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Properties

Label 15.3.293...000.1
Degree $15$
Signature $(3, 6)$
Discriminant $2.939\times 10^{27}$
Root discriminant \(67.80\)
Ramified primes $2,3,5$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_5 \times S_3$ (as 15T29)

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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 - 15*x^13 - 40*x^12 + 720*x^10 - 2480*x^9 + 11700*x^8 - 42150*x^7 + 89040*x^6 - 228150*x^5 + 327600*x^4 - 478980*x^3 + 412200*x^2 - 279300*x + 94480)
 
Copy content gp:K = bnfinit(y^15 - 15*y^13 - 40*y^12 + 720*y^10 - 2480*y^9 + 11700*y^8 - 42150*y^7 + 89040*y^6 - 228150*y^5 + 327600*y^4 - 478980*y^3 + 412200*y^2 - 279300*y + 94480, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 15*x^13 - 40*x^12 + 720*x^10 - 2480*x^9 + 11700*x^8 - 42150*x^7 + 89040*x^6 - 228150*x^5 + 327600*x^4 - 478980*x^3 + 412200*x^2 - 279300*x + 94480);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^15 - 15*x^13 - 40*x^12 + 720*x^10 - 2480*x^9 + 11700*x^8 - 42150*x^7 + 89040*x^6 - 228150*x^5 + 327600*x^4 - 478980*x^3 + 412200*x^2 - 279300*x + 94480)
 

