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Properties

Label 16.0.131...000.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.315\times 10^{25}$
Root discriminant \(37.15\)
Ramified primes $2,3,5,23$
Class number $24$ (GRH)
Class group [2, 12] (GRH)
Galois group $C_2^4$ (as 16T3)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 52*x^14 + 1112*x^12 - 12702*x^10 + 84248*x^8 - 327816*x^6 + 743633*x^4 - 790564*x^2 + 364816)
 
gp: K = bnfinit(y^16 - 52*y^14 + 1112*y^12 - 12702*y^10 + 84248*y^8 - 327816*y^6 + 743633*y^4 - 790564*y^2 + 364816, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 52*x^14 + 1112*x^12 - 12702*x^10 + 84248*x^8 - 327816*x^6 + 743633*x^4 - 790564*x^2 + 364816);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 52*x^14 + 1112*x^12 - 12702*x^10 + 84248*x^8 - 327816*x^6 + 743633*x^4 - 790564*x^2 + 364816)
 

\( x^{16} - 52x^{14} + 1112x^{12} - 12702x^{10} + 84248x^{8} - 327816x^{6} + 743633x^{4} - 790564x^{2} + 364816 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(13153238385373209600000000\) \(\medspace = 2^{16}\cdot 3^{8}\cdot 5^{8}\cdot 23^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(37.15\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2\cdot 3^{1/2}5^{1/2}23^{1/2}\approx 37.14835124201342$
Ramified primes:   \(2\), \(3\), \(5\), \(23\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1380=2^{2}\cdot 3\cdot 5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{1380}(1,·)$, $\chi_{1380}(1289,·)$, $\chi_{1380}(599,·)$, $\chi_{1380}(139,·)$, $\chi_{1380}(781,·)$, $\chi_{1380}(461,·)$, $\chi_{1380}(919,·)$, $\chi_{1380}(1241,·)$, $\chi_{1380}(91,·)$, $\chi_{1380}(1379,·)$, $\chi_{1380}(229,·)$, $\chi_{1380}(551,·)$, $\chi_{1380}(689,·)$, $\chi_{1380}(691,·)$, $\chi_{1380}(829,·)$, $\chi_{1380}(1151,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{2}a^{5}-\frac{3}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{10}-\frac{1}{2}a^{6}-\frac{3}{8}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{11}-\frac{3}{8}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{24}a^{12}+\frac{1}{24}a^{10}-\frac{7}{24}a^{6}+\frac{3}{8}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{24}a^{13}+\frac{1}{24}a^{11}+\frac{5}{24}a^{7}+\frac{3}{8}a^{5}-\frac{1}{3}a^{3}+\frac{1}{6}a$, $\frac{1}{24\!\cdots\!24}a^{14}+\frac{20984649219917}{81\!\cdots\!08}a^{12}+\frac{514959578320037}{24\!\cdots\!24}a^{10}-\frac{536621817790471}{24\!\cdots\!24}a^{8}-\frac{52\!\cdots\!73}{24\!\cdots\!24}a^{6}+\frac{77\!\cdots\!21}{24\!\cdots\!24}a^{4}+\frac{547108507101485}{20\!\cdots\!52}a^{2}-\frac{540663927666595}{15\!\cdots\!64}$, $\frac{1}{36\!\cdots\!24}a^{15}+\frac{16\!\cdots\!33}{12\!\cdots\!08}a^{13}+\frac{79\!\cdots\!65}{36\!\cdots\!24}a^{11}-\frac{40\!\cdots\!35}{36\!\cdots\!24}a^{9}+\frac{12\!\cdots\!59}{36\!\cdots\!24}a^{7}-\frac{10\!\cdots\!43}{36\!\cdots\!24}a^{5}+\frac{63\!\cdots\!53}{30\!\cdots\!52}a^{3}+\frac{78\!\cdots\!33}{22\!\cdots\!64}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$, $3$

