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Properties

Label 16.0.897...896.9
Degree $16$
Signature $[0, 8]$
Discriminant $8.979\times 10^{19}$
Root discriminant \(17.66\)
Ramified primes $2,3,13$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_2^2\wr C_2$ (as 16T39)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 26*x^13 + 82*x^12 - 176*x^11 + 352*x^10 - 652*x^9 + 936*x^8 - 1158*x^7 + 1438*x^6 - 1758*x^5 + 2199*x^4 - 2736*x^3 + 2826*x^2 - 1944*x + 621)
 
gp: K = bnfinit(y^16 - 4*y^15 + 8*y^14 - 26*y^13 + 82*y^12 - 176*y^11 + 352*y^10 - 652*y^9 + 936*y^8 - 1158*y^7 + 1438*y^6 - 1758*y^5 + 2199*y^4 - 2736*y^3 + 2826*y^2 - 1944*y + 621, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 8*x^14 - 26*x^13 + 82*x^12 - 176*x^11 + 352*x^10 - 652*x^9 + 936*x^8 - 1158*x^7 + 1438*x^6 - 1758*x^5 + 2199*x^4 - 2736*x^3 + 2826*x^2 - 1944*x + 621);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 8*x^14 - 26*x^13 + 82*x^12 - 176*x^11 + 352*x^10 - 652*x^9 + 936*x^8 - 1158*x^7 + 1438*x^6 - 1758*x^5 + 2199*x^4 - 2736*x^3 + 2826*x^2 - 1944*x + 621)
 

\( x^{16} - 4 x^{15} + 8 x^{14} - 26 x^{13} + 82 x^{12} - 176 x^{11} + 352 x^{10} - 652 x^{9} + 936 x^{8} + \cdots + 621 \) 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:   \(89791815397090000896\) \(\medspace = 2^{24}\cdot 3^{8}\cdot 13^{8}\) 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:  \(17.66\)
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:  $2^{3/2}3^{1/2}13^{1/2}\approx 17.663521732655695$
Ramified primes:   \(2\), \(3\), \(13\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{6}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{9}a^{12}+\frac{1}{9}a^{11}-\frac{1}{3}a^{9}-\frac{2}{9}a^{8}-\frac{4}{9}a^{5}+\frac{1}{9}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{9}a^{13}-\frac{1}{9}a^{11}+\frac{4}{9}a^{9}+\frac{2}{9}a^{8}-\frac{1}{9}a^{6}-\frac{4}{9}a^{5}-\frac{4}{9}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{27}a^{14}-\frac{1}{27}a^{12}+\frac{1}{9}a^{11}-\frac{2}{27}a^{10}-\frac{7}{27}a^{9}+\frac{1}{3}a^{8}+\frac{2}{27}a^{7}+\frac{5}{27}a^{6}-\frac{4}{27}a^{5}-\frac{2}{9}a^{4}+\frac{1}{3}a$, $\frac{1}{71\!\cdots\!51}a^{15}-\frac{106083272434247}{71\!\cdots\!51}a^{14}-\frac{154507712614735}{71\!\cdots\!51}a^{13}-\frac{115538569526650}{71\!\cdots\!51}a^{12}-\frac{479833357122173}{71\!\cdots\!51}a^{11}+\frac{93426000567836}{799684236961539}a^{10}-\frac{11\!\cdots\!99}{71\!\cdots\!51}a^{9}-\frac{5325975297242}{135795436465167}a^{8}+\frac{840217898005099}{71\!\cdots\!51}a^{7}+\frac{25\!\cdots\!91}{71\!\cdots\!51}a^{6}+\frac{20\!\cdots\!37}{71\!\cdots\!51}a^{5}-\frac{34757453394745}{266561412320513}a^{4}+\frac{6277099134561}{266561412320513}a^{3}+\frac{5176300999017}{11589626622631}a^{2}+\frac{25679238455446}{799684236961539}a-\frac{1623014730626}{11589626622631}$ 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:  $3$

