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Properties

Label 12.0.131...000.1
Degree 1212
Signature [0,6][0, 6]
Discriminant 1.311×10271.311\times 10^{27}
Root discriminant 181.89181.89
Ramified primes 2,3,5,372,3,5,37
Class number 3096430964 (GRH)
Class group [2, 15482] (GRH)
Galois group C12C_{12} (as 12T1)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 48*x^10 - 204*x^9 + 2323*x^8 - 3348*x^7 + 45974*x^6 - 100824*x^5 + 707976*x^4 + 849892*x^3 + 16302428*x^2 + 25990548*x + 84787321)
 
Copy content gp:K = bnfinit(y^12 - 4*y^11 + 48*y^10 - 204*y^9 + 2323*y^8 - 3348*y^7 + 45974*y^6 - 100824*y^5 + 707976*y^4 + 849892*y^3 + 16302428*y^2 + 25990548*y + 84787321, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 48*x^10 - 204*x^9 + 2323*x^8 - 3348*x^7 + 45974*x^6 - 100824*x^5 + 707976*x^4 + 849892*x^3 + 16302428*x^2 + 25990548*x + 84787321);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 48*x^10 - 204*x^9 + 2323*x^8 - 3348*x^7 + 45974*x^6 - 100824*x^5 + 707976*x^4 + 849892*x^3 + 16302428*x^2 + 25990548*x + 84787321)
 

x12−4x11+48x10−204x9+2323x8−3348x7+45974x6−100824x5+⋯+84787321 x^{12} - 4 x^{11} + 48 x^{10} - 204 x^{9} + 2323 x^{8} - 3348 x^{7} + 45974 x^{6} - 100824 x^{5} + \cdots + 84787321 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:  1212
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:  [0,6][0, 6]
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:   13110259312171054080000000001311025931217105408000000000  =218⋅36⋅59⋅378\medspace = 2^{18}\cdot 3^{6}\cdot 5^{9}\cdot 37^{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:  181.89181.89
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:  23/231/253/4372/3≈181.886698576817682^{3/2}3^{1/2}5^{3/4}37^{2/3}\approx 181.88669857681768
Ramified primes:   22, 33, 55, 3737 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(5)\Q(\sqrt{5})
Aut⁡(K/Q)\Aut(K/\Q) == Gal⁡(K/Q)\Gal(K/\Q):   C12C_{12}
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois and abelian over Q\Q.
Conductor:  4440=23⋅3⋅5⋅374440=2^{3}\cdot 3\cdot 5\cdot 37
Dirichlet character group:    {\lbraceχ4440(1,⋅)\chi_{4440}(1,·), χ4440(803,⋅)\chi_{4440}(803,·), χ4440(2209,⋅)\chi_{4440}(2209,·), χ4440(1321,⋅)\chi_{4440}(1321,·), χ4440(4043,⋅)\chi_{4440}(4043,·), χ4440(1009,⋅)\chi_{4440}(1009,·), χ4440(2147,⋅)\chi_{4440}(2147,·), χ4440(121,⋅)\chi_{4440}(121,·), χ4440(889,⋅)\chi_{4440}(889,·), χ4440(3923,⋅)\chi_{4440}(3923,·), χ4440(2267,⋅)\chi_{4440}(2267,·), χ4440(3467,⋅)\chi_{4440}(3467,·)}\rbrace
This is a CM field.
Reflex fields:  4.0.72000.22^{2}, 12.0.1311025931217105408000000000.130^{30}

Integral basis (with respect to field generator aa)

