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Properties

Label 22.14.101...568.1
Degree $22$
Signature $[14, 4]$
Discriminant $1.015\times 10^{41}$
Root discriminant \(73.10\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{11}\wr C_2$ (as 22T57)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 - 21*x^20 + 32*x^19 + 193*x^18 - 192*x^17 - 983*x^16 + 496*x^15 + 2915*x^14 - 344*x^13 - 4994*x^12 - 730*x^11 + 4915*x^10 + 1626*x^9 - 2700*x^8 - 1304*x^7 + 754*x^6 + 502*x^5 - 77*x^4 - 90*x^3 - 5*x^2 + 6*x + 1)
 
Copy content gp:K = bnfinit(y^22 - 2*y^21 - 21*y^20 + 32*y^19 + 193*y^18 - 192*y^17 - 983*y^16 + 496*y^15 + 2915*y^14 - 344*y^13 - 4994*y^12 - 730*y^11 + 4915*y^10 + 1626*y^9 - 2700*y^8 - 1304*y^7 + 754*y^6 + 502*y^5 - 77*y^4 - 90*y^3 - 5*y^2 + 6*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 2*x^21 - 21*x^20 + 32*x^19 + 193*x^18 - 192*x^17 - 983*x^16 + 496*x^15 + 2915*x^14 - 344*x^13 - 4994*x^12 - 730*x^11 + 4915*x^10 + 1626*x^9 - 2700*x^8 - 1304*x^7 + 754*x^6 + 502*x^5 - 77*x^4 - 90*x^3 - 5*x^2 + 6*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^22 - 2*x^21 - 21*x^20 + 32*x^19 + 193*x^18 - 192*x^17 - 983*x^16 + 496*x^15 + 2915*x^14 - 344*x^13 - 4994*x^12 - 730*x^11 + 4915*x^10 + 1626*x^9 - 2700*x^8 - 1304*x^7 + 754*x^6 + 502*x^5 - 77*x^4 - 90*x^3 - 5*x^2 + 6*x + 1)
 