\( x^{15} - 15 x^{13} - 40 x^{12} + 720 x^{10} - 2480 x^{9} + 11700 x^{8} - 42150 x^{7} + 89040 x^{6} + \cdots + 94480 \) 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:   \(2938656153600000000000000000\) \(\medspace = 2^{28}\cdot 3^{15}\cdot 5^{17}\) 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:  \(67.80\)
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:  $2^{7/3}3^{7/6}5^{71/60}\approx 121.94367865849007$
Ramified primes:   \(2\), \(3\), \(5\) 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:  \(\Q(\sqrt{15}) \)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}$, $\frac{1}{19\cdots 76}a^{14}+\frac{87\cdots 63}{49\cdots 69}a^{13}-\frac{35\cdots 83}{19\cdots 76}a^{12}-\frac{29\cdots 93}{49\cdots 69}a^{11}-\frac{17\cdots 84}{49\cdots 69}a^{10}-\frac{17\cdots 09}{49\cdots 69}a^{9}+\frac{16\cdots 29}{49\cdots 69}a^{8}-\frac{23\cdots 91}{49\cdots 69}a^{7}+\frac{52\cdots 45}{14\cdots 34}a^{6}+\frac{26\cdots 90}{70\cdots 67}a^{5}+\frac{37\cdots 35}{98\cdots 38}a^{4}-\frac{25\cdots 89}{49\cdots 69}a^{3}+\frac{22\cdots 22}{49\cdots 69}a^{2}+\frac{13\cdots 34}{49\cdots 69}a+\frac{66\cdots 29}{49\cdots 69}$ 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:  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:  $8$
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:   $\frac{13\cdots 10}{70\cdots 59}a^{14}-\frac{20\cdots 67}{14\cdots 18}a^{13}+\frac{49\cdots 47}{14\cdots 18}a^{12}+\frac{71\cdots 80}{70\cdots 59}a^{11}-\frac{10\cdots 17}{14\cdots 18}a^{10}-\frac{24\cdots 49}{14\cdots 18}a^{9}+\frac{27\cdots 15}{70\cdots 59}a^{8}-\frac{78\cdots 70}{70\cdots 59}a^{7}+\frac{45\cdots 92}{10\cdots 37}a^{6}-\frac{65\cdots 70}{10\cdots 37}a^{5}+\frac{78\cdots 35}{70\cdots 59}a^{4}-\frac{35\cdots 61}{70\cdots 59}a^{3}+\frac{89\cdots 40}{70\cdots 59}a^{2}+\frac{50\cdots 15}{70\cdots 59}a-\frac{24\cdots 71}{70\cdots 59}$, $\frac{23\cdots 94}{49\cdots 69}a^{14}+\frac{42\cdots 15}{98\cdots 38}a^{13}+\frac{52\cdots 29}{98\cdots 38}a^{12}-\frac{75\cdots 13}{98\cdots 38}a^{11}-\frac{28\cdots 41}{98\cdots 38}a^{10}-\frac{57\cdots 43}{49\cdots 69}a^{9}+\frac{18\cdots 10}{49\cdots 69}a^{8}-\frac{26\cdots 62}{49\cdots 69}a^{7}+\frac{15\cdots 39}{70\cdots 67}a^{6}-\frac{69\cdots 91}{70\cdots 67}a^{5}+\frac{46\cdots 06}{49\cdots 69}a^{4}-\frac{13\cdots 19}{49\cdots 69}a^{3}+\frac{61\cdots 56}{49\cdots 69}a^{2}-\frac{10\cdots 61}{49\cdots 69}a+\frac{34\cdots 39}{49\cdots 69}$, $\frac{19\cdots 67}{98\cdots 38}a^{14}+\frac{38\cdots 49}{98\cdots 38}a^{13}+\frac{36\cdots 35}{98\cdots 38}a^{12}+\frac{12\cdots 72}{49\cdots 69}a^{11}-\frac{12\cdots 07}{49\cdots 69}a^{10}-\frac{17\cdots 99}{98\cdots 38}a^{9}+\frac{38\cdots 27}{49\cdots 69}a^{8}-\frac{13\cdots 32}{49\cdots 69}a^{7}+\frac{81\cdots 27}{70\cdots 67}a^{6}-\frac{19\cdots 42}{70\cdots 67}a^{5}+\frac{27\cdots 91}{49\cdots 69}a^{4}-\frac{53\cdots 28}{49\cdots 69}a^{3}+\frac{53\cdots 85}{49\cdots 69}a^{2}-\frac{60\cdots 03}{49\cdots 69}a+\frac{16\cdots 01}{49\cdots 69}$, $\frac{19\cdots 15}{70\cdots 59}a^{14}-\frac{21\cdots 93}{14\cdots 18}a^{13}+\frac{28\cdots 02}{70\cdots 59}a^{12}+\frac{18\cdots 25}{14\cdots 18}a^{11}+\frac{50\cdots 55}{70\cdots 59}a^{10}-\frac{13\cdots 97}{70\cdots 59}a^{9}+\frac{41\cdots 65}{70\cdots 59}a^{8}-\frac{20\cdots 80}{70\cdots 59}a^{7}+\frac{10\cdots 29}{10\cdots 37}a^{6}-\frac{19\cdots 15}{10\cdots 37}a^{5}+\frac{36\cdots 00}{70\cdots 59}a^{4}-\frac{42\cdots 15}{70\cdots 59}a^{3}+\frac{68\cdots 50}{70\cdots 59}a^{2}-\frac{39\cdots 90}{70\cdots 59}a+\frac{30\cdots 39}{70\cdots 59}$, $\frac{89\cdots 93}{15\cdots 06}a^{14}-\frac{86\cdots 40}{76\cdots 03}a^{13}+\frac{13\cdots 25}{15\cdots 06}a^{12}+\frac{33\cdots 69}{76\cdots 03}a^{11}+\frac{31\cdots 60}{76\cdots 03}a^{10}-\frac{37\cdots 40}{76\cdots 03}a^{9}+\frac{39\cdots 09}{76\cdots 03}a^{8}-\frac{28\cdots 60}{76\cdots 03}a^{7}+\frac{10\cdots 05}{76\cdots 03}a^{6}-\frac{68\cdots 68}{76\cdots 03}a^{5}+\frac{37\cdots 00}{76\cdots 03}a^{4}-\frac{14\cdots 00}{76\cdots 03}a^{3}+\frac{42\cdots 70}{76\cdots 03}a^{2}-\frac{70\cdots 00}{76\cdots 03}a-\frac{12\cdots 97}{76\cdots 03}$, $\frac{75\cdots 52}{76\cdots 03}a^{14}+\frac{61\cdots 30}{76\cdots 03}a^{13}-\frac{11\cdots 75}{76\cdots 03}a^{12}-\frac{40\cdots 32}{76\cdots 03}a^{11}-\frac{21\cdots 40}{76\cdots 03}a^{10}+\frac{56\cdots 20}{76\cdots 03}a^{9}-\frac{13\cdots 27}{76\cdots 03}a^{8}+\frac{70\cdots 80}{76\cdots 03}a^{7}-\frac{24\cdots 40}{76\cdots 03}a^{6}+\frac{39\cdots 04}{76\cdots 03}a^{5}-\frac{11\cdots 50}{76\cdots 03}a^{4}+\frac{11\cdots 00}{76\cdots 03}a^{3}-\frac{15\cdots 60}{76\cdots 03}a^{2}+\frac{70\cdots 00}{76\cdots 03}a-\frac{44\cdots 77}{76\cdots 03}$, $\frac{32\cdots 56}{49\cdots 69}a^{14}+\frac{28\cdots 51}{98\cdots 38}a^{13}+\frac{56\cdots 45}{49\cdots 69}a^{12}+\frac{22\cdots 71}{98\cdots 38}a^{11}-\frac{19\cdots 36}{49\cdots 69}a^{10}-\frac{27\cdots 70}{49\cdots 69}a^{9}+\frac{91\cdots 47}{49\cdots 69}a^{8}-\frac{35\cdots 57}{49\cdots 69}a^{7}+\frac{19\cdots 71}{70\cdots 67}a^{6}-\frac{39\cdots 96}{70\cdots 67}a^{5}+\frac{59\cdots 07}{49\cdots 69}a^{4}-\frac{91\cdots 90}{49\cdots 69}a^{3}+\frac{89\cdots 67}{49\cdots 69}a^{2}-\frac{58\cdots 66}{49\cdots 69}a+\frac{16\cdots 91}{49\cdots 69}$, $\frac{10\cdots 89}{98\cdots 38}a^{14}+\frac{15\cdots 15}{49\cdots 69}a^{13}-\frac{10\cdots 09}{98\cdots 38}a^{12}-\frac{26\cdots 79}{49\cdots 69}a^{11}-\frac{25\cdots 33}{49\cdots 69}a^{10}+\frac{10\cdots 36}{49\cdots 69}a^{9}+\frac{11\cdots 42}{49\cdots 69}a^{8}-\frac{60\cdots 66}{49\cdots 69}a^{7}+\frac{30\cdots 69}{70\cdots 67}a^{6}-\frac{84\cdots 64}{70\cdots 67}a^{5}+\frac{91\cdots 34}{49\cdots 69}a^{4}-\frac{13\cdots 12}{49\cdots 69}a^{3}+\frac{10\cdots 29}{49\cdots 69}a^{2}-\frac{70\cdots 33}{49\cdots 69}a+\frac{14\cdots 19}{49\cdots 69}$ 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:  \( 1044480146.3607405 \) (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)
 