11, aa, a2a^{2}, a3a^{3}, a4a^{4}, a5a^{5}, 16a6−13a5+16a4+13a3−13a2+16\frac{1}{6}a^{6}-\frac{1}{3}a^{5}+\frac{1}{6}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{6}, 16a7−12a5−13a4+13a3+13a2+16a+13\frac{1}{6}a^{7}-\frac{1}{2}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{6}a+\frac{1}{3}, 16a8−13a5−16a4+13a3+16a2+13a−12\frac{1}{6}a^{8}-\frac{1}{3}a^{5}-\frac{1}{6}a^{4}+\frac{1}{3}a^{3}+\frac{1}{6}a^{2}+\frac{1}{3}a-\frac{1}{2}, 16a9+16a5−13a4−16a3−13a2−12a+13\frac{1}{6}a^{9}+\frac{1}{6}a^{5}-\frac{1}{3}a^{4}-\frac{1}{6}a^{3}-\frac{1}{3}a^{2}-\frac{1}{2}a+\frac{1}{3}, 119505211846a10−2152601793250868641a9−3625224916501737282a8−2292892953250868641a7+153169321319505211846a6−44867019289752605923a5+8297013529752605923a4−5726383849752605923a3+13611867153250868641a2+25199036359752605923a+1770358436501737282\frac{1}{19505211846}a^{10}-\frac{215260179}{3250868641}a^{9}-\frac{362522491}{6501737282}a^{8}-\frac{229289295}{3250868641}a^{7}+\frac{1531693213}{19505211846}a^{6}-\frac{4486701928}{9752605923}a^{5}+\frac{829701352}{9752605923}a^{4}-\frac{572638384}{9752605923}a^{3}+\frac{1361186715}{3250868641}a^{2}+\frac{2519903635}{9752605923}a+\frac{177035843}{6501737282}, 142⋯46a11+1015022764771⋯41a10+51⋯8671⋯41a9−40⋯0771⋯41a8+10⋯7914⋯82a7−17⋯4921⋯23a6+64⋯8714⋯82a5−68⋯8121⋯23a4−35⋯2314⋯82a3+43⋯4221⋯23a2−61⋯5521⋯23a−67⋯7521⋯23\frac{1}{42\cdots 46}a^{11}+\frac{10150227647}{71\cdots 41}a^{10}+\frac{51\cdots 86}{71\cdots 41}a^{9}-\frac{40\cdots 07}{71\cdots 41}a^{8}+\frac{10\cdots 79}{14\cdots 82}a^{7}-\frac{17\cdots 49}{21\cdots 23}a^{6}+\frac{64\cdots 87}{14\cdots 82}a^{5}-\frac{68\cdots 81}{21\cdots 23}a^{4}-\frac{35\cdots 23}{14\cdots 82}a^{3}+\frac{43\cdots 42}{21\cdots 23}a^{2}-\frac{61\cdots 55}{21\cdots 23}a-\frac{67\cdots 75}{21\cdots 23} 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:  11
Inessential primes:  None

