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Properties

Label 15.3.283...000.1
Degree $15$
Signature $(3, 6)$
Discriminant $2.834\times 10^{24}$
Root discriminant \(42.67\)
Ramified primes $2,5,11$
Class number $1$
Class group trivial
Galois group $C_3^4:(S_3\times F_5)$ (as 15T64)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 10*x^13 - 20*x^12 + 15*x^11 - 21*x^10 + 175*x^9 - 295*x^8 + 785*x^7 - 845*x^6 + 1247*x^5 - 2065*x^4 + 2675*x^3 - 1225*x^2 + 40*x + 72)
 
Copy content gp:K = bnfinit(y^15 - 5*y^14 + 10*y^13 - 20*y^12 + 15*y^11 - 21*y^10 + 175*y^9 - 295*y^8 + 785*y^7 - 845*y^6 + 1247*y^5 - 2065*y^4 + 2675*y^3 - 1225*y^2 + 40*y + 72, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 5*x^14 + 10*x^13 - 20*x^12 + 15*x^11 - 21*x^10 + 175*x^9 - 295*x^8 + 785*x^7 - 845*x^6 + 1247*x^5 - 2065*x^4 + 2675*x^3 - 1225*x^2 + 40*x + 72);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^15 - 5*x^14 + 10*x^13 - 20*x^12 + 15*x^11 - 21*x^10 + 175*x^9 - 295*x^8 + 785*x^7 - 845*x^6 + 1247*x^5 - 2065*x^4 + 2675*x^3 - 1225*x^2 + 40*x + 72)
 