Unit signature rank:  \( 3 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 1044480146.3607405 \cdot 1}{2\cdot\sqrt{2938656153600000000000000000}}\cr\approx \mathstrut & 4.74203771982278 \end{aligned}\] (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^15 - 15*x^13 - 40*x^12 + 720*x^10 - 2480*x^9 + 11700*x^8 - 42150*x^7 + 89040*x^6 - 228150*x^5 + 327600*x^4 - 478980*x^3 + 412200*x^2 - 279300*x + 94480) 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 - 15*x^13 - 40*x^12 + 720*x^10 - 2480*x^9 + 11700*x^8 - 42150*x^7 + 89040*x^6 - 228150*x^5 + 327600*x^4 - 478980*x^3 + 412200*x^2 - 279300*x + 94480, 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 - 15*x^13 - 40*x^12 + 720*x^10 - 2480*x^9 + 11700*x^8 - 42150*x^7 + 89040*x^6 - 228150*x^5 + 327600*x^4 - 478980*x^3 + 412200*x^2 - 279300*x + 94480); 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 - 15*x^13 - 40*x^12 + 720*x^10 - 2480*x^9 + 11700*x^8 - 42150*x^7 + 89040*x^6 - 228150*x^5 + 327600*x^4 - 478980*x^3 + 412200*x^2 - 279300*x + 94480); 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

$S_3\times S_5$ (as 15T29):

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 non-solvable group of order 720
The 21 conjugacy class representatives for $S_5 \times S_3$
Character table for $S_5 \times S_3$

Intermediate fields

3.3.2700.1, 5.1.800000.1

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 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 siblings: data not computed
Degree 45 sibling: 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 ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ $15$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{3}$ ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ $15$

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$$[\ ]$$
2.1.2.2a1.2$x^{2} + 2 x + 6$$2$$1$$2$$C_2$$$[2]$$
2.1.4.8a1.1$x^{4} + 4 x + 2$$4$$1$$8$$S_4$$$[\frac{8}{3}, \frac{8}{3}]_{3}^{2}$$
2.1.8.18c1.5$x^{8} + 4 x^{5} + 2 x^{4} + 4 x^{3} + 6$$8$$1$$18$$S_4\times C_2$$$[2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$$
\(3\) Copy content Toggle raw display 3.2.3.6a1.1$x^{6} + 6 x^{5} + 18 x^{4} + 32 x^{3} + 42 x^{2} + 36 x + 23$$3$$2$$6$$D_{6}$$$[\frac{3}{2}]_{2}^{2}$$
3.3.3.9a1.1$x^{9} + 6 x^{7} + 3 x^{6} + 12 x^{5} + 12 x^{4} + 17 x^{3} + 12 x^{2} + 18 x + 10$$3$$3$$9$$S_3\times C_3$$$[\frac{3}{2}]_{2}^{3}$$
\(5\) Copy content Toggle raw display 5.1.15.17a1.4$x^{15} + 20 x^{3} + 5$$15$$1$$17$$F_5 \times S_3$$$[\frac{5}{4}]_{12}^{2}$$

Spectrum of ring of integers

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