Class group and class number

$C_{2}\times C_{12}$, which has order $24$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{6333766631}{2790647176550272} a^{15} + \frac{1044882293773}{8371941529650816} a^{13} - \frac{23935887073577}{8371941529650816} a^{11} + \frac{99253106765185}{2790647176550272} a^{9} - \frac{2191284577843063}{8371941529650816} a^{7} + \frac{3179970662639065}{2790647176550272} a^{5} - \frac{5667730224543071}{2092985382412704} a^{3} + \frac{1334662852651915}{523246345603176} a \)  (order $12$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{575257433395}{30\!\cdots\!52}a^{15}-\frac{78762727058857}{91\!\cdots\!56}a^{13}+\frac{14\!\cdots\!41}{91\!\cdots\!56}a^{11}-\frac{44\!\cdots\!01}{30\!\cdots\!52}a^{9}+\frac{73\!\cdots\!03}{91\!\cdots\!56}a^{7}-\frac{82\!\cdots\!93}{30\!\cdots\!52}a^{5}+\frac{63\!\cdots\!43}{11\!\cdots\!32}a^{3}+\frac{11\!\cdots\!95}{57\!\cdots\!66}a$, $\frac{52511132933}{83\!\cdots\!16}a^{15}-\frac{2673639160717}{83\!\cdots\!16}a^{13}+\frac{18417706016227}{27\!\cdots\!72}a^{11}-\frac{594674521150115}{83\!\cdots\!16}a^{9}+\frac{11\!\cdots\!65}{27\!\cdots\!72}a^{7}-\frac{11\!\cdots\!03}{83\!\cdots\!16}a^{5}+\frac{41\!\cdots\!23}{20\!\cdots\!04}a^{3}-\frac{36627082795921}{174415448534392}a-1$, $\frac{1015093190269}{11\!\cdots\!32}a^{15}+\frac{216822823}{130858161771584}a^{14}-\frac{8584493397357}{19\!\cdots\!22}a^{13}-\frac{34590306317}{392574485314752}a^{12}+\frac{535123618262161}{57\!\cdots\!66}a^{11}+\frac{765134976313}{392574485314752}a^{10}-\frac{59\!\cdots\!43}{57\!\cdots\!66}a^{9}-\frac{3075718181041}{130858161771584}a^{8}+\frac{37\!\cdots\!31}{57\!\cdots\!66}a^{7}+\frac{67350961390775}{392574485314752}a^{6}-\frac{14\!\cdots\!77}{57\!\cdots\!66}a^{5}-\frac{107020706629225}{130858161771584}a^{4}+\frac{20\!\cdots\!51}{38\!\cdots\!44}a^{3}+\frac{251246419052251}{98143621328688}a^{2}-\frac{11\!\cdots\!74}{28\!\cdots\!83}a-\frac{74540171942579}{24535905332172}$, $\frac{31949138939401}{36\!\cdots\!24}a^{15}-\frac{15\!\cdots\!85}{36\!\cdots\!24}a^{13}+\frac{31\!\cdots\!93}{36\!\cdots\!24}a^{11}-\frac{32\!\cdots\!35}{36\!\cdots\!24}a^{9}+\frac{17\!\cdots\!63}{36\!\cdots\!24}a^{7}-\frac{45\!\cdots\!19}{36\!\cdots\!24}a^{5}+\frac{12\!\cdots\!39}{91\!\cdots\!56}a^{3}-\frac{90\!\cdots\!23}{22\!\cdots\!64}a$, $\frac{11355657536059}{30\!\cdots\!52}a^{15}+\frac{6168652781}{144021408813696}a^{14}-\frac{577501107627471}{30\!\cdots\!52}a^{13}-\frac{105760839751}{48007136271232}a^{12}+\frac{11\!\cdots\!75}{30\!\cdots\!52}a^{11}+\frac{6640328422561}{144021408813696}a^{10}-\frac{12\!\cdots\!85}{30\!\cdots\!52}a^{9}-\frac{72775041827963}{144021408813696}a^{8}+\frac{78\!\cdots\!97}{30\!\cdots\!52}a^{7}+\frac{446731608920015}{144021408813696}a^{6}-\frac{26\!\cdots\!73}{30\!\cdots\!52}a^{5}-\frac{15\!\cdots\!59}{144021408813696}a^{4}+\frac{11\!\cdots\!65}{76\!\cdots\!88}a^{3}+\frac{224647940396441}{12001784067808}a^{2}-\frac{75\!\cdots\!33}{95\!\cdots\!61}a-\frac{99377473703639}{9001338050856}$, $\frac{151419566273161}{18\!\cdots\!12}a^{15}-\frac{1200488796881}{12\!\cdots\!12}a^{14}-\frac{72\!\cdots\!13}{18\!\cdots\!12}a^{13}+\frac{68895411130769}{12\!\cdots\!12}a^{12}+\frac{46\!\cdots\!03}{61\!\cdots\!04}a^{11}-\frac{16\!\cdots\!53}{12\!\cdots\!12}a^{10}-\frac{13\!\cdots\!59}{18\!\cdots\!12}a^{9}+\frac{20\!\cdots\!19}{12\!\cdots\!12}a^{8}+\frac{25\!\cdots\!37}{61\!\cdots\!04}a^{7}-\frac{14\!\cdots\!63}{12\!\cdots\!12}a^{6}-\frac{25\!\cdots\!39}{18\!\cdots\!12}a^{5}+\frac{57\!\cdots\!43}{12\!\cdots\!12}a^{4}+\frac{13\!\cdots\!03}{45\!\cdots\!28}a^{3}-\frac{25\!\cdots\!11}{30\!\cdots\!28}a^{2}-\frac{46\!\cdots\!27}{38\!\cdots\!44}a+\frac{34\!\cdots\!43}{760613065297332}$, $\frac{980340748514791}{36\!\cdots\!24}a^{15}+\frac{7001525734105}{24\!\cdots\!24}a^{14}-\frac{48\!\cdots\!39}{36\!\cdots\!24}a^{13}-\frac{115114790293883}{81\!\cdots\!08}a^{12}+\frac{32\!\cdots\!21}{12\!\cdots\!08}a^{11}+\frac{68\!\cdots\!21}{24\!\cdots\!24}a^{10}-\frac{10\!\cdots\!45}{36\!\cdots\!24}a^{9}-\frac{71\!\cdots\!95}{24\!\cdots\!24}a^{8}+\frac{18\!\cdots\!31}{12\!\cdots\!08}a^{7}+\frac{41\!\cdots\!31}{24\!\cdots\!24}a^{6}-\frac{15\!\cdots\!37}{36\!\cdots\!24}a^{5}-\frac{13\!\cdots\!43}{24\!\cdots\!24}a^{4}+\frac{42\!\cdots\!97}{91\!\cdots\!56}a^{3}+\frac{19\!\cdots\!05}{20\!\cdots\!52}a^{2}+\frac{30\!\cdots\!93}{76\!\cdots\!88}a+\frac{31\!\cdots\!97}{15\!\cdots\!64}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 612241.341839 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 612241.341839 \cdot 24}{12\cdot\sqrt{13153238385373209600000000}}\cr\approx \mathstrut & 0.820115932515 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 52*x^14 + 1112*x^12 - 12702*x^10 + 84248*x^8 - 327816*x^6 + 743633*x^4 - 790564*x^2 + 364816)
 
