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Properties

Label 16.0.205...000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2.054\times 10^{38}$
Root discriminant \(248.05\)
Ramified primes $2,5,13,199$
Class number $8$ (GRH)
Class group [2, 4] (GRH)
Galois group $C_2^3:F_8:C_6$ (as 16T1501)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 22*x^14 - 182*x^13 + 2850*x^12 - 16562*x^11 + 47520*x^10 - 112130*x^9 + 848111*x^8 - 2942361*x^7 - 378872*x^6 + 40135950*x^5 - 70072682*x^4 + 26808900*x^3 + 286993746*x^2 - 457552166*x + 337333286)
 
Copy content gp:K = bnfinit(y^16 - 7*y^15 + 22*y^14 - 182*y^13 + 2850*y^12 - 16562*y^11 + 47520*y^10 - 112130*y^9 + 848111*y^8 - 2942361*y^7 - 378872*y^6 + 40135950*y^5 - 70072682*y^4 + 26808900*y^3 + 286993746*y^2 - 457552166*y + 337333286, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 7*x^15 + 22*x^14 - 182*x^13 + 2850*x^12 - 16562*x^11 + 47520*x^10 - 112130*x^9 + 848111*x^8 - 2942361*x^7 - 378872*x^6 + 40135950*x^5 - 70072682*x^4 + 26808900*x^3 + 286993746*x^2 - 457552166*x + 337333286);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 7*x^15 + 22*x^14 - 182*x^13 + 2850*x^12 - 16562*x^11 + 47520*x^10 - 112130*x^9 + 848111*x^8 - 2942361*x^7 - 378872*x^6 + 40135950*x^5 - 70072682*x^4 + 26808900*x^3 + 286993746*x^2 - 457552166*x + 337333286)
 