Class group and class number

$C_{2}$, which has order $2$ (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:   \( -1 \)  (order $2$) 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:   $\frac{147276263029235}{71\!\cdots\!51}a^{15}-\frac{302913130342556}{71\!\cdots\!51}a^{14}+\frac{494759824199344}{71\!\cdots\!51}a^{13}-\frac{27\!\cdots\!73}{71\!\cdots\!51}a^{12}+\frac{65\!\cdots\!29}{71\!\cdots\!51}a^{11}-\frac{11\!\cdots\!75}{71\!\cdots\!51}a^{10}+\frac{25\!\cdots\!36}{71\!\cdots\!51}a^{9}-\frac{752671606491364}{135795436465167}a^{8}+\frac{15\!\cdots\!75}{23\!\cdots\!17}a^{7}-\frac{67\!\cdots\!72}{799684236961539}a^{6}+\frac{72\!\cdots\!85}{71\!\cdots\!51}a^{5}-\frac{29\!\cdots\!84}{23\!\cdots\!17}a^{4}+\frac{42\!\cdots\!62}{266561412320513}a^{3}-\frac{212066491196941}{11589626622631}a^{2}+\frac{11\!\cdots\!41}{799684236961539}a-\frac{38853487586674}{11589626622631}$, $\frac{2966072019727}{135795436465167}a^{15}-\frac{7063069632733}{135795436465167}a^{14}+\frac{11379591982454}{135795436465167}a^{13}-\frac{56778048607994}{135795436465167}a^{12}+\frac{148491919546789}{135795436465167}a^{11}-\frac{264763450608722}{135795436465167}a^{10}+\frac{573774899294581}{135795436465167}a^{9}-\frac{938580914785522}{135795436465167}a^{8}+\frac{122438286571139}{15088381829463}a^{7}-\frac{466941451205929}{45265145488389}a^{6}+\frac{17\!\cdots\!53}{135795436465167}a^{5}-\frac{692895854004533}{45265145488389}a^{4}+\frac{308779058638483}{15088381829463}a^{3}-\frac{5279933537888}{218672200427}a^{2}+\frac{286428456714710}{15088381829463}a-\frac{1385121286647}{218672200427}$, $\frac{2055660550751}{799684236961539}a^{15}+\frac{24688818786952}{71\!\cdots\!51}a^{14}+\frac{16862426555137}{23\!\cdots\!17}a^{13}-\frac{182917940182330}{71\!\cdots\!51}a^{12}-\frac{20464102766419}{23\!\cdots\!17}a^{11}-\frac{358768785834902}{71\!\cdots\!51}a^{10}+\frac{758122954291958}{71\!\cdots\!51}a^{9}+\frac{7197624942773}{45265145488389}a^{8}+\frac{778981906858871}{71\!\cdots\!51}a^{7}+\frac{864544377197027}{71\!\cdots\!51}a^{6}-\frac{235341955870546}{71\!\cdots\!51}a^{5}-\frac{85795925044399}{23\!\cdots\!17}a^{4}+\frac{50871871289082}{266561412320513}a^{3}+\frac{12825991308014}{34768879867893}a^{2}-\frac{761889563606957}{799684236961539}a+\frac{711721937016}{11589626622631}$, $\frac{281590462049476}{71\!\cdots\!51}a^{15}-\frac{633504295400089}{71\!\cdots\!51}a^{14}+\frac{10\!\cdots\!11}{71\!\cdots\!51}a^{13}-\frac{52\!\cdots\!91}{71\!\cdots\!51}a^{12}+\frac{13\!\cdots\!16}{71\!\cdots\!51}a^{11}-\frac{23\!\cdots\!84}{71\!\cdots\!51}a^{10}+\frac{51\!\cdots\!39}{71\!\cdots\!51}a^{9}-\frac{15\!\cdots\!49}{135795436465167}a^{8}+\frac{32\!\cdots\!55}{23\!\cdots\!17}a^{7}-\frac{41\!\cdots\!68}{23\!\cdots\!17}a^{6}+\frac{15\!\cdots\!26}{71\!\cdots\!51}a^{5}-\frac{21\!\cdots\!87}{799684236961539}a^{4}+\frac{90\!\cdots\!23}{266561412320513}a^{3}-\frac{445643923945127}{11589626622631}a^{2}+\frac{25\!\cdots\!50}{799684236961539}a-\frac{129670966475355}{11589626622631}$, $\frac{5155610429456}{266561412320513}a^{15}-\frac{416315635215509}{71\!\cdots\!51}a^{14}+\frac{72290535222140}{799684236961539}a^{13}-\frac{29\!\cdots\!46}{71\!\cdots\!51}a^{12}+\frac{934068768251141}{799684236961539}a^{11}-\frac{15\!\cdots\!34}{71\!\cdots\!51}a^{10}+\frac{32\!\cdots\!77}{71\!\cdots\!51}a^{9}-\frac{355083560007275}{45265145488389}a^{8}+\frac{68\!\cdots\!11}{71\!\cdots\!51}a^{7}-\frac{86\!\cdots\!14}{71\!\cdots\!51}a^{6}+\frac{10\!\cdots\!48}{71\!\cdots\!51}a^{5}-\frac{14\!\cdots\!41}{799684236961539}a^{4}+\frac{18\!\cdots\!66}{799684236961539}a^{3}-\frac{960846313370440}{34768879867893}a^{2}+\frac{19\!\cdots\!72}{799684236961539}a-\frac{120879181407159}{11589626622631}$, $\frac{11027014745380}{23\!\cdots\!17}a^{15}-\frac{25335229186259}{23\!\cdots\!17}a^{14}+\frac{56139796049698}{23\!\cdots\!17}a^{13}-\frac{243969667294984}{23\!\cdots\!17}a^{12}+\frac{568533157801922}{23\!\cdots\!17}a^{11}-\frac{132931748394556}{266561412320513}a^{10}+\frac{27\!\cdots\!58}{23\!\cdots\!17}a^{9}-\frac{27022327258231}{15088381829463}a^{8}+\frac{59\!\cdots\!39}{23\!\cdots\!17}a^{7}-\frac{83\!\cdots\!16}{23\!\cdots\!17}a^{6}+\frac{29\!\cdots\!48}{799684236961539}a^{5}-\frac{10\!\cdots\!73}{23\!\cdots\!17}a^{4}+\frac{50\!\cdots\!52}{799684236961539}a^{3}-\frac{234137269734881}{34768879867893}a^{2}+\frac{20\!\cdots\!60}{266561412320513}a-\frac{51855887401739}{11589626622631}$, $\frac{14184678294782}{312919918811037}a^{15}-\frac{42058081326130}{312919918811037}a^{14}+\frac{66567000816664}{312919918811037}a^{13}-\frac{295643882673086}{312919918811037}a^{12}+\frac{848457456333005}{312919918811037}a^{11}-\frac{521153866154213}{104306639603679}a^{10}+\frac{32\!\cdots\!85}{312919918811037}a^{9}-\frac{107093732464693}{5904149411529}a^{8}+\frac{69\!\cdots\!40}{312919918811037}a^{7}-\frac{87\!\cdots\!64}{312919918811037}a^{6}+\frac{10\!\cdots\!95}{312919918811037}a^{5}-\frac{14\!\cdots\!67}{34768879867893}a^{4}+\frac{19\!\cdots\!94}{34768879867893}a^{3}-\frac{22\!\cdots\!58}{34768879867893}a^{2}+\frac{19\!\cdots\!81}{34768879867893}a-\frac{285723617865410}{11589626622631}$ 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:  \( 4396.32686314 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 4396.32686314 \cdot 2}{2\cdot\sqrt{89791815397090000896}}\cr\approx \mathstrut & 1.12696530292 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 26*x^13 + 82*x^12 - 176*x^11 + 352*x^10 - 652*x^9 + 936*x^8 - 1158*x^7 + 1438*x^6 - 1758*x^5 + 2199*x^4 - 2736*x^3 + 2826*x^2 - 1944*x + 621)
 