Class group and class number

Ideal class group:  C2×C15482C_{2}\times C_{15482}, which has order 3096430964 (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:  C2×C15482C_{2}\times C_{15482}, which has order 3096430964 (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:   3096430964 (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:  55
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 -1  (order 22) 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:   8149752605923a11−20673250868641a10+430559752605923a9−38406519505211846a8+17295209752605923a7−47393639752605923a6+56984283250868641a5−11277357519505211846a4+4409269609752605923a3+80417461919505211846a2+31155945689752605923a−115262204436501737282\frac{814}{9752605923}a^{11}-\frac{2067}{3250868641}a^{10}+\frac{43055}{9752605923}a^{9}-\frac{384065}{19505211846}a^{8}+\frac{1729520}{9752605923}a^{7}-\frac{4739363}{9752605923}a^{6}+\frac{5698428}{3250868641}a^{5}-\frac{112773575}{19505211846}a^{4}+\frac{440926960}{9752605923}a^{3}+\frac{804174619}{19505211846}a^{2}+\frac{3115594568}{9752605923}a-\frac{11526220443}{6501737282}, 2454357166326071⋯41a11−14⋯8671⋯41a10+10⋯0071⋯41a9−85⋯0571⋯41a8+45⋯2071⋯41a7−34⋯7071⋯41a6+11⋯0871⋯41a5−56⋯0571⋯41a4+17⋯6071⋯41a3−77⋯4071⋯41a2+12⋯4071⋯41a−99⋯5471⋯41\frac{24543571663260}{71\cdots 41}a^{11}-\frac{14\cdots 86}{71\cdots 41}a^{10}+\frac{10\cdots 00}{71\cdots 41}a^{9}-\frac{85\cdots 05}{71\cdots 41}a^{8}+\frac{45\cdots 20}{71\cdots 41}a^{7}-\frac{34\cdots 70}{71\cdots 41}a^{6}+\frac{11\cdots 08}{71\cdots 41}a^{5}-\frac{56\cdots 05}{71\cdots 41}a^{4}+\frac{17\cdots 60}{71\cdots 41}a^{3}-\frac{77\cdots 40}{71\cdots 41}a^{2}+\frac{12\cdots 40}{71\cdots 41}a-\frac{99\cdots 54}{71\cdots 41}, 1430640485320071⋯41a11−23385148644444471⋯41a10+25⋯0071⋯41a9−15⋯5571⋯41a8+10⋯8071⋯41a7−55⋯0071⋯41a6+28⋯4871⋯41a5−80⋯9071⋯41a4+42⋯0071⋯41a3−13⋯4071⋯41a2+45⋯0071⋯41a−17⋯7071⋯41\frac{14306404853200}{71\cdots 41}a^{11}-\frac{233851486444444}{71\cdots 41}a^{10}+\frac{25\cdots 00}{71\cdots 41}a^{9}-\frac{15\cdots 55}{71\cdots 41}a^{8}+\frac{10\cdots 80}{71\cdots 41}a^{7}-\frac{55\cdots 00}{71\cdots 41}a^{6}+\frac{28\cdots 48}{71\cdots 41}a^{5}-\frac{80\cdots 90}{71\cdots 41}a^{4}+\frac{42\cdots 00}{71\cdots 41}a^{3}-\frac{13\cdots 40}{71\cdots 41}a^{2}+\frac{45\cdots 00}{71\cdots 41}a-\frac{17\cdots 70}{71\cdots 41}, 28644283199284871⋯41a11−62⋯6114⋯82a10+21⋯3171⋯41a9−89⋯8742⋯46a8+33⋯1821⋯23a7−17⋯7721⋯23a6+64⋯5521⋯23a5−65⋯4942⋯46a4+42⋯1271⋯41a3−27⋯2221⋯23a2+18⋯6571⋯41a−17⋯0471⋯41\frac{286442831992848}{71\cdots 41}a^{11}-\frac{62\cdots 61}{14\cdots 82}a^{10}+\frac{21\cdots 31}{71\cdots 41}a^{9}-\frac{89\cdots 87}{42\cdots 46}a^{8}+\frac{33\cdots 18}{21\cdots 23}a^{7}-\frac{17\cdots 77}{21\cdots 23}a^{6}+\frac{64\cdots 55}{21\cdots 23}a^{5}-\frac{65\cdots 49}{42\cdots 46}a^{4}+\frac{42\cdots 12}{71\cdots 41}a^{3}-\frac{27\cdots 22}{21\cdots 23}a^{2}+\frac{18\cdots 65}{71\cdots 41}a-\frac{17\cdots 04}{71\cdots 41}, 21495957717815221⋯23a11+13⋯8114⋯82a10−13⋯0321⋯23a9+13⋯3121⋯23a8−13⋯6421⋯23a7+30⋯9914⋯82a6−58⋯7321⋯23a5+11⋯3942⋯46a4+36⋯0421⋯23a3+28⋯4342⋯46a2+23⋯5621⋯23a+29⋯6571⋯41\frac{214959577178152}{21\cdots 23}a^{11}+\frac{13\cdots 81}{14\cdots 82}a^{10}-\frac{13\cdots 03}{21\cdots 23}a^{9}+\frac{13\cdots 31}{21\cdots 23}a^{8}-\frac{13\cdots 64}{21\cdots 23}a^{7}+\frac{30\cdots 99}{14\cdots 82}a^{6}-\frac{58\cdots 73}{21\cdots 23}a^{5}+\frac{11\cdots 39}{42\cdots 46}a^{4}+\frac{36\cdots 04}{21\cdots 23}a^{3}+\frac{28\cdots 43}{42\cdots 46}a^{2}+\frac{23\cdots 56}{21\cdots 23}a+\frac{29\cdots 65}{71\cdots 41} 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:  5133.821582106669 5133.821582106669 (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

