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Properties

Label 16.8.115...000.4
Degree $16$
Signature $[8, 4]$
Discriminant $1.157\times 10^{22}$
Root discriminant \(23.93\)
Ramified primes $2,5,41$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^5:C_4$ (as 16T227)

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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^16 - 4*x^15 + 4*x^14 + 16*x^13 - 65*x^12 + 86*x^11 - 8*x^10 - 212*x^9 + 348*x^8 + 138*x^7 - 688*x^6 + 392*x^5 + 164*x^4 - 198*x^3 + 8*x^2 + 22*x + 1)
 
Copy content gp:K = bnfinit(y^16 - 4*y^15 + 4*y^14 + 16*y^13 - 65*y^12 + 86*y^11 - 8*y^10 - 212*y^9 + 348*y^8 + 138*y^7 - 688*y^6 + 392*y^5 + 164*y^4 - 198*y^3 + 8*y^2 + 22*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 4*x^14 + 16*x^13 - 65*x^12 + 86*x^11 - 8*x^10 - 212*x^9 + 348*x^8 + 138*x^7 - 688*x^6 + 392*x^5 + 164*x^4 - 198*x^3 + 8*x^2 + 22*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 4*x^14 + 16*x^13 - 65*x^12 + 86*x^11 - 8*x^10 - 212*x^9 + 348*x^8 + 138*x^7 - 688*x^6 + 392*x^5 + 164*x^4 - 198*x^3 + 8*x^2 + 22*x + 1)
 