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 - 4*x^15 + 8*x^14 - 26*x^13 + 82*x^12 - 176*x^11 + 352*x^10 - 652*x^9 + 936*x^8 - 1158*x^7 + 1438*x^6 - 1758*x^5 + 2199*x^4 - 2736*x^3 + 2826*x^2 - 1944*x + 621, 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 - 4*x^15 + 8*x^14 - 26*x^13 + 82*x^12 - 176*x^11 + 352*x^10 - 652*x^9 + 936*x^8 - 1158*x^7 + 1438*x^6 - 1758*x^5 + 2199*x^4 - 2736*x^3 + 2826*x^2 - 1944*x + 621);
 
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 - 4*x^15 + 8*x^14 - 26*x^13 + 82*x^12 - 176*x^11 + 352*x^10 - 652*x^9 + 936*x^8 - 1158*x^7 + 1438*x^6 - 1758*x^5 + 2199*x^4 - 2736*x^3 + 2826*x^2 - 1944*x + 621);
 
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

$C_2^2\wr C_2$ (as 16T39):

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)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{39}) \), 4.2.2704.1, 4.0.8112.1 x2, 4.0.7488.1 x2, 4.2.24336.2, 4.2.8112.1, 4.2.507.1, \(\Q(\sqrt{3}, \sqrt{13})\), 8.4.592240896.2, 8.0.1052872704.1, 8.0.9475854336.5, 8.0.65804544.1, 8.0.592240896.2, 8.0.9475854336.3, 8.4.9475854336.2

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]
 

Sibling fields

Galois closure: deg 32
Degree 8 siblings: 8.4.1052872704.1, 8.4.9475854336.1, 8.4.65804544.1, 8.0.65804544.1, 8.0.9475854336.5, 8.0.1052872704.1, 8.4.592240896.1, 8.0.592240896.2
Degree 16 siblings: 16.0.89791815397090000896.2, 16.0.350749278894882816.3, 16.8.89791815397090000896.1, 16.0.531312517142544384.3, 16.0.1108540930828271616.1, 16.0.89791815397090000896.3
Minimal sibling: 8.4.65804544.1

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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ R ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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

# 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 Deg $16$$4$$4$$24$
\(3\) Copy content Toggle raw display 3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{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}$
\(13\) Copy content Toggle raw display 13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$