↑

Properties

Label 18.0.337...024.1
Degree $18$
Signature $[0, 9]$
Discriminant $-3.370\times 10^{43}$
Root discriminant \(261.94\)
Ramified primes $2,3,79$
Class number $18957120$ (GRH)
Class group [2, 2, 28, 169260] (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

Related objects

Downloads

Learn more

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 + 474*x^16 + 80343*x^14 + 6224884*x^12 + 246937647*x^10 + 5138452458*x^8 + 53819444489*x^6 + 260620415400*x^4 + 463401439632*x^2 + 2019487744)
 
Copy content gp:K = bnfinit(y^18 + 474*y^16 + 80343*y^14 + 6224884*y^12 + 246937647*y^10 + 5138452458*y^8 + 53819444489*y^6 + 260620415400*y^4 + 463401439632*y^2 + 2019487744, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 474*x^16 + 80343*x^14 + 6224884*x^12 + 246937647*x^10 + 5138452458*x^8 + 53819444489*x^6 + 260620415400*x^4 + 463401439632*x^2 + 2019487744);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 + 474*x^16 + 80343*x^14 + 6224884*x^12 + 246937647*x^10 + 5138452458*x^8 + 53819444489*x^6 + 260620415400*x^4 + 463401439632*x^2 + 2019487744)
 

\( x^{18} + 474 x^{16} + 80343 x^{14} + 6224884 x^{12} + 246937647 x^{10} + 5138452458 x^{8} + \cdots + 2019487744 \) 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:  $[0, 9]$
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:   \(-33703610765963698533467381091270298700673024\) \(\medspace = -\,2^{12}\cdot 3^{24}\cdot 79^{15}\) 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:  \(261.94\)
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:  $2\cdot 3^{4/3}79^{5/6}\approx 330.0255610451065$
Ramified primes:   \(2\), \(3\), \(79\) 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:  \(\Q(\sqrt{-79}) \)
$\Aut(K/\Q)$:   $C_6$
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 a CM field.
Reflex fields:  unavailable$^{256}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{3}$, $\frac{1}{632}a^{6}-\frac{1}{8}a^{5}-\frac{1}{8}a^{4}+\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{2528}a^{7}-\frac{1}{16}a^{5}+\frac{7}{32}a^{3}+\frac{3}{8}a$, $\frac{1}{5056}a^{8}-\frac{1}{5056}a^{7}+\frac{1}{2528}a^{6}+\frac{1}{32}a^{5}+\frac{7}{64}a^{4}+\frac{9}{64}a^{3}+\frac{3}{16}a^{2}+\frac{1}{16}a$, $\frac{1}{10112}a^{9}+\frac{1}{10112}a^{7}+\frac{9}{128}a^{5}-\frac{1}{8}a^{4}-\frac{11}{128}a^{3}+\frac{1}{8}a^{2}-\frac{7}{32}a-\frac{1}{2}$, $\frac{1}{20224}a^{10}+\frac{1}{20224}a^{8}-\frac{9}{20224}a^{6}-\frac{27}{256}a^{4}-\frac{7}{64}a^{2}$, $\frac{1}{3195392}a^{11}-\frac{1}{40448}a^{10}-\frac{1}{40448}a^{9}-\frac{1}{40448}a^{8}-\frac{7}{40448}a^{7}+\frac{9}{40448}a^{6}+\frac{4277}{40448}a^{5}+\frac{27}{512}a^{4}-\frac{25}{128}a^{3}+\frac{7}{128}a^{2}-\frac{1}{4}a$, $\frac{1}{3195392}a^{12}+\frac{1}{20224}a^{8}-\frac{1}{5056}a^{7}-\frac{1}{10112}a^{6}-\frac{3}{32}a^{5}-\frac{7}{512}a^{4}+\frac{1}{64}a^{3}-\frac{15}{128}a^{2}-\frac{3}{16}a$, $\frac{1}{6390784}a^{13}+\frac{1}{40448}a^{9}+\frac{1}{20224}a^{7}+\frac{41}{1024}a^{5}+\frac{15}{256}a^{3}+\frac{15}{32}a-\frac{1}{2}$, $\frac{1}{434573312}a^{14}-\frac{3}{108643328}a^{12}+\frac{9}{2750464}a^{10}+\frac{15}{1375232}a^{8}-\frac{1081}{5500928}a^{6}-\frac{1}{8}a^{5}-\frac{227}{2176}a^{4}-\frac{1}{8}a^{3}+\frac{319}{4352}a^{2}+\frac{1}{4}a-\frac{5}{17}$, $\frac{1}{869146624}a^{15}-\frac{1}{869146624}a^{14}-\frac{3}{217286656}a^{13}-\frac{31}{217286656}a^{12}+\frac{31}{434573312}a^{11}+\frac{127}{5500928}a^{10}+\frac{83}{2750464}a^{9}+\frac{257}{2750464}a^{8}+\frac{823}{11001856}a^{7}+\frac{1353}{11001856}a^{6}-\frac{82463}{687616}a^{5}+\frac{1061}{17408}a^{4}-\frac{1449}{8704}a^{3}-\frac{1645}{8704}a^{2}-\frac{131}{544}a-\frac{6}{17}$, $\frac{1}{60\cdots 76}a^{16}+\frac{153736189}{76\cdots 44}a^{14}+\frac{48223798439}{38\cdots 72}a^{12}+\frac{3101291757205}{38\cdots 72}a^{10}-\frac{371567124971}{96\cdots 36}a^{8}+\frac{4065752208855}{96\cdots 36}a^{6}-\frac{53029707100143}{12\cdots 92}a^{4}-\frac{1}{4}a^{3}-\frac{1725082204015}{7626088423424}a^{2}+\frac{1}{4}a-\frac{2862869353}{29789407904}$, $\frac{1}{12\cdots 52}a^{17}-\frac{1}{12\cdots 52}a^{16}+\frac{153736189}{15\cdots 88}a^{15}-\frac{153736189}{15\cdots 88}a^{14}+\frac{48223798439}{76\cdots 44}a^{13}+\frac{70933833177}{76\cdots 44}a^{12}+\frac{3193335189}{76\cdots 44}a^{11}+\frac{15725614038123}{76\cdots 44}a^{10}+\frac{105063401493}{19\cdots 72}a^{9}-\frac{581693927957}{19\cdots 72}a^{8}-\frac{223922529321}{19\cdots 72}a^{7}+\frac{2130444635177}{19\cdots 72}a^{6}-\frac{33309119067695}{24\cdots 84}a^{5}+\frac{229531948931343}{24\cdots 84}a^{4}+\frac{3517853587089}{15252176846848}a^{3}+\frac{473927072047}{15252176846848}a^{2}-\frac{12172059323}{59578815808}a+\frac{2862869353}{59578815808}$ 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:  $C_{2}\times C_{2}\times C_{28}\times C_{169260}$, which has order $18957120$ (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:  $C_{2}\times C_{2}\times C_{28}\times C_{169260}$, which has order $18957120$ (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);
 
