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Properties

Label 16.0.396...000.1
Degree $16$
Signature $[0, 8]$
Discriminant $3.962\times 10^{25}$
Root discriminant \(39.80\)
Ramified primes $2,5,7$
Class number $128$ (GRH)
Class group [2, 8, 8] (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 20*x^14 - 34*x^12 - 160*x^11 + 988*x^10 - 2520*x^9 + 6291*x^8 - 13520*x^7 + 25876*x^6 - 37312*x^5 + 55602*x^4 - 61552*x^3 + 73496*x^2 - 47168*x + 55036)
 
gp: K = bnfinit(y^16 - 8*y^15 + 20*y^14 - 34*y^12 - 160*y^11 + 988*y^10 - 2520*y^9 + 6291*y^8 - 13520*y^7 + 25876*y^6 - 37312*y^5 + 55602*y^4 - 61552*y^3 + 73496*y^2 - 47168*y + 55036, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 20*x^14 - 34*x^12 - 160*x^11 + 988*x^10 - 2520*x^9 + 6291*x^8 - 13520*x^7 + 25876*x^6 - 37312*x^5 + 55602*x^4 - 61552*x^3 + 73496*x^2 - 47168*x + 55036);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 20*x^14 - 34*x^12 - 160*x^11 + 988*x^10 - 2520*x^9 + 6291*x^8 - 13520*x^7 + 25876*x^6 - 37312*x^5 + 55602*x^4 - 61552*x^3 + 73496*x^2 - 47168*x + 55036)
 

\( x^{16} - 8 x^{15} + 20 x^{14} - 34 x^{12} - 160 x^{11} + 988 x^{10} - 2520 x^{9} + 6291 x^{8} + \cdots + 55036 \) 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:   \(39615410820716953600000000\) \(\medspace = 2^{44}\cdot 5^{8}\cdot 7^{8}\) 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:  \(39.80\)
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:  $2^{11/4}5^{1/2}7^{1/2}\approx 39.79848225570752$
Ramified primes:   \(2\), \(5\), \(7\) 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:  \(560=2^{4}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{560}(1,·)$, $\chi_{560}(69,·)$, $\chi_{560}(321,·)$, $\chi_{560}(141,·)$, $\chi_{560}(461,·)$, $\chi_{560}(209,·)$, $\chi_{560}(281,·)$, $\chi_{560}(169,·)$, $\chi_{560}(349,·)$, $\chi_{560}(421,·)$, $\chi_{560}(489,·)$, $\chi_{560}(29,·)$, $\chi_{560}(449,·)$, $\chi_{560}(181,·)$, $\chi_{560}(41,·)$, $\chi_{560}(309,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{5}$, $\frac{1}{15213630121762}a^{14}-\frac{7}{15213630121762}a^{13}+\frac{399050155412}{7606815060881}a^{12}+\frac{1409106598014}{7606815060881}a^{11}+\frac{3454518763187}{15213630121762}a^{10}+\frac{3802478095741}{15213630121762}a^{9}+\frac{777420658819}{15213630121762}a^{8}+\frac{2537896973436}{7606815060881}a^{7}+\frac{2993650546753}{7606815060881}a^{6}-\frac{2177773688846}{7606815060881}a^{5}-\frac{3811350157817}{15213630121762}a^{4}+\frac{403633356699}{7606815060881}a^{3}-\frac{1747672498190}{7606815060881}a^{2}+\frac{1677389937641}{7606815060881}a+\frac{2774098182768}{7606815060881}$, $\frac{1}{82\!\cdots\!82}a^{15}+\frac{27073}{82\!\cdots\!82}a^{14}-\frac{71\!\cdots\!86}{41\!\cdots\!41}a^{13}+\frac{12\!\cdots\!19}{82\!\cdots\!82}a^{12}-\frac{71\!\cdots\!64}{41\!\cdots\!41}a^{11}+\frac{56\!\cdots\!87}{82\!\cdots\!82}a^{10}+\frac{40\!\cdots\!41}{82\!\cdots\!82}a^{9}-\frac{81\!\cdots\!93}{82\!\cdots\!82}a^{8}+\frac{39\!\cdots\!51}{82\!\cdots\!82}a^{7}+\frac{13\!\cdots\!93}{41\!\cdots\!41}a^{6}+\frac{32\!\cdots\!57}{82\!\cdots\!82}a^{5}-\frac{18\!\cdots\!92}{41\!\cdots\!41}a^{4}+\frac{40\!\cdots\!88}{41\!\cdots\!41}a^{3}-\frac{48\!\cdots\!26}{41\!\cdots\!41}a^{2}-\frac{19\!\cdots\!70}{41\!\cdots\!41}a-\frac{75\!\cdots\!77}{41\!\cdots\!41}$ 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:  $2$