\( x^{16} - 7 x^{15} + 22 x^{14} - 182 x^{13} + 2850 x^{12} - 16562 x^{11} + 47520 x^{10} + \cdots + 337333286 \) 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:  $16$
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:  $[0, 8]$
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:   \(205433557246404340023920345190400000000\) \(\medspace = 2^{18}\cdot 5^{8}\cdot 13^{8}\cdot 199^{8}\) 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:  \(248.05\)
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:  $2^{45/28}5^{1/2}13^{1/2}199^{2/3}\approx 837.1891731578766$
Ramified primes:   \(2\), \(5\), \(13\), \(199\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{10\cdots 22}a^{15}+\frac{23\cdots 76}{52\cdots 11}a^{14}+\frac{19\cdots 33}{52\cdots 11}a^{13}+\frac{23\cdots 22}{52\cdots 11}a^{12}-\frac{36\cdots 22}{74\cdots 73}a^{11}-\frac{31\cdots 49}{74\cdots 73}a^{10}-\frac{13\cdots 69}{52\cdots 11}a^{9}-\frac{20\cdots 34}{52\cdots 11}a^{8}+\frac{10\cdots 21}{10\cdots 22}a^{7}-\frac{13\cdots 37}{74\cdots 73}a^{6}-\frac{33\cdots 48}{52\cdots 11}a^{5}-\frac{12\cdots 95}{52\cdots 11}a^{4}+\frac{17\cdots 03}{52\cdots 11}a^{3}-\frac{12\cdots 53}{74\cdots 73}a^{2}-\frac{66\cdots 25}{52\cdots 11}a-\frac{57\cdots 02}{52\cdots 11}$ 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:  $C_{2}\times C_{4}$, which has order $8$ (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:  $C_{2}\times C_{4}$, which has order $8$ (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:  $7$
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:   $\frac{41\cdots 31}{56\cdots 67}a^{15}-\frac{70\cdots 98}{56\cdots 67}a^{14}+\frac{32\cdots 44}{56\cdots 67}a^{13}-\frac{12\cdots 30}{56\cdots 67}a^{12}+\frac{18\cdots 92}{56\cdots 67}a^{11}-\frac{17\cdots 30}{56\cdots 67}a^{10}+\frac{71\cdots 02}{56\cdots 67}a^{9}-\frac{14\cdots 98}{56\cdots 67}a^{8}+\frac{55\cdots 73}{56\cdots 67}a^{7}-\frac{42\cdots 90}{56\cdots 67}a^{6}+\frac{71\cdots 42}{56\cdots 67}a^{5}+\frac{36\cdots 20}{56\cdots 67}a^{4}-\frac{18\cdots 46}{56\cdots 67}a^{3}+\frac{11\cdots 58}{56\cdots 67}a^{2}+\frac{25\cdots 96}{56\cdots 67}a-\frac{32\cdots 27}{56\cdots 67}$, $\frac{22\cdots 36}{52\cdots 11}a^{15}-\frac{15\cdots 70}{52\cdots 11}a^{14}+\frac{44\cdots 88}{52\cdots 11}a^{13}-\frac{38\cdots 55}{52\cdots 11}a^{12}+\frac{89\cdots 75}{74\cdots 73}a^{11}-\frac{51\cdots 44}{74\cdots 73}a^{10}+\frac{93\cdots 80}{52\cdots 11}a^{9}-\frac{19\cdots 54}{52\cdots 11}a^{8}+\frac{17\cdots 26}{52\cdots 11}a^{7}-\frac{89\cdots 22}{74\cdots 73}a^{6}-\frac{40\cdots 92}{52\cdots 11}a^{5}+\frac{95\cdots 92}{52\cdots 11}a^{4}-\frac{13\cdots 19}{52\cdots 11}a^{3}-\frac{11\cdots 03}{74\cdots 73}a^{2}+\frac{62\cdots 98}{52\cdots 11}a-\frac{78\cdots 99}{52\cdots 11}$, $\frac{51\cdots 19}{52\cdots 11}a^{15}-\frac{11\cdots 29}{52\cdots 11}a^{14}-\frac{67\cdots 92}{52\cdots 11}a^{13}-\frac{10\cdots 32}{52\cdots 11}a^{12}+\frac{14\cdots 22}{74\cdots 73}a^{11}-\frac{34\cdots 06}{74\cdots 73}a^{10}+\frac{41\cdots 34}{52\cdots 11}a^{9}-\frac{39\cdots 49}{52\cdots 11}a^{8}+\frac{24\cdots 33}{52\cdots 11}a^{7}+\frac{31\cdots 77}{74\cdots 73}a^{6}-\frac{17\cdots 74}{52\cdots 11}a^{5}+\frac{44\cdots 48}{52\cdots 11}a^{4}-\frac{13\cdots 40}{52\cdots 11}a^{3}-\frac{16\cdots 29}{74\cdots 73}a^{2}+\frac{21\cdots 08}{52\cdots 11}a-\frac{13\cdots 07}{52\cdots 11}$, $\frac{13\cdots 07}{52\cdots 11}a^{15}-\frac{23\cdots 02}{52\cdots 11}a^{14}-\frac{63\cdots 76}{52\cdots 11}a^{13}-\frac{22\cdots 85}{52\cdots 11}a^{12}+\frac{98\cdots 03}{74\cdots 73}a^{11}-\frac{52\cdots 99}{74\cdots 73}a^{10}+\frac{27\cdots 24}{52\cdots 11}a^{9}-\frac{17\cdots 79}{52\cdots 11}a^{8}+\frac{19\cdots 93}{52\cdots 11}a^{7}-\frac{89\cdots 21}{74\cdots 73}a^{6}-\frac{23\cdots 20}{52\cdots 11}a^{5}+\frac{53\cdots 87}{52\cdots 11}a^{4}-\frac{75\cdots 89}{52\cdots 11}a^{3}-\frac{26\cdots 79}{74\cdots 73}a^{2}+\frac{36\cdots 00}{52\cdots 11}a-\frac{31\cdots 45}{52\cdots 11}$, $\frac{89\cdots 54}{52\cdots 11}a^{15}-\frac{55\cdots 84}{52\cdots 11}a^{14}+\frac{14\cdots 41}{52\cdots 11}a^{13}-\frac{14\cdots 98}{52\cdots 11}a^{12}+\frac{34\cdots 55}{74\cdots 73}a^{11}-\frac{18\cdots 89}{74\cdots 73}a^{10}+\frac{30\cdots 85}{52\cdots 11}a^{9}-\frac{68\cdots 90}{52\cdots 11}a^{8}+\frac{69\cdots 58}{52\cdots 11}a^{7}-\frac{29\cdots 42}{74\cdots 73}a^{6}-\frac{23\cdots 88}{52\cdots 11}a^{5}+\frac{34\cdots 46}{52\cdots 11}a^{4}-\frac{32\cdots 95}{52\cdots 11}a^{3}-\frac{31\cdots 41}{74\cdots 73}a^{2}+\frac{24\cdots 10}{52\cdots 11}a-\frac{20\cdots 03}{52\cdots 11}$, $\frac{50\cdots 05}{52\cdots 11}a^{15}-\frac{58\cdots 26}{52\cdots 11}a^{14}-\frac{19\cdots 44}{52\cdots 11}a^{13}-\frac{88\cdots 81}{52\cdots 11}a^{12}+\frac{27\cdots 92}{74\cdots 73}a^{11}-\frac{13\cdots 94}{74\cdots 73}a^{10}+\frac{74\cdots 94}{52\cdots 11}a^{9}-\frac{52\cdots 83}{52\cdots 11}a^{8}+\frac{53\cdots 21}{52\cdots 11}a^{7}-\frac{19\cdots 63}{74\cdots 73}a^{6}-\frac{61\cdots 94}{52\cdots 11}a^{5}+\frac{13\cdots 36}{52\cdots 11}a^{4}-\frac{17\cdots 54}{52\cdots 11}a^{3}-\frac{67\cdots 65}{74\cdots 73}a^{2}+\frac{91\cdots 50}{52\cdots 11}a-\frac{78\cdots 19}{52\cdots 11}$, $\frac{25\cdots 41}{52\cdots 11}a^{15}-\frac{17\cdots 76}{52\cdots 11}a^{14}+\frac{49\cdots 46}{52\cdots 11}a^{13}-\frac{43\cdots 25}{52\cdots 11}a^{12}+\frac{10\cdots 53}{74\cdots 73}a^{11}-\frac{56\cdots 55}{74\cdots 73}a^{10}+\frac{10\cdots 30}{52\cdots 11}a^{9}-\frac{22\cdots 22}{52\cdots 11}a^{8}+\frac{20\cdots 54}{52\cdots 11}a^{7}-\frac{97\cdots 61}{74\cdots 73}a^{6}-\frac{43\cdots 24}{52\cdots 11}a^{5}+\frac{10\cdots 04}{52\cdots 11}a^{4}-\frac{14\cdots 94}{52\cdots 11}a^{3}-\frac{82\cdots 47}{74\cdots 73}a^{2}+\frac{71\cdots 60}{52\cdots 11}a-\frac{92\cdots 33}{52\cdots 11}$ 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:  \( 4844961142290 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 4844961142290 \cdot 8}{2\cdot\sqrt{205433557246404340023920345190400000000}}\cr\approx \mathstrut & 3.28438109207257 \end{aligned}\] (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^16 - 7*x^15 + 22*x^14 - 182*x^13 + 2850*x^12 - 16562*x^11 + 47520*x^10 - 112130*x^9 + 848111*x^8 - 2942361*x^7 - 378872*x^6 + 40135950*x^5 - 70072682*x^4 + 26808900*x^3 + 286993746*x^2 - 457552166*x + 337333286) 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^16 - 7*x^15 + 22*x^14 - 182*x^13 + 2850*x^12 - 16562*x^11 + 47520*x^10 - 112130*x^9 + 848111*x^8 - 2942361*x^7 - 378872*x^6 + 40135950*x^5 - 70072682*x^4 + 26808900*x^3 + 286993746*x^2 - 457552166*x + 337333286, 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^16 - 7*x^15 + 22*x^14 - 182*x^13 + 2850*x^12 - 16562*x^11 + 47520*x^10 - 112130*x^9 + 848111*x^8 - 2942361*x^7 - 378872*x^6 + 40135950*x^5 - 70072682*x^4 + 26808900*x^3 + 286993746*x^2 - 457552166*x + 337333286); 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^16 - 7*x^15 + 22*x^14 - 182*x^13 + 2850*x^12 - 16562*x^11 + 47520*x^10 - 112130*x^9 + 848111*x^8 - 2942361*x^7 - 378872*x^6 + 40135950*x^5 - 70072682*x^4 + 26808900*x^3 + 286993746*x^2 - 457552166*x + 337333286); 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

