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Properties

Label 20.10.599...000.1
Degree $20$
Signature $[10, 5]$
Discriminant $-5.992\times 10^{27}$
Root discriminant \(24.48\)
Ramified primes $2,5,29,13345751$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^8.S_5^2:D_4$ (as 20T1045)

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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^20 - 4*x^18 + 11*x^14 - x^12 - 4*x^10 - 9*x^8 - 9*x^6 + 10*x^4 + 7*x^2 - 1)
 
Copy content gp:K = bnfinit(y^20 - 4*y^18 + 11*y^14 - y^12 - 4*y^10 - 9*y^8 - 9*y^6 + 10*y^4 + 7*y^2 - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^18 + 11*x^14 - x^12 - 4*x^10 - 9*x^8 - 9*x^6 + 10*x^4 + 7*x^2 - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 4*x^18 + 11*x^14 - x^12 - 4*x^10 - 9*x^8 - 9*x^6 + 10*x^4 + 7*x^2 - 1)
 

\( x^{20} - 4x^{18} + 11x^{14} - x^{12} - 4x^{10} - 9x^{8} - 9x^{6} + 10x^{4} + 7x^{2} - 1 \) 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:  $20$
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:  $[10, 5]$
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:   \(-5991589106524593640000000000\) \(\medspace = -\,2^{12}\cdot 5^{10}\cdot 29^{2}\cdot 13345751^{2}\) 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:  \(24.48\)
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:  $2^{63/32}5^{1/2}29^{1/2}13345751^{1/2}\approx 172190.15379250972$
Ramified primes:   \(2\), \(5\), \(29\), \(13345751\) 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:  \(\Q(\sqrt{-1}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{82}a^{18}-\frac{3}{41}a^{16}-\frac{29}{82}a^{14}-\frac{1}{2}a^{13}+\frac{14}{41}a^{12}-\frac{1}{2}a^{11}+\frac{25}{82}a^{10}-\frac{1}{2}a^{9}-\frac{13}{82}a^{8}-\frac{12}{41}a^{6}-\frac{1}{41}a^{4}-\frac{1}{2}a^{3}-\frac{27}{82}a^{2}-\frac{1}{2}a+\frac{10}{41}$, $\frac{1}{82}a^{19}-\frac{3}{41}a^{17}-\frac{29}{82}a^{15}-\frac{1}{2}a^{14}+\frac{14}{41}a^{13}-\frac{1}{2}a^{12}+\frac{25}{82}a^{11}-\frac{1}{2}a^{10}-\frac{13}{82}a^{9}-\frac{12}{41}a^{7}-\frac{1}{41}a^{5}-\frac{1}{2}a^{4}-\frac{27}{82}a^{3}-\frac{1}{2}a^{2}+\frac{10}{41}a$ 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 $1$ (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:  $14$
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:   $a$, $\frac{14}{41}a^{18}-\frac{43}{41}a^{16}-\frac{37}{41}a^{14}+\frac{105}{41}a^{12}+\frac{104}{41}a^{10}+\frac{23}{41}a^{8}-\frac{90}{41}a^{6}-\frac{151}{41}a^{4}-\frac{50}{41}a^{2}+\frac{75}{41}$, $\frac{51}{41}a^{18}-\frac{224}{41}a^{16}+\frac{120}{41}a^{14}+\frac{403}{41}a^{12}-\frac{242}{41}a^{10}+\frac{116}{41}a^{8}-\frac{404}{41}a^{6}-\frac{143}{41}a^{4}+\frac{345}{41}a^{2}-\frac{87}{41}$, $\frac{20}{41}a^{19}-\frac{120}{41}a^{17}+\frac{158}{41}a^{15}+\frac{191}{41}a^{13}-\frac{320}{41}a^{11}-\frac{55}{41}a^{9}-\frac{316}{41}a^{7}+\frac{124}{41}a^{5}+\frac{485}{41}a^{3}+\frac{72}{41}a$, $\frac{60}{41}a^{19}-\frac{278}{41}a^{17}+\frac{187}{41}a^{15}+\frac{491}{41}a^{13}-\frac{345}{41}a^{11}+\frac{81}{41}a^{9}-\frac{620}{41}a^{7}-\frac{161}{41}a^{5}+\frac{553}{41}a^{3}+\frac{52}{41}a$, $\frac{8}{41}a^{18}-\frac{89}{41}a^{16}+\frac{219}{41}a^{14}+\frac{60}{41}a^{12}-\frac{415}{41}a^{10}-\frac{22}{41}a^{8}-\frac{274}{41}a^{6}+\frac{353}{41}a^{4}+\frac{399}{41}a^{2}-\frac{86}{41}$, $\frac{26}{41}a^{19}-\frac{156}{41}a^{17}+\frac{230}{41}a^{15}+\frac{154}{41}a^{13}-\frac{416}{41}a^{11}+\frac{113}{41}a^{9}-\frac{337}{41}a^{7}+\frac{235}{41}a^{5}+\frac{364}{41}a^{3}-\frac{136}{41}a$, $\frac{32}{41}a^{19}-\frac{27}{82}a^{18}-\frac{151}{41}a^{17}+\frac{121}{82}a^{16}+\frac{235}{82}a^{15}-\frac{39}{41}a^{14}+\frac{240}{41}a^{13}-\frac{91}{41}a^{12}-\frac{225}{41}a^{11}+\frac{52}{41}a^{10}+\frac{193}{82}a^{9}-\frac{50}{41}a^{8}-\frac{317}{41}a^{7}+\frac{279}{82}a^{6}-\frac{5}{82}a^{5}+\frac{95}{82}a^{4}+\frac{284}{41}a^{3}-\frac{66}{41}a^{2}-\frac{155}{82}a+\frac{75}{82}$, $\frac{32}{41}a^{19}+\frac{27}{82}a^{18}-\frac{151}{41}a^{17}-\frac{121}{82}a^{16}+\frac{235}{82}a^{15}+\frac{39}{41}a^{14}+\frac{240}{41}a^{13}+\frac{91}{41}a^{12}-\frac{225}{41}a^{11}-\frac{52}{41}a^{10}+\frac{193}{82}a^{9}+\frac{50}{41}a^{8}-\frac{317}{41}a^{7}-\frac{279}{82}a^{6}-\frac{5}{82}a^{5}-\frac{95}{82}a^{4}+\frac{284}{41}a^{3}+\frac{66}{41}a^{2}-\frac{155}{82}a-\frac{75}{82}$, $\frac{7}{82}a^{19}-\frac{32}{41}a^{18}-\frac{1}{82}a^{17}+\frac{151}{41}a^{16}-\frac{121}{82}a^{15}-\frac{235}{82}a^{14}+\frac{155}{82}a^{13}-\frac{240}{41}a^{12}+\frac{175}{82}a^{11}+\frac{225}{41}a^{10}-\frac{66}{41}a^{9}-\frac{193}{82}a^{8}+\frac{37}{82}a^{7}+\frac{317}{41}a^{6}-\frac{171}{41}a^{5}+\frac{5}{82}a^{4}-\frac{25}{82}a^{3}-\frac{284}{41}a^{2}+\frac{181}{82}a+\frac{155}{82}$, $\frac{3}{41}a^{19}+\frac{47}{82}a^{18}+\frac{23}{41}a^{17}-\frac{141}{41}a^{16}-\frac{128}{41}a^{15}+\frac{359}{82}a^{14}+\frac{45}{82}a^{13}+\frac{248}{41}a^{12}+\frac{519}{82}a^{11}-\frac{793}{82}a^{10}+\frac{45}{82}a^{9}-\frac{119}{82}a^{8}+\frac{92}{41}a^{7}-\frac{359}{41}a^{6}-\frac{252}{41}a^{5}+\frac{117}{41}a^{4}-\frac{449}{82}a^{3}+\frac{1109}{82}a^{2}+\frac{161}{82}a-\frac{22}{41}$, $\frac{27}{82}a^{19}+\frac{65}{82}a^{18}-\frac{121}{82}a^{17}-\frac{267}{82}a^{16}-\frac{2}{41}a^{15}+\frac{83}{82}a^{14}+\frac{469}{82}a^{13}+\frac{254}{41}a^{12}-\frac{63}{82}a^{11}-\frac{69}{41}a^{10}-\frac{433}{82}a^{9}+\frac{139}{82}a^{8}-\frac{443}{82}a^{7}-\frac{535}{82}a^{6}-\frac{423}{82}a^{5}-\frac{147}{41}a^{4}+\frac{583}{82}a^{3}+\frac{209}{41}a^{2}+\frac{270}{41}a+\frac{35}{41}$, $\frac{35}{41}a^{19}-\frac{9}{41}a^{18}-\frac{128}{41}a^{17}+\frac{67}{82}a^{16}-\frac{21}{82}a^{15}-\frac{11}{82}a^{14}+\frac{525}{82}a^{13}-\frac{47}{41}a^{12}+\frac{69}{82}a^{11}+\frac{1}{82}a^{10}+\frac{119}{41}a^{9}-\frac{135}{82}a^{8}-\frac{225}{41}a^{7}+\frac{63}{82}a^{6}-\frac{509}{82}a^{5}+\frac{77}{82}a^{4}+\frac{119}{82}a^{3}+\frac{117}{82}a^{2}-\frac{38}{41}a+\frac{173}{82}$, $\frac{11}{82}a^{19}+\frac{17}{82}a^{18}-\frac{33}{41}a^{17}-\frac{61}{82}a^{16}+\frac{25}{41}a^{15}-\frac{1}{82}a^{14}+\frac{113}{41}a^{13}+\frac{107}{82}a^{12}-\frac{135}{82}a^{11}+\frac{15}{82}a^{10}-\frac{133}{41}a^{9}+\frac{33}{41}a^{8}-\frac{132}{41}a^{7}-\frac{121}{82}a^{6}-\frac{63}{82}a^{5}-\frac{17}{41}a^{4}+\frac{441}{82}a^{3}+\frac{33}{82}a^{2}+\frac{261}{82}a-\frac{29}{82}$ 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:  \( 2835601.07584 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{5}\cdot 2835601.07584 \cdot 1}{2\cdot\sqrt{5991589106524593640000000000}}\cr\approx \mathstrut & 0.183672198319 \end{aligned}\] (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^20 - 4*x^18 + 11*x^14 - x^12 - 4*x^10 - 9*x^8 - 9*x^6 + 10*x^4 + 7*x^2 - 1) 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^20 - 4*x^18 + 11*x^14 - x^12 - 4*x^10 - 9*x^8 - 9*x^6 + 10*x^4 + 7*x^2 - 1, 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^20 - 4*x^18 + 11*x^14 - x^12 - 4*x^10 - 9*x^8 - 9*x^6 + 10*x^4 + 7*x^2 - 1); 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^20 - 4*x^18 + 11*x^14 - x^12 - 4*x^10 - 9*x^8 - 9*x^6 + 10*x^4 + 7*x^2 - 1); 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