Relative class number:   $1579760$ (assuming GRH)

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:  $8$
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:   $\frac{50386809}{65\cdots 28}a^{16}+\frac{297224613}{82\cdots 32}a^{14}+\frac{24448613583}{41\cdots 16}a^{12}+\frac{1795985428893}{41\cdots 16}a^{10}+\frac{1652733987885}{104775823556608}a^{8}+\frac{29528400529807}{104775823556608}a^{6}+\frac{28339980338601}{13096977944576}a^{4}+\frac{390418351929}{82892265472}a^{2}-\frac{1045415721}{323797912}$, $\frac{50386809}{65\cdots 28}a^{16}+\frac{297224613}{82\cdots 32}a^{14}+\frac{24448613583}{41\cdots 16}a^{12}+\frac{1795985428893}{41\cdots 16}a^{10}+\frac{1652733987885}{104775823556608}a^{8}+\frac{29528400529807}{104775823556608}a^{6}+\frac{28339980338601}{13096977944576}a^{4}+\frac{390418351929}{82892265472}a^{2}-\frac{1693011545}{323797912}$, $\frac{11108992819}{44\cdots 16}a^{16}+\frac{60832542895}{55\cdots 04}a^{14}+\frac{4385738389877}{27\cdots 52}a^{12}+\frac{254416270808871}{27\cdots 52}a^{10}+\frac{166998060698255}{70877762994176}a^{8}+\frac{19\cdots 29}{70877762994176}a^{6}+\frac{12\cdots 63}{8859720374272}a^{4}+\frac{13744516924235}{56074179584}a^{2}+\frac{297975921}{219039764}$, $\frac{120451908003}{37\cdots 36}a^{16}+\frac{678086844403}{47\cdots 84}a^{14}+\frac{51535727176621}{23\cdots 92}a^{12}+\frac{33\cdots 43}{23\cdots 92}a^{10}+\frac{26\cdots 63}{602460985450496}a^{8}+\frac{40\cdots 25}{602460985450496}a^{6}+\frac{36\cdots 27}{75307623181312}a^{4}+\frac{603354369151055}{476630526464}a^{2}+\frac{10285255711}{1861837994}$, $\frac{2402805989}{19\cdots 36}a^{16}+\frac{1046701570143}{19\cdots 36}a^{14}+\frac{76450631626829}{95\cdots 68}a^{12}+\frac{57593908931241}{12\cdots 92}a^{10}+\frac{31\cdots 31}{24\cdots 84}a^{8}+\frac{37\cdots 17}{24\cdots 84}a^{6}+\frac{312979943096053}{3813044211712}a^{4}+\frac{288142166035195}{1906522105856}a^{2}+\frac{4906357605}{7447351976}$, $\frac{358989449}{95\cdots 68}a^{16}+\frac{162252713915}{95\cdots 68}a^{14}+\frac{12592099637297}{47\cdots 84}a^{12}+\frac{10387784232013}{602460985450496}a^{10}+\frac{585651771981203}{12\cdots 92}a^{8}+\frac{48\cdots 01}{12\cdots 92}a^{6}-\frac{83870623169527}{1906522105856}a^{4}-\frac{542445962179321}{953261052928}a^{2}-\frac{5538255465187}{3723675988}$, $\frac{10954838739}{95\cdots 68}a^{16}+\frac{4749121790009}{95\cdots 68}a^{14}+\frac{343764507805995}{47\cdots 84}a^{12}+\frac{254464092099711}{602460985450496}a^{10}+\frac{13\cdots 77}{12\cdots 92}a^{8}+\frac{15\cdots 83}{12\cdots 92}a^{6}+\frac{12\cdots 71}{1906522105856}a^{4}+\frac{10\cdots 33}{953261052928}a^{2}+\frac{20193923567}{3723675988}$, $\frac{691679282051}{19\cdots 36}a^{16}+\frac{323969558549881}{19\cdots 36}a^{14}+\frac{26\cdots 63}{95\cdots 68}a^{12}+\frac{25\cdots 51}{12\cdots 92}a^{10}+\frac{18\cdots 97}{24\cdots 84}a^{8}+\frac{34\cdots 95}{24\cdots 84}a^{6}+\frac{43\cdots 75}{3813044211712}a^{4}+\frac{57\cdots 69}{1906522105856}a^{2}+\frac{9798611969411}{7447351976}$ 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:  \( 32038743174.66701 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 32038743174.66701 \cdot 18957120}{2\cdot\sqrt{33703610765963698533467381091270298700673024}}\cr\approx \mathstrut & 798.359384337643 \end{aligned}\] (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^18 + 474*x^16 + 80343*x^14 + 6224884*x^12 + 246937647*x^10 + 5138452458*x^8 + 53819444489*x^6 + 260620415400*x^4 + 463401439632*x^2 + 2019487744) 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 + 474*x^16 + 80343*x^14 + 6224884*x^12 + 246937647*x^10 + 5138452458*x^8 + 53819444489*x^6 + 260620415400*x^4 + 463401439632*x^2 + 2019487744, 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 + 474*x^16 + 80343*x^14 + 6224884*x^12 + 246937647*x^10 + 5138452458*x^8 + 53819444489*x^6 + 260620415400*x^4 + 463401439632*x^2 + 2019487744); 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^18 + 474*x^16 + 80343*x^14 + 6224884*x^12 + 246937647*x^10 + 5138452458*x^8 + 53819444489*x^6 + 260620415400*x^4 + 463401439632*x^2 + 2019487744); 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

