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Properties

Label 20.0.591...625.1
Degree $20$
Signature $(0, 10)$
Discriminant $5.913\times 10^{23}$
Root discriminant \(15.44\)
Ramified primes $5,7,11$
Class number $1$
Class group trivial
Galois group $D_5^2$ (as 20T28)

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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 - 3*x^19 + 5*x^18 - 9*x^17 + 18*x^16 - 29*x^15 + 37*x^14 - 24*x^13 + 26*x^12 - 13*x^11 + 31*x^10 - 13*x^9 + 26*x^8 - 24*x^7 + 37*x^6 - 29*x^5 + 18*x^4 - 9*x^3 + 5*x^2 - 3*x + 1)
 
Copy content gp:K = bnfinit(y^20 - 3*y^19 + 5*y^18 - 9*y^17 + 18*y^16 - 29*y^15 + 37*y^14 - 24*y^13 + 26*y^12 - 13*y^11 + 31*y^10 - 13*y^9 + 26*y^8 - 24*y^7 + 37*y^6 - 29*y^5 + 18*y^4 - 9*y^3 + 5*y^2 - 3*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 3*x^19 + 5*x^18 - 9*x^17 + 18*x^16 - 29*x^15 + 37*x^14 - 24*x^13 + 26*x^12 - 13*x^11 + 31*x^10 - 13*x^9 + 26*x^8 - 24*x^7 + 37*x^6 - 29*x^5 + 18*x^4 - 9*x^3 + 5*x^2 - 3*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 3*x^19 + 5*x^18 - 9*x^17 + 18*x^16 - 29*x^15 + 37*x^14 - 24*x^13 + 26*x^12 - 13*x^11 + 31*x^10 - 13*x^9 + 26*x^8 - 24*x^7 + 37*x^6 - 29*x^5 + 18*x^4 - 9*x^3 + 5*x^2 - 3*x + 1)
 