\( x^{15} - 5 x^{14} + 10 x^{13} - 20 x^{12} + 15 x^{11} - 21 x^{10} + 175 x^{9} - 295 x^{8} + 785 x^{7} + \cdots + 72 \) 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:  $15$
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:  $(3, 6)$
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:   \(2834497600000000000000000\) \(\medspace = 2^{21}\cdot 5^{17}\cdot 11^{6}\) 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:  \(42.67\)
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:   \(2\), \(5\), \(11\) 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:  \(\Q(\sqrt{10}) \)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{58\cdots 96}a^{14}-\frac{84\cdots 01}{29\cdots 98}a^{13}+\frac{15\cdots 11}{14\cdots 99}a^{12}-\frac{69\cdots 65}{29\cdots 98}a^{11}-\frac{13\cdots 49}{58\cdots 96}a^{10}-\frac{46\cdots 17}{29\cdots 98}a^{9}+\frac{98\cdots 15}{58\cdots 96}a^{8}+\frac{48\cdots 31}{29\cdots 98}a^{7}+\frac{12\cdots 87}{58\cdots 96}a^{6}+\frac{86\cdots 23}{29\cdots 98}a^{5}+\frac{13\cdots 15}{58\cdots 96}a^{4}+\frac{13\cdots 15}{14\cdots 99}a^{3}+\frac{24\cdots 51}{58\cdots 96}a^{2}+\frac{30\cdots 45}{29\cdots 98}a+\frac{54\cdots 55}{14\cdots 99}$ 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:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$
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 $1$
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:  $8$
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:   $\frac{83\cdots 55}{29\cdots 98}a^{14}-\frac{35\cdots 65}{29\cdots 98}a^{13}+\frac{25\cdots 39}{14\cdots 99}a^{12}-\frac{10\cdots 09}{29\cdots 98}a^{11}+\frac{39\cdots 27}{29\cdots 98}a^{10}-\frac{54\cdots 42}{14\cdots 99}a^{9}+\frac{13\cdots 77}{29\cdots 98}a^{8}-\frac{15\cdots 81}{29\cdots 98}a^{7}+\frac{46\cdots 91}{29\cdots 98}a^{6}-\frac{16\cdots 30}{14\cdots 99}a^{5}+\frac{61\cdots 69}{29\cdots 98}a^{4}-\frac{10\cdots 49}{29\cdots 98}a^{3}+\frac{14\cdots 91}{29\cdots 98}a^{2}-\frac{23\cdots 16}{14\cdots 99}a+\frac{14\cdots 81}{14\cdots 99}$, $\frac{19\cdots 74}{14\cdots 99}a^{14}-\frac{81\cdots 40}{14\cdots 99}a^{13}+\frac{24\cdots 65}{29\cdots 98}a^{12}-\frac{29\cdots 81}{14\cdots 99}a^{11}+\frac{18\cdots 89}{29\cdots 98}a^{10}-\frac{22\cdots 98}{14\cdots 99}a^{9}+\frac{31\cdots 66}{14\cdots 99}a^{8}-\frac{28\cdots 53}{14\cdots 99}a^{7}+\frac{24\cdots 37}{29\cdots 98}a^{6}-\frac{73\cdots 32}{14\cdots 99}a^{5}+\frac{16\cdots 46}{14\cdots 99}a^{4}-\frac{28\cdots 51}{14\cdots 99}a^{3}+\frac{72\cdots 59}{29\cdots 98}a^{2}-\frac{47\cdots 12}{14\cdots 99}a-\frac{27\cdots 93}{14\cdots 99}$, $\frac{91\cdots 87}{58\cdots 96}a^{14}-\frac{75\cdots 39}{29\cdots 98}a^{13}-\frac{11\cdots 57}{29\cdots 98}a^{12}-\frac{12\cdots 91}{14\cdots 99}a^{11}-\frac{19\cdots 57}{58\cdots 96}a^{10}-\frac{63\cdots 15}{14\cdots 99}a^{9}+\frac{10\cdots 57}{58\cdots 96}a^{8}+\frac{10\cdots 65}{29\cdots 98}a^{7}+\frac{47\cdots 63}{58\cdots 96}a^{6}+\frac{21\cdots 47}{14\cdots 99}a^{5}+\frac{10\cdots 09}{58\cdots 96}a^{4}+\frac{12\cdots 47}{14\cdots 99}a^{3}-\frac{11\cdots 01}{58\cdots 96}a^{2}+\frac{41\cdots 13}{14\cdots 99}a+\frac{25\cdots 97}{14\cdots 99}$, $\frac{75\cdots 18}{14\cdots 99}a^{14}-\frac{30\cdots 26}{14\cdots 99}a^{13}+\frac{75\cdots 75}{29\cdots 98}a^{12}-\frac{15\cdots 11}{29\cdots 98}a^{11}-\frac{15\cdots 31}{29\cdots 98}a^{10}-\frac{15\cdots 65}{29\cdots 98}a^{9}+\frac{11\cdots 75}{14\cdots 99}a^{8}-\frac{91\cdots 77}{14\cdots 99}a^{7}+\frac{77\cdots 87}{29\cdots 98}a^{6}-\frac{38\cdots 95}{29\cdots 98}a^{5}+\frac{55\cdots 46}{14\cdots 99}a^{4}-\frac{78\cdots 33}{14\cdots 99}a^{3}+\frac{17\cdots 11}{29\cdots 98}a^{2}+\frac{44\cdots 39}{29\cdots 98}a-\frac{13\cdots 49}{14\cdots 99}$, $\frac{93\cdots 01}{14\cdots 99}a^{14}-\frac{10\cdots 85}{29\cdots 98}a^{13}+\frac{11\cdots 61}{14\cdots 99}a^{12}-\frac{21\cdots 49}{14\cdots 99}a^{11}+\frac{19\cdots 61}{14\cdots 99}a^{10}-\frac{40\cdots 75}{29\cdots 98}a^{9}+\frac{16\cdots 55}{14\cdots 99}a^{8}-\frac{67\cdots 99}{29\cdots 98}a^{7}+\frac{80\cdots 92}{14\cdots 99}a^{6}-\frac{20\cdots 37}{29\cdots 98}a^{5}+\frac{14\cdots 86}{14\cdots 99}a^{4}-\frac{46\cdots 29}{29\cdots 98}a^{3}+\frac{31\cdots 51}{14\cdots 99}a^{2}-\frac{38\cdots 35}{29\cdots 98}a+\frac{43\cdots 13}{14\cdots 99}$, $\frac{26\cdots 16}{14\cdots 99}a^{14}-\frac{11\cdots 33}{14\cdots 99}a^{13}+\frac{18\cdots 93}{14\cdots 99}a^{12}-\frac{45\cdots 37}{14\cdots 99}a^{11}+\frac{23\cdots 35}{14\cdots 99}a^{10}-\frac{38\cdots 53}{14\cdots 99}a^{9}+\frac{42\cdots 37}{14\cdots 99}a^{8}-\frac{50\cdots 07}{14\cdots 99}a^{7}+\frac{18\cdots 18}{14\cdots 99}a^{6}-\frac{14\cdots 62}{14\cdots 99}a^{5}+\frac{23\cdots 73}{14\cdots 99}a^{4}-\frac{41\cdots 18}{14\cdots 99}a^{3}+\frac{54\cdots 53}{14\cdots 99}a^{2}-\frac{11\cdots 55}{14\cdots 99}a-\frac{43\cdots 53}{14\cdots 99}$, $\frac{10\cdots 85}{14\cdots 99}a^{14}-\frac{90\cdots 63}{29\cdots 98}a^{13}+\frac{76\cdots 70}{14\cdots 99}a^{12}-\frac{16\cdots 47}{14\cdots 99}a^{11}+\frac{15\cdots 06}{14\cdots 99}a^{10}-\frac{32\cdots 41}{29\cdots 98}a^{9}+\frac{16\cdots 09}{14\cdots 99}a^{8}-\frac{41\cdots 31}{29\cdots 98}a^{7}+\frac{78\cdots 20}{14\cdots 99}a^{6}-\frac{57\cdots 99}{29\cdots 98}a^{5}+\frac{10\cdots 62}{14\cdots 99}a^{4}-\frac{19\cdots 67}{29\cdots 98}a^{3}+\frac{19\cdots 64}{14\cdots 99}a^{2}-\frac{31\cdots 05}{29\cdots 98}a+\frac{39\cdots 49}{14\cdots 99}$, $\frac{69\cdots 79}{29\cdots 98}a^{14}-\frac{31\cdots 49}{29\cdots 98}a^{13}+\frac{53\cdots 45}{29\cdots 98}a^{12}-\frac{11\cdots 63}{29\cdots 98}a^{11}+\frac{21\cdots 48}{14\cdots 99}a^{10}-\frac{58\cdots 07}{14\cdots 99}a^{9}+\frac{11\cdots 75}{29\cdots 98}a^{8}-\frac{14\cdots 27}{29\cdots 98}a^{7}+\frac{23\cdots 22}{14\cdots 99}a^{6}-\frac{16\cdots 06}{14\cdots 99}a^{5}+\frac{66\cdots 95}{29\cdots 98}a^{4}-\frac{10\cdots 09}{29\cdots 98}a^{3}+\frac{61\cdots 96}{14\cdots 99}a^{2}-\frac{72\cdots 52}{14\cdots 99}a-\frac{39\cdots 41}{14\cdots 99}$ Copy content Toggle raw display
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:  \( 25661881.7232 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  \( 3 \)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 25661881.7232 \cdot 1}{2\cdot\sqrt{2834497600000000000000000}}\cr\approx \mathstrut & 3.75136786328 \end{aligned}\]