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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^16 - 52*x^14 + 1112*x^12 - 12702*x^10 + 84248*x^8 - 327816*x^6 + 743633*x^4 - 790564*x^2 + 364816, 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 - 52*x^14 + 1112*x^12 - 12702*x^10 + 84248*x^8 - 327816*x^6 + 743633*x^4 - 790564*x^2 + 364816);
 
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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 52*x^14 + 1112*x^12 - 12702*x^10 + 84248*x^8 - 327816*x^6 + 743633*x^4 - 790564*x^2 + 364816);
 
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

$C_2^4$ (as 16T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_2^4$
Character table for $C_2^4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{115}) \), \(\Q(\sqrt{-115}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{345}) \), \(\Q(\sqrt{-345}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{69}) \), \(\Q(\sqrt{-69}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{23}) \), \(\Q(\sqrt{-23}) \), \(\Q(i, \sqrt{115})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{345})\), \(\Q(\sqrt{3}, \sqrt{115})\), \(\Q(\sqrt{-3}, \sqrt{115})\), \(\Q(\sqrt{3}, \sqrt{-115})\), \(\Q(\sqrt{-3}, \sqrt{-115})\), \(\Q(i, \sqrt{15})\), \(\Q(i, \sqrt{69})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{23})\), \(\Q(\sqrt{15}, \sqrt{69})\), \(\Q(\sqrt{-15}, \sqrt{-69})\), \(\Q(\sqrt{5}, \sqrt{23})\), \(\Q(\sqrt{-5}, \sqrt{-23})\), \(\Q(\sqrt{15}, \sqrt{-69})\), \(\Q(\sqrt{-15}, \sqrt{69})\), \(\Q(\sqrt{5}, \sqrt{-23})\), \(\Q(\sqrt{-5}, \sqrt{23})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{3}, \sqrt{23})\), \(\Q(\sqrt{3}, \sqrt{-23})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-23})\), \(\Q(\sqrt{-3}, \sqrt{23})\), \(\Q(\sqrt{15}, \sqrt{23})\), \(\Q(\sqrt{-15}, \sqrt{-23})\), \(\Q(\sqrt{5}, \sqrt{69})\), \(\Q(\sqrt{-5}, \sqrt{-69})\), \(\Q(\sqrt{15}, \sqrt{-23})\), \(\Q(\sqrt{-15}, \sqrt{23})\), \(\Q(\sqrt{-5}, \sqrt{69})\), \(\Q(\sqrt{5}, \sqrt{-69})\), 8.0.3626739360000.10, 8.0.3626739360000.7, 8.0.44774560000.1, 8.0.12960000.1, 8.0.5802782976.1, 8.0.3626739360000.6, 8.0.3626739360000.8, 8.8.3626739360000.1, 8.0.3626739360000.3, 8.0.3626739360000.1, 8.0.3626739360000.9, 8.0.3626739360000.5, 8.0.3626739360000.2, 8.0.3626739360000.4, 8.0.14166950625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R ${\href{/padicField/7.2.0.1}{2} }^{8}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ R ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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 2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
\(3\) Copy content Toggle raw display 3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(5\) Copy content Toggle raw display 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(23\) Copy content Toggle raw display 23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$