }$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.8.0.1}{8} }$ ${\href{/padicField/41.3.0.1}{3} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{8}$ ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ R $16{,}\,{\href{/padicField/53.4.0.1}{4} }$ ${\href{/padicField/59.10.0.1}{10} }^{2}$

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 2.2.4.12a6.2$x^{8} + 6 x^{7} + 18 x^{6} + 36 x^{5} + 49 x^{4} + 48 x^{3} + 32 x^{2} + 18 x + 5$$4$$2$$12$$(C_8:C_2):C_2$$$[2, 2, 2]^{4}$$
2.2.6.16a1.8$x^{12} + 6 x^{11} + 23 x^{10} + 60 x^{9} + 120 x^{8} + 186 x^{7} + 233 x^{6} + 234 x^{5} + 192 x^{4} + 124 x^{3} + 63 x^{2} + 26 x + 7$$6$$2$$16$12T30$$[\frac{4}{3}, \frac{4}{3}, 2]_{3}^{2}$$
\(5\) Copy content Toggle raw display 5.1.8.7a1.1$x^{8} + 5$$8$$1$$7$$C_8:C_2$$$[\ ]_{8}^{2}$$
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}$$$[\ ]_{4}^{3}$$
\(47\) Copy content Toggle raw display 47.4.1.0a1.1$x^{4} + 8 x^{2} + 40 x + 5$$1$$4$$0$$C_4$$$[\ ]^{4}$$
47.4.1.0a1.1$x^{4} + 8 x^{2} + 40 x + 5$$1$$4$$0$$C_4$$$[\ ]^{4}$$
47.2.3.4a1.1$x^{6} + 135 x^{5} + 6090 x^{4} + 92475 x^{3} + 30450 x^{2} + 3422 x + 125$$3$$2$$4$$S_3\times C_3$$$[\ ]_{3}^{6}$$
47.2.3.4a1.1$x^{6} + 135 x^{5} + 6090 x^{4} + 92475 x^{3} + 30450 x^{2} + 3422 x + 125$$3$$2$$4$$S_3\times C_3$$$[\ ]_{3}^{6}$$

Spectrum of ring of integers

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