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Properties

Label 20.0.333...328.5
Degree $20$
Signature $[0, 10]$
Discriminant $3.330\times 10^{26}$
Root discriminant \(21.19\)
Ramified primes $2,13$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^9.(C_2\times F_5)$ (as 20T514)

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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 - 2*x^18 - 15*x^16 + 60*x^14 + 58*x^12 - 488*x^10 + 770*x^8 - 572*x^6 + 273*x^4 - 78*x^2 + 13)
 
Copy content gp:K = bnfinit(y^20 - 2*y^18 - 15*y^16 + 60*y^14 + 58*y^12 - 488*y^10 + 770*y^8 - 572*y^6 + 273*y^4 - 78*y^2 + 13, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^18 - 15*x^16 + 60*x^14 + 58*x^12 - 488*x^10 + 770*x^8 - 572*x^6 + 273*x^4 - 78*x^2 + 13);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 2*x^18 - 15*x^16 + 60*x^14 + 58*x^12 - 488*x^10 + 770*x^8 - 572*x^6 + 273*x^4 - 78*x^2 + 13)
 

\( x^{20} - 2x^{18} - 15x^{16} + 60x^{14} + 58x^{12} - 488x^{10} + 770x^{8} - 572x^{6} + 273x^{4} - 78x^{2} + 13 \) 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:  $[0, 10]$
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:   \(333014701462077604341219328\) \(\medspace = 2^{40}\cdot 13^{13}\) 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:  \(21.19\)
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:  not computed
Ramified primes:   \(2\), \(13\) 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:  \(\Q(\sqrt{13}) \)
$\Aut(K/\Q)$:   $C_2$
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.
Maximal CM subfield:  \(\Q(\sqrt{-1}) \)