$C_2^3:F_8:C_6$ (as 16T1501):

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 2688
The 20 conjugacy class representatives for $C_2^3:F_8:C_6$
Character table for $C_2^3:F_8:C_6$

Intermediate fields

\(\Q(\sqrt{65}) \)

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 28 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 ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ R ${\href{/padicField/7.3.0.1}{3} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ R ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ ${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.4.0.1}{4} }$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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$$[\ ]$$
2.1.7.6a1.1$x^{7} + 2$$7$$1$$6$$C_7:C_3$$$[\ ]_{7}^{3}$$
2.1.8.12a1.1$x^{8} + 2 x^{5} + 2$$8$$1$$12$$C_2^3:(C_7: C_3)$$$[\frac{12}{7}, \frac{12}{7}, \frac{12}{7}]_{7}^{3}$$
\(5\) Copy content Toggle raw display 5.1.2.1a1.2$x^{2} + 10$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.7.2.7a1.1$x^{14} + 6 x^{8} + 6 x^{7} + 9 x^{2} + 23 x + 9$$2$$7$$7$$C_{14}$$$[\ ]_{2}^{7}$$
\(13\) Copy content Toggle raw display 13.2.2.2a1.2$x^{4} + 24 x^{3} + 148 x^{2} + 48 x + 17$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
13.6.2.6a1.2$x^{12} + 20 x^{9} + 22 x^{8} + 22 x^{7} + 104 x^{6} + 220 x^{5} + 341 x^{4} + 282 x^{3} + 165 x^{2} + 44 x + 17$$2$$6$$6$$C_6\times C_2$$$[\ ]_{2}^{6}$$
\(199\) Copy content Toggle raw display $\Q_{199}$$x + 196$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{199}$$x + 196$$1$$1$$0$Trivial$$[\ ]$$
199.2.1.0a1.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
199.1.3.2a1.1$x^{3} + 199$$3$$1$$2$$C_3$$$[\ ]_{3}$$
199.1.3.2a1.1$x^{3} + 199$$3$$1$$2$$C_3$$$[\ ]_{3}$$
199.2.3.4a1.2$x^{6} + 579 x^{5} + 111756 x^{4} + 7192531 x^{3} + 335268 x^{2} + 5211 x + 226$$3$$2$$4$$C_6$$$[\ ]_{3}^{2}$$

Spectrum of ring of integers

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