Class group and class number

$C_{2}\times C_{8}\times C_{8}$, which has order $128$ (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:   $\frac{89214}{424463761}a^{14}-\frac{624498}{424463761}a^{13}+\frac{966104}{424463761}a^{12}+\frac{2321850}{424463761}a^{11}-\frac{4026778}{424463761}a^{10}-\frac{16033604}{424463761}a^{9}+\frac{146792427}{848927522}a^{8}-\frac{146327676}{424463761}a^{7}+\frac{270380585}{424463761}a^{6}-\frac{418667340}{424463761}a^{5}+\frac{1027171111}{848927522}a^{4}-\frac{441739660}{424463761}a^{3}+\frac{184508634}{424463761}a^{2}-\frac{17828600}{424463761}a-\frac{819590072}{424463761}$, $\frac{150159}{402882001}a^{14}-\frac{1051113}{402882001}a^{13}+\frac{3400733}{805764002}a^{12}+\frac{3462270}{402882001}a^{11}-\frac{12014477}{805764002}a^{10}-\frac{26752809}{402882001}a^{9}+\frac{246526257}{805764002}a^{8}-\frac{251505216}{402882001}a^{7}+\frac{1010005683}{805764002}a^{6}-\frac{830279784}{402882001}a^{5}+\frac{1250885453}{402882001}a^{4}-\frac{1316071800}{402882001}a^{3}+\frac{1235424986}{402882001}a^{2}-\frac{688221244}{402882001}a-\frac{74489741}{402882001}$, $\frac{143988294336}{17\!\cdots\!19}a^{15}-\frac{1079912207520}{17\!\cdots\!19}a^{14}+\frac{2151457485403}{17\!\cdots\!19}a^{13}+\frac{4788389651201}{34\!\cdots\!38}a^{12}-\frac{6650690785925}{17\!\cdots\!19}a^{11}-\frac{25790414416602}{17\!\cdots\!19}a^{10}+\frac{3264013565077}{43640984112322}a^{9}-\frac{540064731215667}{34\!\cdots\!38}a^{8}+\frac{631011956597736}{17\!\cdots\!19}a^{7}-\frac{12\!\cdots\!30}{17\!\cdots\!19}a^{6}+\frac{37\!\cdots\!51}{34\!\cdots\!38}a^{5}-\frac{17\!\cdots\!32}{17\!\cdots\!19}a^{4}+\frac{21\!\cdots\!07}{17\!\cdots\!19}a^{3}-\frac{19\!\cdots\!54}{17\!\cdots\!19}a^{2}+\frac{283646734526511}{17\!\cdots\!19}a-\frac{15\!\cdots\!51}{17\!\cdots\!19}$, $\frac{2104584495}{7606815060881}a^{14}-\frac{14732091465}{7606815060881}a^{13}+\frac{43800571151}{15213630121762}a^{12}+\frac{60115475592}{7606815060881}a^{11}-\frac{208143434215}{15213630121762}a^{10}-\frac{381814787305}{7606815060881}a^{9}+\frac{1738187238039}{7606815060881}a^{8}-\frac{3398678831244}{7606815060881}a^{7}+\frac{11621106633099}{15213630121762}a^{6}-\frac{8433183102336}{7606815060881}a^{5}+\frac{15272132531455}{15213630121762}a^{4}-\frac{3768492658572}{7606815060881}a^{3}-\frac{6416635821157}{7606815060881}a^{2}+\frac{7248681843208}{7606815060881}a-\frac{25023668308017}{7606815060881}$, $\frac{312567947962}{17\!\cdots\!19}a^{15}-\frac{473685218186751}{41\!\cdots\!41}a^{14}+\frac{673281038519708}{41\!\cdots\!41}a^{13}+\frac{22\!\cdots\!95}{82\!\cdots\!82}a^{12}-\frac{53414850421974}{41\!\cdots\!41}a^{11}-\frac{11\!\cdots\!74}{41\!\cdots\!41}a^{10}+\frac{95\!\cdots\!07}{82\!\cdots\!82}a^{9}-\frac{11\!\cdots\!20}{41\!\cdots\!41}a^{8}+\frac{29\!\cdots\!93}{41\!\cdots\!41}a^{7}-\frac{53\!\cdots\!71}{41\!\cdots\!41}a^{6}+\frac{19\!\cdots\!07}{82\!\cdots\!82}a^{5}-\frac{24\!\cdots\!49}{82\!\cdots\!82}a^{4}+\frac{17\!\cdots\!14}{41\!\cdots\!41}a^{3}-\frac{17\!\cdots\!42}{41\!\cdots\!41}a^{2}+\frac{13\!\cdots\!35}{41\!\cdots\!41}a-\frac{16\!\cdots\!81}{41\!\cdots\!41}$, $\frac{143988294336}{17\!\cdots\!19}a^{15}-\frac{437424041199}{17\!\cdots\!19}a^{14}-\frac{2345959678844}{17\!\cdots\!19}a^{13}+\frac{9669585276114}{17\!\cdots\!19}a^{12}+\frac{8163389646205}{17\!\cdots\!19}a^{11}-\frac{102987399848167}{34\!\cdots\!38}a^{10}+\frac{366095788579}{43640984112322}a^{9}+\frac{257375924304558}{17\!\cdots\!19}a^{8}-\frac{445108189700568}{17\!\cdots\!19}a^{7}+\frac{17\!\cdots\!17}{34\!\cdots\!38}a^{6}-\frac{33\!\cdots\!41}{34\!\cdots\!38}a^{5}+\frac{35\!\cdots\!75}{17\!\cdots\!19}a^{4}-\frac{34\!\cdots\!93}{17\!\cdots\!19}a^{3}+\frac{33\!\cdots\!80}{17\!\cdots\!19}a^{2}-\frac{26\!\cdots\!25}{17\!\cdots\!19}a+\frac{32\!\cdots\!27}{17\!\cdots\!19}$, $\frac{143988294336}{17\!\cdots\!19}a^{15}-\frac{344691846132414}{41\!\cdots\!41}a^{14}+\frac{11\!\cdots\!55}{41\!\cdots\!41}a^{13}-\frac{731013213065409}{82\!\cdots\!82}a^{12}-\frac{38\!\cdots\!25}{41\!\cdots\!41}a^{11}-\frac{22\!\cdots\!60}{41\!\cdots\!41}a^{10}+\frac{92\!\cdots\!85}{82\!\cdots\!82}a^{9}-\frac{13\!\cdots\!00}{41\!\cdots\!41}a^{8}+\frac{29\!\cdots\!60}{41\!\cdots\!41}a^{7}-\frac{57\!\cdots\!55}{41\!\cdots\!41}a^{6}+\frac{17\!\cdots\!69}{82\!\cdots\!82}a^{5}-\frac{18\!\cdots\!87}{82\!\cdots\!82}a^{4}+\frac{94\!\cdots\!33}{41\!\cdots\!41}a^{3}-\frac{65\!\cdots\!60}{41\!\cdots\!41}a^{2}+\frac{85\!\cdots\!29}{41\!\cdots\!41}a+\frac{12\!\cdots\!25}{41\!\cdots\!41}$ 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:  \( 12198.9512748 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 12198.9512748 \cdot 128}{2\cdot\sqrt{39615410820716953600000000}}\cr\approx \mathstrut & 0.301307079171 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 20*x^14 - 34*x^12 - 160*x^11 + 988*x^10 - 2520*x^9 + 6291*x^8 - 13520*x^7 + 25876*x^6 - 37312*x^5 + 55602*x^4 - 61552*x^3 + 73496*x^2 - 47168*x + 55036)
 