\( x^{22} - 2 x^{21} - 21 x^{20} + 32 x^{19} + 193 x^{18} - 192 x^{17} - 983 x^{16} + 496 x^{15} + 2915 x^{14} + \cdots + 1 \) 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:  $22$
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:  $[14, 4]$
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:   \(101482802839751113302452158783383633133568\) \(\medspace = 2^{33}\cdot 1721\cdot 9473\cdot 146628265753\cdot 4942156467121\) 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:  \(73.10\)
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:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(1721\), \(9473\), \(146628265753\), \(4942156467121\) 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:  $\Q(\sqrt{23628\!\cdots\!47458}$)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$ 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:  Yes
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:  $17$
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:   $a^{21}-2a^{20}-21a^{19}+32a^{18}+193a^{17}-192a^{16}-983a^{15}+496a^{14}+2915a^{13}-344a^{12}-4994a^{11}-730a^{10}+4915a^{9}+1626a^{8}-2700a^{7}-1304a^{6}+754a^{5}+502a^{4}-77a^{3}-90a^{2}-5a+6$, $a^{21}-3a^{20}-18a^{19}+50a^{18}+143a^{17}-335a^{16}-648a^{15}+1144a^{14}+1771a^{13}-2115a^{12}-2879a^{11}+2149a^{10}+2766a^{9}-1140a^{8}-1560a^{7}+256a^{6}+498a^{5}+4a^{4}-81a^{3}-9a^{2}+4a+2$, $a^{21}-2a^{20}-21a^{19}+32a^{18}+193a^{17}-192a^{16}-983a^{15}+496a^{14}+2915a^{13}-344a^{12}-4994a^{11}-730a^{10}+4915a^{9}+1626a^{8}-2700a^{7}-1304a^{6}+754a^{5}+502a^{4}-77a^{3}-90a^{2}-4a+6$, $3a^{21}-7a^{20}-61a^{19}+117a^{18}+547a^{17}-769a^{16}-2757a^{15}+2471a^{14}+8249a^{13}-3947a^{12}-14638a^{11}+2804a^{10}+15474a^{9}-36a^{8}-9715a^{7}-1217a^{6}+3525a^{5}+752a^{4}-680a^{3}-183a^{2}+51a+17$, $a^{21}-3a^{20}-18a^{19}+50a^{18}+143a^{17}-335a^{16}-648a^{15}+1144a^{14}+1771a^{13}-2115a^{12}-2879a^{11}+2149a^{10}+2766a^{9}-1140a^{8}-1560a^{7}+256a^{6}+498a^{5}+4a^{4}-81a^{3}-9a^{2}+3a+2$, $14a^{21}-30a^{20}-289a^{19}+488a^{18}+2617a^{17}-3042a^{16}-13185a^{15}+8718a^{14}+38835a^{13}-10150a^{12}-66313a^{11}-576a^{10}+65276a^{9}+12205a^{8}-36139a^{7}-11239a^{6}+10477a^{5}+4246a^{4}-1350a^{3}-645a^{2}+41a+27$, $3a^{20}-7a^{19}-60a^{18}+115a^{17}+526a^{16}-737a^{15}-2564a^{14}+2278a^{13}+7268a^{12}-3435a^{11}-11749a^{10}+2354a^{9}+10607a^{8}-402a^{7}-5092a^{6}-262a^{5}+1162a^{4}+107a^{3}-108a^{2}-10a+3$, $4a^{21}-10a^{20}-79a^{19}+167a^{18}+690a^{17}-1104a^{16}-3405a^{15}+3615a^{14}+10020a^{13}-6062a^{12}-17517a^{11}+4953a^{10}+18240a^{9}-1176a^{8}-11275a^{7}-961a^{6}+4023a^{5}+756a^{4}-761a^{3}-191a^{2}+54a+16$, $9a^{21}-16a^{20}-191a^{19}+244a^{18}+1754a^{17}-1315a^{16}-8769a^{15}+2416a^{14}+24827a^{13}+2351a^{12}-38728a^{11}-13661a^{10}+31985a^{9}+17945a^{8}-12186a^{7}-10304a^{6}+834a^{5}+2498a^{4}+527a^{3}-151a^{2}-75a-9$, $5a^{21}-9a^{20}-108a^{19}+142a^{18}+1015a^{17}-817a^{16}-5251a^{15}+1836a^{14}+15731a^{13}-4a^{12}-27145a^{11}-6232a^{10}+26899a^{9}+10114a^{8}-14969a^{7}-7058a^{6}+4369a^{5}+2356a^{4}-569a^{3}-348a^{2}+18a+18$, $3a^{20}-7a^{19}-59a^{18}+112a^{17}+509a^{16}-690a^{15}-2437a^{14}+1986a^{13}+6736a^{12}-2528a^{11}-10469a^{10}+847a^{9}+8962a^{8}+1059a^{7}-4033a^{6}-1139a^{5}+843a^{4}+403a^{3}-57a^{2}-43a-4$, $16a^{21}-25a^{20}-347a^{19}+360a^{18}+3248a^{17}-1648a^{16}-16481a^{15}+702a^{14}+47145a^{13}+15244a^{12}-73900a^{11}-44426a^{10}+60444a^{9}+53297a^{8}-21129a^{7}-31038a^{6}-889a^{5}+8187a^{4}+2230a^{3}-616a^{2}-356a-44$, $11a^{21}-22a^{20}-231a^{19}+353a^{18}+2121a^{17}-2133a^{16}-10780a^{15}+5645a^{14}+31860a^{13}-4710a^{12}-54361a^{11}-5440a^{10}+53431a^{9}+13815a^{8}-29710a^{7}-10771a^{6}+8884a^{5}+3829a^{4}-1313a^{3}-600a^{2}+78a+33$, $11a^{21}-21a^{20}-230a^{19}+325a^{18}+2093a^{17}-1825a^{16}-10447a^{15}+3930a^{14}+29801a^{13}+415a^{12}-47461a^{11}-13498a^{10}+40964a^{9}+20070a^{8}-17467a^{7}-12459a^{6}+2510a^{5}+3352a^{4}+274a^{3}-291a^{2}-64a$, $32a^{21}-75a^{20}-648a^{19}+1250a^{18}+5784a^{17}-8182a^{16}-28990a^{15}+26108a^{14}+86051a^{13}-41101a^{12}-150702a^{11}+28006a^{10}+155726a^{9}+1349a^{8}-93894a^{7}-13385a^{6}+31734a^{5}+7561a^{4}-5473a^{3}-1647a^{2}+344a+130$, $16a^{21}-30a^{20}-337a^{19}+465a^{18}+3088a^{17}-2613a^{16}-15521a^{15}+5616a^{14}+44667a^{13}+685a^{12}-72204a^{11}-19565a^{10}+64194a^{9}+29113a^{8}-29408a^{7}-18280a^{6}+5615a^{5}+5111a^{4}-78a^{3}-518a^{2}-53a+11$, $a^{21}-12a^{20}-a^{19}+239a^{18}-122a^{17}-2058a^{16}+863a^{15}+9738a^{14}-1675a^{13}-26571a^{12}-2031a^{11}+41047a^{10}+10618a^{9}-35081a^{8}-13484a^{7}+15652a^{6}+7462a^{5}-3159a^{4}-1753a^{3}+224a^{2}+130a+4$ 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:  \( 5288131548180 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{4}\cdot 5288131548180 \cdot 1}{2\cdot\sqrt{101482802839751113302452158783383633133568}}\cr\approx \mathstrut & 0.211941225161792 \end{aligned}\] (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^22 - 2*x^21 - 21*x^20 + 32*x^19 + 193*x^18 - 192*x^17 - 983*x^16 + 496*x^15 + 2915*x^14 - 344*x^13 - 4994*x^12 - 730*x^11 + 4915*x^10 + 1626*x^9 - 2700*x^8 - 1304*x^7 + 754*x^6 + 502*x^5 - 77*x^4 - 90*x^3 - 5*x^2 + 6*x + 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^22 - 2*x^21 - 21*x^20 + 32*x^19 + 193*x^18 - 192*x^17 - 983*x^16 + 496*x^15 + 2915*x^14 - 344*x^13 - 4994*x^12 - 730*x^11 + 4915*x^10 + 1626*x^9 - 2700*x^8 - 1304*x^7 + 754*x^6 + 502*x^5 - 77*x^4 - 90*x^3 - 5*x^2 + 6*x + 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^22 - 2*x^21 - 21*x^20 + 32*x^19 + 193*x^18 - 192*x^17 - 983*x^16 + 496*x^15 + 2915*x^14 - 344*x^13 - 4994*x^12 - 730*x^11 + 4915*x^10 + 1626*x^9 - 2700*x^8 - 1304*x^7 + 754*x^6 + 502*x^5 - 77*x^4 - 90*x^3 - 5*x^2 + 6*x + 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 = PolynomialRing(QQ); K, a = NumberField(x^22 - 2*x^21 - 21*x^20 + 32*x^19 + 193*x^18 - 192*x^17 - 983*x^16 + 496*x^15 + 2915*x^14 - 344*x^13 - 4994*x^12 - 730*x^11 + 4915*x^10 + 1626*x^9 - 2700*x^8 - 1304*x^7 + 754*x^6 + 502*x^5 - 77*x^4 - 90*x^3 - 5*x^2 + 6*x + 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

