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Properties

Label 18.18.220...000.1
Degree $18$
Signature $[18, 0]$
Discriminant $2.206\times 10^{67}$
Root discriminant \(5512.08\)
Ramified primes $2,3,5,11,2339,34819$
Class number not computed
Class group not computed
Galois group $C_3^4:(C_6^2:C_4)$ (as 18T576)

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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^18 - 19674*x^16 - 236892*x^15 + 141285519*x^14 + 2591050644*x^13 - 483401755344*x^12 - 9309346836120*x^11 + 919061549546175*x^10 + 14739425235311240*x^9 - 1067582767106931570*x^8 - 10423786760736518700*x^7 + 770353183144777007265*x^6 + 1519419179629720980900*x^5 - 315065054456543206397700*x^4 + 1832965846899730969466400*x^3 + 51184572251157572179388400*x^2 - 717390367091535907200369600*x + 2636432883430939313935566400)
 
Copy content gp:K = bnfinit(y^18 - 19674*y^16 - 236892*y^15 + 141285519*y^14 + 2591050644*y^13 - 483401755344*y^12 - 9309346836120*y^11 + 919061549546175*y^10 + 14739425235311240*y^9 - 1067582767106931570*y^8 - 10423786760736518700*y^7 + 770353183144777007265*y^6 + 1519419179629720980900*y^5 - 315065054456543206397700*y^4 + 1832965846899730969466400*y^3 + 51184572251157572179388400*y^2 - 717390367091535907200369600*y + 2636432883430939313935566400, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 19674*x^16 - 236892*x^15 + 141285519*x^14 + 2591050644*x^13 - 483401755344*x^12 - 9309346836120*x^11 + 919061549546175*x^10 + 14739425235311240*x^9 - 1067582767106931570*x^8 - 10423786760736518700*x^7 + 770353183144777007265*x^6 + 1519419179629720980900*x^5 - 315065054456543206397700*x^4 + 1832965846899730969466400*x^3 + 51184572251157572179388400*x^2 - 717390367091535907200369600*x + 2636432883430939313935566400);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 19674*x^16 - 236892*x^15 + 141285519*x^14 + 2591050644*x^13 - 483401755344*x^12 - 9309346836120*x^11 + 919061549546175*x^10 + 14739425235311240*x^9 - 1067582767106931570*x^8 - 10423786760736518700*x^7 + 770353183144777007265*x^6 + 1519419179629720980900*x^5 - 315065054456543206397700*x^4 + 1832965846899730969466400*x^3 + 51184572251157572179388400*x^2 - 717390367091535907200369600*x + 2636432883430939313935566400)
 