\( x^{20} - 3 x^{19} + 5 x^{18} - 9 x^{17} + 18 x^{16} - 29 x^{15} + 37 x^{14} - 24 x^{13} + 26 x^{12} + \cdots + 1 \) 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:  $(0, 10)$
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:   \(591319123885120791015625\) \(\medspace = 5^{10}\cdot 7^{10}\cdot 11^{8}\) 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:  \(15.44\)
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:  $5^{1/2}7^{1/2}11^{4/5}\approx 40.28544546409263$
Ramified primes:   \(5\), \(7\), \(11\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $D_5$
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.
Maximal CM subfield:  \(\Q(\sqrt{5}, \sqrt{-7})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5}a^{16}+\frac{2}{5}a^{15}-\frac{2}{5}a^{14}+\frac{2}{5}a^{13}+\frac{1}{5}a^{12}+\frac{2}{5}a^{10}-\frac{1}{5}a^{9}+\frac{1}{5}a^{8}-\frac{1}{5}a^{7}+\frac{2}{5}a^{6}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{2}{5}a^{2}+\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{17}-\frac{1}{5}a^{15}+\frac{1}{5}a^{14}+\frac{2}{5}a^{13}-\frac{2}{5}a^{12}+\frac{2}{5}a^{11}-\frac{2}{5}a^{9}+\frac{2}{5}a^{8}-\frac{1}{5}a^{7}+\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{1}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{10955}a^{18}-\frac{456}{10955}a^{17}+\frac{618}{10955}a^{16}-\frac{250}{2191}a^{15}+\frac{2563}{10955}a^{14}-\frac{5161}{10955}a^{13}+\frac{4183}{10955}a^{12}+\frac{779}{1565}a^{11}-\frac{5109}{10955}a^{10}-\frac{957}{2191}a^{9}-\frac{5109}{10955}a^{8}+\frac{779}{1565}a^{7}+\frac{4183}{10955}a^{6}-\frac{5161}{10955}a^{5}+\frac{2563}{10955}a^{4}-\frac{250}{2191}a^{3}+\frac{618}{10955}a^{2}-\frac{456}{10955}a+\frac{1}{10955}$, $\frac{1}{208145}a^{19}-\frac{4}{208145}a^{18}+\frac{9224}{208145}a^{17}-\frac{6744}{208145}a^{16}-\frac{8634}{41629}a^{15}+\frac{97253}{208145}a^{14}+\frac{83702}{208145}a^{13}+\frac{64493}{208145}a^{12}-\frac{1313}{10955}a^{11}-\frac{1929}{29735}a^{10}-\frac{13352}{41629}a^{9}-\frac{3516}{10955}a^{8}-\frac{13464}{208145}a^{7}-\frac{3571}{29735}a^{6}+\frac{9222}{29735}a^{5}+\frac{11948}{29735}a^{4}+\frac{2780}{5947}a^{3}-\frac{6171}{29735}a^{2}-\frac{1346}{41629}a+\frac{9216}{208145}$ 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:  $9$
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:   \( -1 \)  (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:   $\frac{262}{208145}a^{19}-\frac{128671}{208145}a^{18}+\frac{29871}{29735}a^{17}-\frac{4831}{5947}a^{16}+\frac{74074}{29735}a^{15}-\frac{162763}{29735}a^{14}+\frac{166474}{29735}a^{13}-\frac{958388}{208145}a^{12}-\frac{57839}{10955}a^{11}-\frac{384675}{41629}a^{10}-\frac{128932}{29735}a^{9}-\frac{135103}{10955}a^{8}-\frac{409030}{41629}a^{7}-\frac{2341398}{208145}a^{6}+\frac{519587}{208145}a^{5}-\frac{233706}{41629}a^{4}-\frac{144442}{208145}a^{3}+\frac{271189}{208145}a^{2}+\frac{192049}{208145}a-\frac{2626}{208145}$, $\frac{14941}{29735}a^{19}-\frac{61946}{208145}a^{18}-\frac{124541}{208145}a^{17}-\frac{100442}{208145}a^{16}+\frac{321382}{208145}a^{15}+\frac{77453}{41629}a^{14}-\frac{1117047}{208145}a^{13}+\frac{2825111}{208145}a^{12}+\frac{12026}{1565}a^{11}+\frac{1893511}{208145}a^{10}+\frac{1859254}{208145}a^{9}+\frac{40853}{2191}a^{8}+\frac{197}{19}a^{7}+\frac{148546}{41629}a^{6}-\frac{852846}{208145}a^{5}+\frac{1936078}{208145}a^{4}-\frac{881988}{208145}a^{3}-\frac{70284}{41629}a^{2}-\frac{131961}{208145}a+\frac{352328}{208145}$, $a$, $\frac{91002}{208145}a^{19}-\frac{43116}{41629}a^{18}+\frac{292281}{208145}a^{17}-\frac{585759}{208145}a^{16}+\frac{1185084}{208145}a^{15}-\frac{1674634}{208145}a^{14}+\frac{1918013}{208145}a^{13}-\frac{427822}{208145}a^{12}+\frac{67703}{10955}a^{11}+\frac{35686}{29735}a^{10}+\frac{1753918}{208145}a^{9}+\frac{3783}{10955}a^{8}+\frac{258233}{41629}a^{7}-\frac{211592}{29735}a^{6}+\frac{202693}{29735}a^{5}-\frac{215987}{29735}a^{4}-\frac{4268}{29735}a^{3}-\frac{13142}{29735}a^{2}-\frac{200171}{208145}a+\frac{81768}{208145}$, $\frac{146}{10955}a^{19}-\frac{13168}{10955}a^{18}+\frac{4597}{1565}a^{17}-\frac{5908}{1565}a^{16}+\frac{11482}{1565}a^{15}-\frac{24729}{1565}a^{14}+\frac{34984}{1565}a^{13}-\frac{261766}{10955}a^{12}+\frac{55523}{10955}a^{11}-\frac{178057}{10955}a^{10}+\frac{8449}{1565}a^{9}-\frac{254256}{10955}a^{8}+\frac{16488}{10955}a^{7}-\frac{174733}{10955}a^{6}+\frac{229659}{10955}a^{5}-\frac{246251}{10955}a^{4}+\frac{136009}{10955}a^{3}-\frac{48128}{10955}a^{2}+\frac{27546}{10955}a-\frac{3779}{2191}$, $\frac{246333}{208145}a^{19}-\frac{495151}{208145}a^{18}+\frac{109577}{41629}a^{17}-\frac{1232494}{208145}a^{16}+\frac{2649326}{208145}a^{15}-\frac{3351679}{208145}a^{14}+\frac{3312978}{208145}a^{13}+\frac{827094}{208145}a^{12}+\frac{35538}{2191}a^{11}+\frac{1029237}{208145}a^{10}+\frac{5119367}{208145}a^{9}+\frac{134783}{10955}a^{8}+\frac{4243237}{208145}a^{7}-\frac{2081559}{208145}a^{6}+\frac{693811}{41629}a^{5}-\frac{747209}{208145}a^{4}-\frac{101139}{208145}a^{3}-\frac{143544}{208145}a^{2}-\frac{45858}{208145}a+\frac{166191}{208145}$, $\frac{29662}{41629}a^{19}-\frac{343713}{208145}a^{18}+\frac{75452}{29735}a^{17}-\frac{168004}{29735}a^{16}+\frac{335153}{29735}a^{15}-\frac{97259}{5947}a^{14}+\frac{638966}{29735}a^{13}-\frac{617327}{41629}a^{12}+\frac{286912}{10955}a^{11}-\frac{1968766}{208145}a^{10}+\frac{673001}{29735}a^{9}-\frac{5113}{2191}a^{8}+\frac{4578459}{208145}a^{7}-\frac{3357271}{208145}a^{6}+\frac{4480004}{208145}a^{5}-\frac{3601803}{208145}a^{4}+\frac{3467591}{208145}a^{3}-\frac{507197}{41629}a^{2}+\frac{1290477}{208145}a-\frac{188389}{208145}$, $\frac{2626}{208145}a^{19}-\frac{1088}{29735}a^{18}-\frac{115541}{208145}a^{17}+\frac{185463}{208145}a^{16}-\frac{121817}{208145}a^{15}+\frac{442364}{208145}a^{14}-\frac{1042179}{208145}a^{13}+\frac{1102294}{208145}a^{12}-\frac{46848}{10955}a^{11}-\frac{1133079}{208145}a^{10}-\frac{1841969}{208145}a^{9}-\frac{49298}{10955}a^{8}-\frac{2498681}{208145}a^{7}-\frac{2108174}{208145}a^{6}-\frac{2244236}{208145}a^{5}+\frac{443433}{208145}a^{4}-\frac{1121262}{208145}a^{3}-\frac{168076}{208145}a^{2}+\frac{10882}{29735}a+\frac{184171}{208145}$, $\frac{445027}{208145}a^{19}-\frac{956724}{208145}a^{18}+\frac{1047817}{208145}a^{17}-\frac{2330512}{208145}a^{16}+\frac{1043712}{41629}a^{15}-\frac{6713057}{208145}a^{14}+\frac{6726519}{208145}a^{13}+\frac{135476}{208145}a^{12}+\frac{368521}{10955}a^{11}-\frac{1277449}{208145}a^{10}+\frac{1721711}{41629}a^{9}+\frac{94194}{10955}a^{8}+\frac{6787642}{208145}a^{7}-\frac{6692706}{208145}a^{6}+\frac{6431162}{208145}a^{5}-\frac{2507657}{208145}a^{4}+\frac{320471}{41629}a^{3}-\frac{1703782}{208145}a^{2}+\frac{215905}{41629}a-\frac{54331}{208145}$ 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:  \( 4090.28443721 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 4090.28443721 \cdot 1}{2\cdot\sqrt{591319123885120791015625}}\cr\approx \mathstrut & 0.255041723934 \end{aligned}\]