lim⁡s→1(s−1)ζK(s)=(2r1⋅(2π)r2⋅R⋅hw⋅∣D∣≈(20⋅(2π)6⋅5133.821582106669⋅309642⋅1311025931217105408000000000≈(0.135064558634208 \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)^{6}\cdot 5133.821582106669 \cdot 30964}{2\cdot\sqrt{1311025931217105408000000000}}\cr\approx \mathstrut & 0.135064558634208 \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^12 - 4*x^11 + 48*x^10 - 204*x^9 + 2323*x^8 - 3348*x^7 + 45974*x^6 - 100824*x^5 + 707976*x^4 + 849892*x^3 + 16302428*x^2 + 25990548*x + 84787321) 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^12 - 4*x^11 + 48*x^10 - 204*x^9 + 2323*x^8 - 3348*x^7 + 45974*x^6 - 100824*x^5 + 707976*x^4 + 849892*x^3 + 16302428*x^2 + 25990548*x + 84787321, 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^12 - 4*x^11 + 48*x^10 - 204*x^9 + 2323*x^8 - 3348*x^7 + 45974*x^6 - 100824*x^5 + 707976*x^4 + 849892*x^3 + 16302428*x^2 + 25990548*x + 84787321); 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^12 - 4*x^11 + 48*x^10 - 204*x^9 + 2323*x^8 - 3348*x^7 + 45974*x^6 - 100824*x^5 + 707976*x^4 + 849892*x^3 + 16302428*x^2 + 25990548*x + 84787321); 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

C12C_{12} (as 12T1):

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 cyclic group of order 12
The 12 conjugacy class representatives for C12C_{12}
Character table for C12C_{12}

Intermediate fields

Q(5)\Q(\sqrt{5}) , 3.3.1369.1, 4.0.72000.2, 6.6.234270125.1

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]
 

Frobenius cycle types

pp 22 33 55 77 1111 1313 1717 1919 2323 2929 3131 3737 4141 4343 4747 5353 5959
Cycle type R R R 12{\href{/padicField/7.12.0.1}{12} } 26{\href{/padicField/11.2.0.1}{2} }^{6} 12{\href{/padicField/13.12.0.1}{12} } 12{\href{/padicField/17.12.0.1}{12} } 62{\href{/padicField/19.6.0.1}{6} }^{2} 43{\href{/padicField/23.4.0.1}{4} }^{3} 26{\href{/padicField/29.2.0.1}{2} }^{6} 26{\href{/padicField/31.2.0.1}{2} }^{6} R 62{\href{/padicField/41.6.0.1}{6} }^{2} 43{\href{/padicField/43.4.0.1}{4} }^{3} 43{\href{/padicField/47.4.0.1}{4} }^{3} 12{\href{/padicField/53.12.0.1}{12} } 34{\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

ppLabelPolynomial ee ff cc Galois group Slope content
22 Copy content Toggle raw display 2.6.2.18a1.2x12+2x10+2x9+x8+4x7+3x6+2x5+4x4+10x3+x2+2x+3x^{12} + 2 x^{10} + 2 x^{9} + x^{8} + 4 x^{7} + 3 x^{6} + 2 x^{5} + 4 x^{4} + 10 x^{3} + x^{2} + 2 x + 322661818C12C_{12}[3]6[3]^{6}
33 Copy content Toggle raw display 3.6.2.6a1.1x12+4x10+6x8+4x7+8x6+8x5+9x4+4x3+8x2+11x+4x^{12} + 4 x^{10} + 6 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} + 9 x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4226666C12C_{12}[ ]26[\ ]_{2}^{6}
55 Copy content Toggle raw display 5.3.4.9a1.3x12+12x10+12x9+54x8+108x7+162x6+324x5+405x4+432x3+486x2+324x+86x^{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 + 86443399C12C_{12}[ ]43[\ ]_{4}^{3}
3737 Copy content Toggle raw display 37.4.3.8a1.3x12+18x10+72x9+114x8+864x7+2016x6+2880x5+10596x4+15552x3+3528x2+288x+45x^{12} + 18 x^{10} + 72 x^{9} + 114 x^{8} + 864 x^{7} + 2016 x^{6} + 2880 x^{5} + 10596 x^{4} + 15552 x^{3} + 3528 x^{2} + 288 x + 45334488C12C_{12}[ ]34[\ ]_{3}^{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)