\( x^{18} - 19674 x^{16} - 236892 x^{15} + 141285519 x^{14} + 2591050644 x^{13} - 483401755344 x^{12} + \cdots + 26\!\cdots\!00 \) 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:  $18$
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:  $[18, 0]$
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:   \(22063911869341701750760207002328140551944852253885577375000000000000\) \(\medspace = 2^{12}\cdot 3^{30}\cdot 5^{15}\cdot 11^{7}\cdot 2339^{4}\cdot 34819^{4}\) 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:  \(5512.08\)
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:   \(2\), \(3\), \(5\), \(11\), \(2339\), \(34819\) 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:  \(\Q(\sqrt{55}) \)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $\frac{1}{6}a^{2}-\frac{1}{6}a-\frac{1}{3}$, $\frac{1}{12}a^{3}-\frac{1}{4}a-\frac{1}{6}$, $\frac{1}{72}a^{4}+\frac{1}{72}a^{3}-\frac{1}{24}a^{2}-\frac{5}{72}a-\frac{1}{36}$, $\frac{1}{72}a^{5}+\frac{1}{36}a^{3}-\frac{1}{36}a^{2}-\frac{5}{24}a-\frac{5}{36}$, $\frac{1}{432}a^{6}+\frac{1}{216}a^{3}-\frac{1}{48}a^{2}-\frac{1}{24}a-\frac{1}{54}$, $\frac{1}{864}a^{7}-\frac{1}{864}a^{6}-\frac{1}{144}a^{5}+\frac{1}{432}a^{4}+\frac{13}{864}a^{3}+\frac{1}{288}a^{2}-\frac{1}{108}a-\frac{1}{216}$, $\frac{1}{5184}a^{8}+\frac{1}{2592}a^{7}-\frac{5}{5184}a^{6}-\frac{1}{324}a^{5}-\frac{5}{5184}a^{4}+\frac{13}{2592}a^{3}+\frac{37}{5184}a^{2}+\frac{5}{1296}a+\frac{1}{1296}$, $\frac{1}{15552}a^{9}-\frac{1}{1728}a^{7}-\frac{1}{2592}a^{6}+\frac{1}{576}a^{5}+\frac{1}{432}a^{4}-\frac{5}{5184}a^{3}-\frac{1}{288}a^{2}-\frac{1}{432}a-\frac{1}{1944}$, $\frac{1}{31104}a^{10}-\frac{1}{31104}a^{9}-\frac{1}{10368}a^{8}+\frac{5}{10368}a^{7}+\frac{1}{10368}a^{6}-\frac{29}{10368}a^{5}-\frac{1}{384}a^{4}+\frac{13}{3456}a^{3}+\frac{5}{648}a^{2}+\frac{37}{7776}a+\frac{1}{972}$, $\frac{1}{933120}a^{11}+\frac{7}{466560}a^{10}-\frac{1}{116640}a^{9}-\frac{1}{38880}a^{8}-\frac{47}{155520}a^{7}-\frac{47}{77760}a^{6}+\frac{23}{15552}a^{5}+\frac{1}{972}a^{4}-\frac{2257}{62208}a^{3}-\frac{2897}{93312}a^{2}+\frac{15955}{46656}a+\frac{7873}{23328}$, $\frac{1}{1866240}a^{12}-\frac{1}{1866240}a^{11}+\frac{11}{933120}a^{10}+\frac{1}{51840}a^{9}-\frac{17}{311040}a^{8}+\frac{41}{311040}a^{7}+\frac{5}{31104}a^{6}+\frac{31}{7776}a^{5}+\frac{59}{124416}a^{4}-\frac{6505}{373248}a^{3}+\frac{3743}{186624}a^{2}+\frac{16229}{93312}a+\frac{2177}{15552}$, $\frac{1}{5598720}a^{13}+\frac{1}{5598720}a^{12}+\frac{1}{139968}a^{10}+\frac{13}{559872}a^{9}+\frac{1}{34560}a^{8}+\frac{11}{29160}a^{7}-\frac{11}{23328}a^{6}-\frac{47}{13824}a^{5}-\frac{4735}{1119744}a^{4}-\frac{1871}{139968}a^{3}-\frac{5}{288}a^{2}-\frac{20989}{69984}a-\frac{20593}{69984}$, $\frac{1}{22394880}a^{14}-\frac{1}{11197440}a^{13}-\frac{1}{7464960}a^{12}-\frac{1}{2799360}a^{11}+\frac{29}{11197440}a^{10}+\frac{43}{1866240}a^{9}-\frac{77}{3732480}a^{8}-\frac{79}{186624}a^{7}-\frac{809}{829440}a^{6}+\frac{10651}{2239488}a^{5}-\frac{17971}{4478976}a^{4}+\frac{575}{46656}a^{3}+\frac{8743}{559872}a^{2}-\frac{48821}{139968}a+\frac{14675}{93312}$, $\frac{1}{54\cdots 00}a^{15}+\frac{1}{74649600}a^{14}-\frac{1058747891}{18\cdots 00}a^{13}+\frac{13274750591}{54\cdots 00}a^{12}-\frac{1173054599}{30\cdots 00}a^{11}+\frac{52310167091}{91\cdots 00}a^{10}+\frac{90550399027}{54\cdots 40}a^{9}-\frac{1993025683}{40530572826624}a^{8}+\frac{45701254229}{243183436959744}a^{7}+\frac{2946268480097}{10\cdots 80}a^{6}-\frac{2598288473081}{12\cdots 20}a^{5}+\frac{22867288994813}{36\cdots 60}a^{4}+\frac{517435454839}{273581366579712}a^{3}-\frac{17875}{373248}a^{2}-\frac{85459}{559872}a+\frac{482279}{1679616}$, $\frac{1}{54\cdots 