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^15 - 5*x^14 + 10*x^13 - 20*x^12 + 15*x^11 - 21*x^10 + 175*x^9 - 295*x^8 + 785*x^7 - 845*x^6 + 1247*x^5 - 2065*x^4 + 2675*x^3 - 1225*x^2 + 40*x + 72) 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^15 - 5*x^14 + 10*x^13 - 20*x^12 + 15*x^11 - 21*x^10 + 175*x^9 - 295*x^8 + 785*x^7 - 845*x^6 + 1247*x^5 - 2065*x^4 + 2675*x^3 - 1225*x^2 + 40*x + 72, 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^15 - 5*x^14 + 10*x^13 - 20*x^12 + 15*x^11 - 21*x^10 + 175*x^9 - 295*x^8 + 785*x^7 - 845*x^6 + 1247*x^5 - 2065*x^4 + 2675*x^3 - 1225*x^2 + 40*x + 72); 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^15 - 5*x^14 + 10*x^13 - 20*x^12 + 15*x^11 - 21*x^10 + 175*x^9 - 295*x^8 + 785*x^7 - 845*x^6 + 1247*x^5 - 2065*x^4 + 2675*x^3 - 1225*x^2 + 40*x + 72); 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

$C_3^4:(S_3\times F_5)$ (as 15T64):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
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 9720
The 45 conjugacy class representatives for $C_3^4:(S_3\times F_5)$
Character table for $C_3^4:(S_3\times F_5)$

Intermediate fields

5.1.200000.1

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 15 sibling: data not computed
Degree 30 siblings: data not computed
Degree 45 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 ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ R ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{3}$ R ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ $15$ ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ $15$ ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }$

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 $\Q_{2}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
2.1.2.3a1.3$x^{2} + 4 x + 2$$2$$1$$3$$C_2$$$[3]$$
2.2.2.6a1.6$x^{4} + 2 x^{3} + 7 x^{2} + 14 x + 7$$2$$2$$6$$C_4$$$[3]^{2}$$
2.4.2.12a1.9$x^{8} + 2 x^{5} + 6 x^{4} + x^{2} + 6 x + 7$$2$$4$$12$$C_4\times C_2$$$[3]^{4}$$
\(5\) Copy content Toggle raw display 5.1.15.17a1.4$x^{15} + 20 x^{3} + 5$$15$$1$$17$$F_5 \times S_3$$$[\frac{5}{4}]_{12}^{2}$$
\(11\) Copy content Toggle raw display 11.1.3.2a1.1$x^{3} + 11$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
11.6.1.0a1.1$x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
11.2.3.4a1.2$x^{6} + 21 x^{5} + 153 x^{4} + 427 x^{3} + 306 x^{2} + 84 x + 19$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$

Spectrum of ring of integers

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