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^20 - 3*x^19 + 5*x^18 - 9*x^17 + 18*x^16 - 29*x^15 + 37*x^14 - 24*x^13 + 26*x^12 - 13*x^11 + 31*x^10 - 13*x^9 + 26*x^8 - 24*x^7 + 37*x^6 - 29*x^5 + 18*x^4 - 9*x^3 + 5*x^2 - 3*x + 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 - 3*x^19 + 5*x^18 - 9*x^17 + 18*x^16 - 29*x^15 + 37*x^14 - 24*x^13 + 26*x^12 - 13*x^11 + 31*x^10 - 13*x^9 + 26*x^8 - 24*x^7 + 37*x^6 - 29*x^5 + 18*x^4 - 9*x^3 + 5*x^2 - 3*x + 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 - 3*x^19 + 5*x^18 - 9*x^17 + 18*x^16 - 29*x^15 + 37*x^14 - 24*x^13 + 26*x^12 - 13*x^11 + 31*x^10 - 13*x^9 + 26*x^8 - 24*x^7 + 37*x^6 - 29*x^5 + 18*x^4 - 9*x^3 + 5*x^2 - 3*x + 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 = polynomial_ring(QQ); K, a = number_field(x^20 - 3*x^19 + 5*x^18 - 9*x^17 + 18*x^16 - 29*x^15 + 37*x^14 - 24*x^13 + 26*x^12 - 13*x^11 + 31*x^10 - 13*x^9 + 26*x^8 - 24*x^7 + 37*x^6 - 29*x^5 + 18*x^4 - 9*x^3 + 5*x^2 - 3*x + 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

$D_5^2$ (as 20T28):

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 100
The 16 conjugacy class representatives for $D_5^2$
Character table for $D_5^2$

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{5}, \sqrt{-7})\), 10.2.109853253125.1 x5

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 10 siblings: data not computed
Degree 20 sibling: data not computed
Degree 25 sibling: data not computed
Minimal sibling: 10.2.109853253125.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.10.0.1}{10} }^{2}$ ${\href{/padicField/3.10.0.1}{10} }^{2}$ R R R ${\href{/padicField/13.10.0.1}{10} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{10}$ ${\href{/padicField/23.10.0.1}{10} }^{2}$ ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{10}$ ${\href{/padicField/31.2.0.1}{2} }^{10}$ ${\href{/padicField/37.10.0.1}{10} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{10}$ ${\href{/padicField/43.10.0.1}{10} }^{2}$ ${\href{/padicField/47.10.0.1}{10} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{10}$

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
\(5\) Copy content Toggle raw display 5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(7\) Copy content Toggle raw display 7.2.2.2a1.2$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
7.2.2.2a1.2$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
7.2.2.2a1.2$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
7.2.2.2a1.2$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
7.2.2.2a1.2$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(11\) Copy content Toggle raw display 11.1.5.4a1.1$x^{5} + 11$$5$$1$$4$$C_5$$$[\ ]_{5}$$
11.1.5.4a1.1$x^{5} + 11$$5$$1$$4$$C_5$$$[\ ]_{5}$$
11.5.1.0a1.1$x^{5} + 10 x^{2} + 9$$1$$5$$0$$C_5$$$[\ ]^{5}$$
11.5.1.0a1.1$x^{5} + 10 x^{2} + 9$$1$$5$$0$$C_5$$$[\ ]^{5}$$

Spectrum of ring of integers

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