00}a^{16}-\frac{1093}{30\cdots 00}a^{14}-\frac{101816857}{34\cdots 00}a^{13}-\frac{1581737647}{18\cdots 00}a^{12}+\frac{26621213}{84438693388800}a^{11}-\frac{117999981401}{13\cdots 00}a^{10}-\frac{3248667247}{113992236074880}a^{9}-\frac{12168276223}{405305728266240}a^{8}-\frac{8918431769}{136790683289856}a^{7}-\frac{44332152833}{607958592399360}a^{6}+\frac{3043300957}{15198964809984}a^{5}-\frac{21687336473711}{10\cdots 80}a^{4}+\frac{27311}{1119744}a^{3}-\frac{9571}{124416}a^{2}-\frac{386999}{839808}a+\frac{192751}{559872}$, $\frac{1}{66\cdots 00}a^{17}-\frac{18\cdots 43}{33\cdots 00}a^{16}+\frac{92\cdots 41}{33\cdots 00}a^{15}-\frac{17\cdots 53}{83\cdots 00}a^{14}-\frac{14\cdots 77}{66\cdots 00}a^{13}-\frac{31\cdots 07}{33\cdots 00}a^{12}+\frac{66\cdots 41}{33\cdots 00}a^{11}+\frac{12\cdots 37}{83\cdots 00}a^{10}+\frac{23\cdots 99}{13\cdots 00}a^{9}-\frac{47\cdots 93}{66\cdots 00}a^{8}+\frac{17\cdots 61}{66\cdots 00}a^{7}-\frac{14\cdots 59}{16\cdots 00}a^{6}+\frac{20\cdots 09}{26\cdots 20}a^{5}+\frac{25\cdots 63}{13\cdots 60}a^{4}+\frac{64\cdots 25}{66\cdots 88}a^{3}-\frac{46\cdots 89}{20\cdots 20}a^{2}+\frac{90\cdots 45}{40\cdots 84}a+\frac{49\cdots 63}{10\cdots 60}$ 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:  not computed
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:  not computed
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:  not computed
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:  not computed
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot R \cdot h}{2\cdot\sqrt{22063911869341701750760207002328140551944852253885577375000000000000}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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^18 - 19674*x^16 - 236892*x^15 + 141285519*x^14 + 2591050644*x^13 - 483401755344*x^12 - 9309346836120*x^11 + 919061549546175*x^10 + 14739425235311240*x^9 - 1067582767106931570*x^8 - 10423786760736518700*x^7 + 770353183144777007265*x^6 + 1519419179629720980900*x^5 - 315065054456543206397700*x^4 + 1832965846899730969466400*x^3 + 51184572251157572179388400*x^2 - 717390367091535907200369600*x + 2636432883430939313935566400) 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^18 - 19674*x^16 - 236892*x^15 + 141285519*x^14 + 2591050644*x^13 - 483401755344*x^12 - 9309346836120*x^11 + 919061549546175*x^10 + 14739425235311240*x^9 - 1067582767106931570*x^8 - 10423786760736518700*x^7 + 770353183144777007265*x^6 + 1519419179629720980900*x^5 - 315065054456543206397700*x^4 + 1832965846899730969466400*x^3 + 51184572251157572179388400*x^2 - 717390367091535907200369600*x + 2636432883430939313935566400, 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^18 - 19674*x^16 - 236892*x^15 + 141285519*x^14 + 2591050644*x^13 - 483401755344*x^12 - 9309346836120*x^11 + 919061549546175*x^10 + 14739425235311240*x^9 - 1067582767106931570*x^8 - 10423786760736518700*x^7 + 770353183144777007265*x^6 + 1519419179629720980900*x^5 - 315065054456543206397700*x^4 + 1832965846899730969466400*x^3 + 51184572251157572179388400*x^2 - 717390367091535907200369600*x + 2636432883430939313935566400); 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^18 - 19674*x^16 - 236892*x^15 + 141285519*x^14 + 2591050644*x^13 - 483401755344*x^12 - 9309346836120*x^11 + 919061549546175*x^10 + 14739425235311240*x^9 - 1067582767106931570*x^8 - 10423786760736518700*x^7 + 770353183144777007265*x^6 + 1519419179629720980900*x^5 - 315065054456543206397700*x^4 + 1832965846899730969466400*x^3 + 51184572251157572179388400*x^2 - 717390367091535907200369600*x + 2636432883430939313935566400); 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