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}$, $\frac{1}{4}a^{13}+\frac{1}{4}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{10}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{11}-\frac{1}{2}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{52}a^{16}+\frac{3}{26}a^{14}+\frac{1}{13}a^{12}-\frac{1}{13}a^{10}+\frac{7}{26}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}$, $\frac{1}{52}a^{17}+\frac{3}{26}a^{15}+\frac{1}{13}a^{13}-\frac{1}{13}a^{11}+\frac{7}{26}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{4}a$, $\frac{1}{79978964}a^{18}-\frac{346657}{39989482}a^{16}+\frac{9342313}{79978964}a^{14}-\frac{660787}{39989482}a^{12}+\frac{4106821}{79978964}a^{10}-\frac{8586879}{39989482}a^{8}-\frac{1}{2}a^{7}-\frac{2751847}{6152228}a^{6}-\frac{1}{2}a^{5}+\frac{666364}{1538057}a^{4}-\frac{1}{2}a^{3}+\frac{1424995}{3076114}a^{2}-\frac{1}{2}a+\frac{83168}{1538057}$, $\frac{1}{79978964}a^{19}-\frac{346657}{39989482}a^{17}+\frac{9342313}{79978964}a^{15}-\frac{660787}{39989482}a^{13}+\frac{4106821}{79978964}a^{11}-\frac{8586879}{39989482}a^{9}-\frac{1}{2}a^{8}-\frac{2751847}{6152228}a^{7}-\frac{1}{2}a^{6}+\frac{666364}{1538057}a^{5}-\frac{1}{2}a^{4}+\frac{1424995}{3076114}a^{3}-\frac{1}{2}a^{2}+\frac{83168}{1538057}a$ 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:  $9$
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:   \( \frac{181755}{3076114} a^{18} - \frac{138030}{1538057} a^{16} - \frac{2828685}{3076114} a^{14} + \frac{9521833}{3076114} a^{12} + \frac{7325349}{1538057} a^{10} - \frac{40216688}{1538057} a^{8} + \frac{104424931}{3076114} a^{6} - \frac{64075245}{3076114} a^{4} + \frac{30284683}{3076114} a^{2} - \frac{3123541}{1538057} \)  (order $4$) 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:   $\frac{6139737}{39989482}a^{18}-\frac{9842135}{39989482}a^{16}-\frac{48134285}{19994741}a^{14}+\frac{165248654}{19994741}a^{12}+\frac{491154193}{39989482}a^{10}-\frac{2814657093}{39989482}a^{8}+\frac{138067622}{1538057}a^{6}-\frac{76079330}{1538057}a^{4}+\frac{28006142}{1538057}a^{2}-\frac{4402140}{1538057}$, $\frac{3836073}{79978964}a^{18}-\frac{5717921}{79978964}a^{16}-\frac{28881409}{39989482}a^{14}+\frac{99942301}{39989482}a^{12}+\frac{70252178}{19994741}a^{10}-\frac{819832457}{39989482}a^{8}+\frac{94118705}{3076114}a^{6}-\frac{67261141}{3076114}a^{4}+\frac{52895575}{6152228}a^{2}-\frac{10267269}{6152228}$, $\frac{10057831}{79978964}a^{18}-\frac{18362303}{79978964}a^{16}-\frac{38966125}{19994741}a^{14}+\frac{11132381}{1538057}a^{12}+\frac{178123631}{19994741}a^{10}-\frac{2437522917}{39989482}a^{8}+\frac{129706653}{1538057}a^{6}-\frac{72991172}{1538057}a^{4}+\frac{83273649}{6152228}a^{2}-\frac{8884275}{6152228}$, $\frac{18102235}{79978964}a^{19}+\frac{271118}{19994741}a^{18}-\frac{29364329}{79978964}a^{17}-\frac{882968}{19994741}a^{16}-\frac{71479619}{19994741}a^{15}-\frac{7641389}{39989482}a^{14}+\frac{982761229}{79978964}a^{13}+\frac{88400745}{79978964}a^{12}+\frac{368383579}{19994741}a^{11}+\frac{4206819}{39989482}a^{10}-\frac{8444604483}{79978964}a^{9}-\frac{54172627}{6152228}a^{8}+\frac{201053356}{1538057}a^{7}+\frac{52396767}{3076114}a^{6}-\frac{381221003}{6152228}a^{5}-\frac{65758563}{6152228}a^{4}+\frac{107401339}{6152228}a^{3}+\frac{665388}{1538057}a^{2}-\frac{1952027}{1538057}a+\frac{5281207}{6152228}$, $\frac{17017763}{79978964}a^{19}-\frac{3953375}{19994741}a^{18}-\frac{25832457}{79978964}a^{17}+\frac{14607691}{39989482}a^{16}-\frac{135317849}{39989482}a^{15}+\frac{243954593}{79978964}a^{14}+\frac{223590121}{19994741}a^{13}-\frac{914087753}{79978964}a^{12}+\frac{732560339}{39989482}a^{11}-\frac{1093367083}{79978964}a^{10}-\frac{1935090083}{19994741}a^{9}+\frac{7654748121}{79978964}a^{8}+\frac{349709945}{3076114}a^{7}-\frac{829310415}{6152228}a^{6}-\frac{78865610}{1538057}a^{5}+\frac{498951721}{6152228}a^{4}+\frac{104739787}{6152228}a^{3}-\frac{181417399}{6152228}a^{2}-\frac{13089315}{6152228}a+\frac{32267717}{6152228}$, $\frac{1979583}{19994741}a^{19}+\frac{1695391}{39989482}a^{18}-\frac{18989013}{79978964}a^{17}-\frac{2969275}{19994741}a^{16}-\frac{28674206}{19994741}a^{15}-\frac{42914037}{79978964}a^{14}+\frac{263649333}{39989482}a^{13}+\frac{284268659}{79978964}a^{12}+\frac{75922401}{19994741}a^{11}-\frac{74934107}{79978964}a^{10}-\frac{1047092043}{19994741}a^{9}-\frac{2082129457}{79978964}a^{8}+\frac{289091739}{3076114}a^{7}+\frac{381647055