$S_{11}\wr C_2$ (as 22T57):

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); G, transitive_group_identification(G)
 
A non-solvable group of order 3186701844480000
The 1652 conjugacy class representatives for $S_{11}\wr C_2$
Character table for $S_{11}\wr C_2$

Intermediate fields

\(\Q(\sqrt{2}) \)

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(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 44 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 ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ ${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ $18{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ $16{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ $16{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{5}$

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 2.11.2.33a1.189$x^{22} + 2 x^{13} + 6 x^{11} + x^{4} + 6 x^{2} + 7$$2$$11$$33$not computednot computed
\(1721\) Copy content Toggle raw display $\Q_{1721}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1721}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1721}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1721}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $7$$1$$7$$0$$C_7$$$[\ ]^{7}$$
\(9473\) Copy content Toggle raw display $\Q_{9473}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{9473}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $7$$1$$7$$0$$C_7$$$[\ ]^{7}$$
Deg $9$$1$$9$$0$$C_9$$$[\ ]^{9}$$
\(146628265753\) Copy content Toggle raw display $\Q_{146628265753}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$
Deg $11$$1$$11$$0$$C_{11}$$$[\ ]^{11}$$
\(4942156467121\) Copy content Toggle raw display $\Q_{4942156467121}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{4942156467121}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{4942156467121}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{4942156467121}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $5$$1$$5$$0$$C_5$$$[\ ]^{5}$$
Deg $11$$1$$11$$0$$C_{11}$$$[\ ]^{11}$$

Spectrum of ring of integers

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