$C_3^4:(C_6^2:C_4)$ (as 18T576):

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 solvable group of order 11664
The 49 conjugacy class representatives for $C_3^4:(C_6^2:C_4)$
Character table for $C_3^4:(C_6^2:C_4)$

Intermediate fields

\(\Q(\sqrt{5}) \), 6.6.55130625.1

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

Sibling fields

Degree 18 sibling: data not computed
Degree 36 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 R R ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ R ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.6.0.1}{6} }$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.6.0.1}{6} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{5}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{4}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{4}$

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 2.2.1.0a1.1$x^{2} + x + 1$$1$$2$$0$$C_2$$$[\ ]^{2}$$
2.4.1.0a1.1$x^{4} + x + 1$$1$$4$$0$$C_4$$$[\ ]^{4}$$
2.2.2.4a2.2$x^{4} + 4 x^{3} + 5 x^{2} + 4 x + 7$$2$$2$$4$$D_{4}$$$[2, 2]^{2}$$
2.4.2.8a3.1$x^{8} + 2 x^{6} + 4 x^{5} + 2 x^{4} + 2 x^{3} + 5 x^{2} + 4 x + 3$$2$$4$$8$$C_2^2:C_4$$$[2, 2]^{4}$$
\(3\) Copy content Toggle raw display 3.2.9.30b15.13$x^{18} + 21 x^{17} + 213 x^{16} + 1398 x^{15} + 6663 x^{14} + 24486 x^{13} + 71862 x^{12} + 172104 x^{11} + 340788 x^{10} + 561899 x^{9} + 773187 x^{8} + 885918 x^{7} + 839364 x^{6} + 649236 x^{5} + 401496 x^{4} + 191928 x^{3} + 66960 x^{2} + 15264 x + 1715$$9$$2$$30$18T49not computed
\(5\) Copy content Toggle raw display 5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.4.3a1.1$x^{4} + 5$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.1.12.11a1.4$x^{12} + 20$$12$$1$$11$$S_3 \times C_4$$$[\ ]_{12}^{2}$$
\(11\) Copy content Toggle raw display $\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
11.2.1.0a1.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
11.2.1.0a1.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
11.2.1.0a1.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
11.1.3.2a1.1$x^{3} + 11$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
11.1.6.5a1.2$x^{6} + 22$$6$$1$$5$$D_{6}$$$[\ ]_{6}^{2}$$
\(2339\) Copy content Toggle raw display $\Q_{2339}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $3$$3$$1$$2$
Deg $3$$3$$1$$2$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
\(34819\) Copy content Toggle raw display $\Q_{34819}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
Deg $6$$3$$2$$4$

Spectrum of ring of integers

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