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 - 8*x^15 + 20*x^14 - 34*x^12 - 160*x^11 + 988*x^10 - 2520*x^9 + 6291*x^8 - 13520*x^7 + 25876*x^6 - 37312*x^5 + 55602*x^4 - 61552*x^3 + 73496*x^2 - 47168*x + 55036, 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 - 8*x^15 + 20*x^14 - 34*x^12 - 160*x^11 + 988*x^10 - 2520*x^9 + 6291*x^8 - 13520*x^7 + 25876*x^6 - 37312*x^5 + 55602*x^4 - 61552*x^3 + 73496*x^2 - 47168*x + 55036);
 
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 - 8*x^15 + 20*x^14 - 34*x^12 - 160*x^11 + 988*x^10 - 2520*x^9 + 6291*x^8 - 13520*x^7 + 25876*x^6 - 37312*x^5 + 55602*x^4 - 61552*x^3 + 73496*x^2 - 47168*x + 55036);
 
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

$C_2^2\times C_4$ (as 16T2):

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 $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{-7}, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{10}, \sqrt{-14})\), \(\Q(\sqrt{2}, \sqrt{-35})\), \(\Q(\sqrt{5}, \sqrt{-14})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{5}, \sqrt{-7})\), 4.0.2508800.1, 4.0.100352.5, \(\Q(\zeta_{16})^+\), 4.4.51200.1, 8.0.6146560000.2, 8.0.6294077440000.7, 8.8.2621440000.1, 8.0.6294077440000.6, 8.0.6294077440000.2, 8.0.6294077440000.5, 8.0.10070523904.2

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{4}$ R R ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.8.22.2$x^{8} + 16 x^{7} + 88 x^{6} + 192 x^{5} + 180 x^{4} + 288 x^{3} + 432 x^{2} + 516$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.8.22.2$x^{8} + 16 x^{7} + 88 x^{6} + 192 x^{5} + 180 x^{4} + 288 x^{3} + 432 x^{2} + 516$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
\(5\) Copy content Toggle raw display 5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(7\) Copy content Toggle raw display 7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$