$C_2^8.S_5^2:D_4$ (as 20T1045):

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 non-solvable group of order 29491200
The 702 conjugacy class representatives for $C_2^8.S_5^2:D_4$
Character table for $C_2^8.S_5^2:D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 10.8.1209458684375.1

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]
 

Sibling fields

Degree 20 siblings: data not computed
Degree 40 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 ${\href{/padicField/3.8.0.1}{8} }^{2}{,}\,{\href{/padicField/3.4.0.1}{4} }$ R $20$ ${\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.10.0.1}{10} }^{2}$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ $16{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ R ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ $20$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.8.0.1}{8} }$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.4.1.0a1.1$x^{4} + x + 1$$1$$4$$0$$C_4$$$[\ ]^{4}$$
2.4.1.0a1.1$x^{4} + x + 1$$1$$4$$0$$C_4$$$[\ ]^{4}$$
2.6.2.12a8.1$x^{12} + 2 x^{11} + 2 x^{10} + 4 x^{9} + 5 x^{8} + 4 x^{7} + 7 x^{6} + 6 x^{5} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 3$$2$$6$$12$12T134$$[2, 2, 2, 2, 2, 2]^{6}$$
\(5\) Copy content Toggle raw display 5.10.2.10a1.2$x^{20} + 6 x^{15} + 6 x^{14} + 4 x^{13} + 8 x^{12} + 2 x^{11} + 13 x^{10} + 18 x^{9} + 21 x^{8} + 36 x^{7} + 34 x^{6} + 34 x^{5} + 32 x^{4} + 16 x^{3} + 17 x^{2} + 4 x + 9$$2$$10$$10$20T3not computed
\(29\) Copy content Toggle raw display $\Q_{29}$$x + 27$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{29}$$x + 27$$1$$1$$0$Trivial$$[\ ]$$
29.3.1.0a1.1$x^{3} + 2 x + 27$$1$$3$$0$$C_3$$$[\ ]^{3}$$
29.3.1.0a1.1$x^{3} + 2 x + 27$$1$$3$$0$$C_3$$$[\ ]^{3}$$
29.2.2.2a1.2$x^{4} + 48 x^{3} + 580 x^{2} + 96 x + 33$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
29.4.1.0a1.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
29.4.1.0a1.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
\(13345751\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$

Spectrum of ring of integers

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