}{6152228}a^{6}-\frac{111536710}{1538057}a^{5}-\frac{367130255}{6152228}a^{4}+\frac{82054655}{3076114}a^{3}+\frac{167254315}{6152228}a^{2}-\frac{18266625}{6152228}a-\frac{34301835}{6152228}$, $\frac{2986455}{19994741}a^{19}-\frac{3197527}{19994741}a^{18}-\frac{14386085}{39989482}a^{17}+\frac{12788657}{39989482}a^{16}-\frac{87019611}{39989482}a^{15}+\frac{193608349}{79978964}a^{14}+\frac{797695641}{79978964}a^{13}-\frac{770544181}{79978964}a^{12}+\frac{236238471}{39989482}a^{11}-\frac{768679643}{79978964}a^{10}-\frac{6365867537}{79978964}a^{9}+\frac{6343493617}{79978964}a^{8}+\frac{215764498}{1538057}a^{7}-\frac{748294613}{6152228}a^{6}-\frac{648965601}{6152228}a^{5}+\frac{496896351}{6152228}a^{4}+\frac{126427415}{3076114}a^{3}-\frac{173497393}{6152228}a^{2}-\frac{51829783}{6152228}a+\frac{30060447}{6152228}$, $\frac{1898029}{39989482}a^{19}-\frac{630425}{79978964}a^{18}-\frac{808103}{39989482}a^{17}-\frac{1021753}{79978964}a^{16}-\frac{5051051}{6152228}a^{15}+\frac{3117384}{19994741}a^{14}+\frac{131979873}{79978964}a^{13}-\frac{1421551}{79978964}a^{12}+\frac{527893739}{79978964}a^{11}-\frac{74855607}{39989482}a^{10}-\frac{1317228345}{79978964}a^{9}+\frac{7349807}{6152228}a^{8}+\frac{18555987}{6152228}a^{7}+\frac{19329967}{3076114}a^{6}+\frac{63765953}{6152228}a^{5}-\frac{56668813}{6152228}a^{4}-\frac{22506689}{6152228}a^{3}+\frac{17866339}{6152228}a^{2}+\frac{12816841}{6152228}a-\frac{1899384}{1538057}$, $\frac{1556163}{39989482}a^{19}-\frac{2229041}{39989482}a^{18}-\frac{554795}{3076114}a^{17}+\frac{9767973}{79978964}a^{16}-\frac{17432195}{39989482}a^{15}+\frac{65050551}{79978964}a^{14}+\frac{159091833}{39989482}a^{13}-\frac{280423677}{79978964}a^{12}-\frac{60385571}{19994741}a^{11}-\frac{204836915}{79978964}a^{10}-\frac{1129550553}{39989482}a^{9}+\frac{2224259607}{79978964}a^{8}+\frac{234199397}{3076114}a^{7}-\frac{298251051}{6152228}a^{6}-\frac{226500589}{3076114}a^{5}+\frac{248494147}{6152228}a^{4}+\frac{90851953}{3076114}a^{3}-\frac{121643503}{6152228}a^{2}-\frac{10602389}{1538057}a+\frac{11584143}{3076114}$ 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:  \( 752160.226838 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 752160.226838 \cdot 1}{4\cdot\sqrt{333014701462077604341219328}}\cr\approx \mathstrut & 0.988137437739 \end{aligned}\] (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 - 2*x^18 - 15*x^16 + 60*x^14 + 58*x^12 - 488*x^10 + 770*x^8 - 572*x^6 + 273*x^4 - 78*x^2 + 13) 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 - 2*x^18 - 15*x^16 + 60*x^14 + 58*x^12 - 488*x^10 + 770*x^8 - 572*x^6 + 273*x^4 - 78*x^2 + 13, 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 - 2*x^18 - 15*x^16 + 60*x^14 + 58*x^12 - 488*x^10 + 770*x^8 - 572*x^6 + 273*x^4 - 78*x^2 + 13); 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 - 2*x^18 - 15*x^16 + 60*x^14 + 58*x^12 - 488*x^10 + 770*x^8 - 572*x^6 + 273*x^4 - 78*x^2 + 13); 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

$C_2^9.(C_2\times F_5)$ (as 20T514):

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 solvable group of order 20480
The 74 conjugacy class representatives for $C_2^9.(C_2\times F_5)$
Character table for $C_2^9.(C_2\times F_5)$

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.1.35152.1, 10.0.79082438656.1

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 ${\href{/padicField/3.10.0.1}{10} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{5}$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ R ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{5}$ ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.10.0.1}{10} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{5}$ ${\href{/padicField/37.8.0.1}{8} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}$ ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }$ ${\href{/padicField/53.5.0.1}{5} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{5}$

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 2.1.20.40d1.22$x^{20} + 4 x^{12} + 2 x^{10} + 4 x^{7} + 4 x^{3} + 4 x + 2$$20$$1$$40$20T514not computed
\(13\) Copy content Toggle raw display 13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.1.4.3a1.1$x^{4} + 13$$4$$1$$3$$C_4$$$[\ ]_{4}$$
13.1.4.3a1.1$x^{4} + 13$$4$$1$$3$$C_4$$$[\ ]_{4}$$
13.1.8.7a1.1$x^{8} + 13$$8$$1$$7$$C_8:C_2$$$[\ ]_{8}^{2}$$

Spectrum of ring of integers

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