$C_6\times S_3$ (as 18T6):

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 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-79}) \), 3.3.505521.1, 3.3.316.1, 6.0.7888624.1, 6.0.20188567033839.2, 9.9.653167654112852807616.1

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

Galois closure: data not computed
Degree 12 sibling: 12.0.1669440466451705194787135164416.3
Degree 18 sibling: 18.0.27304191000274388685340663162548090086621184.1
Minimal sibling: 12.0.1669440466451705194787135164416.3

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 ${\href{/padicField/5.3.0.1}{3} }^{6}$ ${\href{/padicField/7.6.0.1}{6} }^{3}$ ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ ${\href{/padicField/17.2.0.1}{2} }^{9}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{6}$ ${\href{/padicField/29.6.0.1}{6} }^{3}$ ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }^{3}$ ${\href{/padicField/43.2.0.1}{2} }^{9}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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 $\Q_{2}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
2.1.2.2a1.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$$[2]$$
2.1.2.2a1.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$$[2]$$
2.1.2.2a1.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$$[2]$$
2.1.2.2a1.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$$[2]$$
2.1.2.2a1.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$$[2]$$
2.1.2.2a1.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$$[2]$$
\(3\) Copy content Toggle raw display 3.6.3.24a2.1$x^{18} + 6 x^{16} + 15 x^{14} + 6 x^{13} + 32 x^{12} + 24 x^{11} + 63 x^{10} + 36 x^{9} + 90 x^{8} + 72 x^{7} + 109 x^{6} + 102 x^{5} + 96 x^{4} + 56 x^{3} + 84 x^{2} + 72 x + 35$$3$$6$$24$not computednot computed
\(79\) Copy content Toggle raw display 79.1.6.5a1.5$x^{6} + 711$$6$$1$$5$$C_6$$$[\ ]_{6}$$
79.2.6.10a1.3$x^{12} + 468 x^{11} + 91278 x^{10} + 9498060 x^{9} + 556321095 x^{8} + 17408507688 x^{7} + 228535884324 x^{6} + 52225523064 x^{5} + 5006889855 x^{4} + 256447620 x^{3} + 7393518 x^{2} + 119570 x + 1203$$6$$2$$10$$C_6\times C_2$$$[\ ]_{6}^{2}$$

Spectrum of ring of integers

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