\( x^{16} - 4 x^{15} + 4 x^{14} + 16 x^{13} - 65 x^{12} + 86 x^{11} - 8 x^{10} - 212 x^{9} + 348 x^{8} + \cdots + 1 \) 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:  $16$
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:  $[8, 4]$
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:   \(11574317056000000000000\) \(\medspace = 2^{24}\cdot 5^{12}\cdot 41^{4}\) 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:  \(23.93\)
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:  $2^{7/4}5^{3/4}41^{1/2}\approx 72.01482738072863$
Ramified primes:   \(2\), \(5\), \(41\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2^2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{11}a^{11}-\frac{5}{11}a^{10}-\frac{1}{11}a^{8}-\frac{2}{11}a^{7}+\frac{1}{11}a^{6}-\frac{3}{11}a^{5}-\frac{2}{11}a^{2}-\frac{2}{11}a-\frac{2}{11}$, $\frac{1}{55}a^{12}+\frac{2}{55}a^{11}+\frac{4}{11}a^{10}+\frac{2}{11}a^{9}-\frac{9}{55}a^{8}+\frac{4}{11}a^{7}+\frac{26}{55}a^{6}-\frac{2}{11}a^{5}+\frac{1}{5}a^{4}+\frac{4}{11}a^{3}-\frac{1}{11}a^{2}-\frac{27}{55}a-\frac{14}{55}$, $\frac{1}{55}a^{13}+\frac{1}{55}a^{11}-\frac{2}{11}a^{10}+\frac{26}{55}a^{9}-\frac{2}{55}a^{8}+\frac{16}{55}a^{7}-\frac{2}{5}a^{6}+\frac{21}{55}a^{5}-\frac{2}{55}a^{4}+\frac{2}{11}a^{3}+\frac{13}{55}a^{2}+\frac{3}{11}a+\frac{3}{55}$, $\frac{1}{605}a^{14}-\frac{1}{605}a^{12}+\frac{1}{605}a^{11}-\frac{199}{605}a^{10}-\frac{2}{55}a^{9}-\frac{146}{605}a^{8}+\frac{18}{605}a^{7}+\frac{259}{605}a^{6}-\frac{302}{605}a^{5}-\frac{232}{605}a^{4}-\frac{192}{605}a^{3}+\frac{43}{121}a^{2}-\frac{138}{605}a-\frac{2}{605}$, $\frac{1}{931824025}a^{15}-\frac{12546}{37272961}a^{14}-\frac{7066676}{931824025}a^{13}-\frac{1965153}{931824025}a^{12}-\frac{32283842}{931824025}a^{11}+\frac{364054533}{931824025}a^{10}-\frac{312510806}{931824025}a^{9}-\frac{398317166}{931824025}a^{8}+\frac{358158549}{931824025}a^{7}+\frac{246627554}{931824025}a^{6}-\frac{25466957}{931824025}a^{5}-\frac{180412516}{931824025}a^{4}-\frac{56657771}{186364805}a^{3}+\frac{141431517}{931824025}a^{2}-\frac{445897269}{931824025}a-\frac{159985174}{931824025}$ 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:  $C_{2}\times C_{2}$, which has order $4$ (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:  $11$
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:   $\frac{32567134}{186364805}a^{15}-\frac{127038416}{186364805}a^{14}+\frac{24081520}{37272961}a^{13}+\frac{518266347}{186364805}a^{12}-\frac{2046503794}{186364805}a^{11}+\frac{2642879436}{186364805}a^{10}-\frac{236457558}{186364805}a^{9}-\frac{6564096808}{186364805}a^{8}+\frac{10598840812}{186364805}a^{7}+\frac{4856597392}{186364805}a^{6}-\frac{1856533042}{16942255}a^{5}+\frac{10928061003}{186364805}a^{4}+\frac{3535436892}{186364805}a^{3}-\frac{894680760}{37272961}a^{2}+\frac{369430186}{186364805}a+\frac{142934691}{186364805}$, $\frac{13688693}{931824025}a^{15}-\frac{316782}{186364805}a^{14}-\frac{109973233}{931824025}a^{13}+\frac{280699941}{931824025}a^{12}+\frac{8725239}{931824025}a^{11}-\frac{1439648091}{931824025}a^{10}+\frac{2062090097}{931824025}a^{9}-\frac{1495803288}{931824025}a^{8}-\frac{4802275213}{931824025}a^{7}+\frac{11353594922}{931824025}a^{6}+\frac{6436393804}{931824025}a^{5}-\frac{14800836153}{931824025}a^{4}+\frac{563174251}{186364805}a^{3}+\frac{4713141161}{931824025}a^{2}-\frac{661447032}{931824025}a-\frac{896411047}{931824025}$, $\frac{9400111}{186364805}a^{15}-\frac{56988501}{186364805}a^{14}+\frac{101878496}{186364805}a^{13}+\frac{114546403}{186364805}a^{12}-\frac{939053691}{186364805}a^{11}+\frac{1838599112}{186364805}a^{10}-\frac{1071034972}{186364805}a^{9}-\frac{2409031259}{186364805}a^{8}+\frac{6961869063}{186364805}a^{7}-\frac{2905758929}{186364805}a^{6}-\frac{11631963403}{186364805}a^{5}+\frac{13046196417}{186364805}a^{4}+\frac{82324862}{186364805}a^{3}-\frac{5170109837}{186364805}a^{2}+\frac{967174404}{186364805}a+\frac{491614174}{186364805}$, $\frac{57110886}{931824025}a^{15}-\frac{12391901}{186364805}a^{14}-\frac{333048746}{931824025}a^{13}+\frac{113249672}{84711275}a^{12}-\frac{81860342}{84711275}a^{11}-\frac{4113367717}{931824025}a^{10}+\frac{8514258784}{931824025}a^{9}-\frac{8802189881}{931824025}a^{8}-\frac{12383611136}{931824025}a^{7}+\frac{46025972814}{931824025}a^{6}+\frac{3782123198}{931824025}a^{5}-\frac{60496070551}{931824025}a^{4}+\frac{5347031036}{186364805}a^{3}+\frac{1661723737}{84711275}a^{2}-\frac{11715808234}{931824025}a-\frac{2124783864}{931824025}$, $\frac{32567134}{186364805}a^{15}+\frac{127038416}{186364805}a^{14}-\frac{24081520}{37272961}a^{13}-\frac{518266347}{186364805}a^{12}+\frac{2046503794}{186364805}a^{11}-\frac{2642879436}{186364805}a^{10}+\frac{236457558}{186364805}a^{9}+\frac{6564096808}{186364805}a^{8}-\frac{10598840812}{186364805}a^{7}-\frac{4856597392}{186364805}a^{6}+\frac{1856533042}{16942255}a^{5}-\frac{10928061003}{186364805}a^{4}-\frac{3535436892}{186364805}a^{3}+\frac{894680760}{37272961}a^{2}-\frac{555794991}{186364805}a-\frac{142934691}{186364805}$, $a$, $\frac{57967508}{186364805}a^{15}-\frac{41341611}{37272961}a^{14}+\frac{152471226}{186364805}a^{13}+\frac{957057044}{186364805}a^{12}-\frac{3326364391}{186364805}a^{11}+\frac{3714859064}{186364805}a^{10}+\frac{555784126}{186364805}a^{9}-\frac{11401709283}{186364805}a^{8}+\frac{15374408821}{186364805}a^{7}+\frac{12571356012}{186364805}a^{6}-\frac{31677931512}{186364805}a^{5}+\frac{11327502627}{186364805}a^{4}+\frac{1706258795}{37272961}a^{3}-\frac{5811123317}{186364805}a^{2}-\frac{199591433}{186364805}a-\frac{231396762}{186364805}$, $\frac{15138248}{186364805}a^{15}-\frac{9828413}{37272961}a^{14}+\frac{28154001}{186364805}a^{13}+\frac{251829114}{186364805}a^{12}-\frac{796183331}{186364805}a^{11}+\frac{783809284}{186364805}a^{10}+\frac{285949421}{186364805}a^{9}-\frac{2926467413}{186364805}a^{8}+\frac{299966231}{16942255}a^{7}+\frac{3798269007}{186364805}a^{6}-\frac{7212948887}{186364805}a^{5}+\frac{2288137507}{186364805}a^{4}+\frac{604964064}{37272961}a^{3}-\frac{1697058692}{186364805}a^{2}-\frac{30583943}{16942255}a+\frac{443058923}{186364805}$, $\frac{228594726}{931824025}a^{15}-\frac{38435836}{37272961}a^{14}+\frac{1114936749}{931824025}a^{13}+\frac{3398657047}{931824025}a^{12}-\frac{15510919517}{931824025}a^{11}+\frac{22926361158}{931824025}a^{10}-\frac{7017200181}{931824025}a^{9}-\frac{46312650766}{931824025}a^{8}+\frac{89140130599}{931824025}a^{7}+\frac{1063798639}{84711275}a^{6}-\frac{156406780432}{931824025}a^{5}+\frac{122778169834}{931824025}a^{4}+\frac{70641854}{16942255}a^{3}-\frac{45547099408}{931824025}a^{2}+\frac{14765681006}{931824025}a+\frac{1373756301}{931824025}$, $\frac{186470496}{931824025}a^{15}-\frac{139264519}{186364805}a^{14}+\frac{608237484}{931824025}a^{13}+\frac{2985656227}{931824025}a^{12}-\frac{11249572932}{931824025}a^{11}+\frac{13869858498}{931824025}a^{10}-\frac{348513856}{931824025}a^{9}-\frac{37335380531}{931824025}a^{8}+\frac{465289769}{7701025}a^{7}+\frac{31526901539}{931824025}a^{6}-\frac{109725523202}{931824025}a^{5}+\frac{58377333539}{931824025}a^{4}+\frac{4598887962}{186364805}a^{3}-\frac{29977698203}{931824025}a^{2}+\frac{205395536}{84711275}a+\frac{2374703696}{931824025}$, $\frac{49118796}{931824025}a^{15}-\frac{10242362}{37272961}a^{14}+\frac{356866399}{931824025}a^{13}+\frac{795793422}{931824025}a^{12}-\frac{4316919217}{931824025}a^{11}+\frac{6869044143}{931824025}a^{10}-\frac{1536498131}{931824025}a^{9}-\frac{1301819281}{84711275}a^{8}+\frac{29348683049}{931824025}a^{7}-\frac{340726696}{931824025}a^{6}-\frac{58363752027}{931824025}a^{5}+\frac{4154192769}{84711275}a^{4}+\frac{3680498474}{186364805}a^{3}-\frac{32355324983}{931824025}a^{2}+\frac{6921426881}{931824025}a+\frac{3353268566}{931824025}$ 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:  \( 223161.3908924072 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 223161.3908924072 \cdot 1}{2\cdot\sqrt{11574317056000000000000}}\cr\approx \mathstrut & 0.413809841584068 \end{aligned}\] (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 - 4*x^15 + 4*x^14 + 16*x^13 - 65*x^12 + 86*x^11 - 8*x^10 - 212*x^9 + 348*x^8 + 138*x^7 - 688*x^6 + 392*x^5 + 164*x^4 - 198*x^3 + 8*x^2 + 22*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^16 - 4*x^15 + 4*x^14 + 16*x^13 - 65*x^12 + 86*x^11 - 8*x^10 - 212*x^9 + 348*x^8 + 138*x^7 - 688*x^6 + 392*x^5 + 164*x^4 - 198*x^3 + 8*x^2 + 22*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^16 - 4*x^15 + 4*x^14 + 16*x^13 - 65*x^12 + 86*x^11 - 8*x^10 - 212*x^9 + 348*x^8 + 138*x^7 - 688*x^6 + 392*x^5 + 164*x^4 - 198*x^3 + 8*x^2 + 22*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 = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 4*x^14 + 16*x^13 - 65*x^12 + 86*x^11 - 8*x^10 - 212*x^9 + 348*x^8 + 138*x^7 - 688*x^6 + 392*x^5 + 164*x^4 - 198*x^3 + 8*x^2 + 22*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

$C_2^5:C_4$ (as 16T227):

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 128
The 26 conjugacy class representatives for $C_2^5:C_4$
Character table for $C_2^5:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.16400.1, \(\Q(\zeta_{20})^+\), 4.4.5125.1, 8.4.21516800000.4, 8.4.21516800000.1, 8.8.6724000000.1

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 16 siblings: data not computed
Degree 32 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 ${\href{/padicField/3.8.0.1}{8} }^{2}$ R ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ R ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.4.4.24a29.1$x^{16} + 2 x^{14} + 6 x^{13} + 6 x^{12} + 6 x^{11} + 20 x^{10} + 26 x^{9} + 18 x^{8} + 26 x^{7} + 44 x^{6} + 36 x^{5} + 21 x^{4} + 24 x^{3} + 26 x^{2} + 14 x + 5$$4$$4$$24$16T53$$[2, 2, 2]^{4}$$
\(5\) Copy content Toggle raw display 5.1.4.3a1.1$x^{4} + 5$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.1.4.3a1.1$x^{4} + 5$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.1.4.3a1.1$x^{4} + 5$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.1.4.3a1.1$x^{4} + 5$$4$$1$$3$$C_4$$$[\ ]_{4}$$
\(41\) Copy content Toggle raw display 41.2.1.0a1.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
41.2.1.0a1.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
41.2.1.0a1.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
41.2.1.0a1.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
41.2.2.2a1.2$x^{4} + 76 x^{3} + 1456 x^{2} + 456 x + 77$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
41.2.2.2a1.2$x^{4} + 76 x^{3} + 1456 x^{2} + 456 x + 77$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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