Properties

Label 16.0.272...625.1
Degree $16$
Signature $[0, 8]$
Discriminant $2.720\times 10^{22}$
Root discriminant \(25.24\)
Ramified primes $3,5,19$
Class number $8$ (GRH)
Class group [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 - x^15 + 5*x^14 - 14*x^13 + 14*x^12 + 31*x^11 - 30*x^10 + 434*x^9 - 1059*x^8 + 2170*x^7 - 750*x^6 + 3875*x^5 + 8750*x^4 - 43750*x^3 + 78125*x^2 - 78125*x + 390625)
 
gp: K = bnfinit(y^16 - y^15 + 5*y^14 - 14*y^13 + 14*y^12 + 31*y^11 - 30*y^10 + 434*y^9 - 1059*y^8 + 2170*y^7 - 750*y^6 + 3875*y^5 + 8750*y^4 - 43750*y^3 + 78125*y^2 - 78125*y + 390625, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 + 5*x^14 - 14*x^13 + 14*x^12 + 31*x^11 - 30*x^10 + 434*x^9 - 1059*x^8 + 2170*x^7 - 750*x^6 + 3875*x^5 + 8750*x^4 - 43750*x^3 + 78125*x^2 - 78125*x + 390625);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 + 5*x^14 - 14*x^13 + 14*x^12 + 31*x^11 - 30*x^10 + 434*x^9 - 1059*x^8 + 2170*x^7 - 750*x^6 + 3875*x^5 + 8750*x^4 - 43750*x^3 + 78125*x^2 - 78125*x + 390625)
 

\( x^{16} - x^{15} + 5 x^{14} - 14 x^{13} + 14 x^{12} + 31 x^{11} - 30 x^{10} + 434 x^{9} - 1059 x^{8} + 2170 x^{7} - 750 x^{6} + 3875 x^{5} + 8750 x^{4} - 43750 x^{3} + 78125 x^{2} + \cdots + 390625 \) 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:   \(27204384060547119140625\) \(\medspace = 3^{8}\cdot 5^{12}\cdot 19^{8}\) 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:  \(25.24\)
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:  $3^{1/2}5^{3/4}19^{1/2}\approx 25.24439291382227$
Ramified primes:   \(3\), \(5\), \(19\) 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:  \(285=3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{285}(1,·)$, $\chi_{285}(134,·)$, $\chi_{285}(77,·)$, $\chi_{285}(208,·)$, $\chi_{285}(248,·)$, $\chi_{285}(151,·)$, $\chi_{285}(284,·)$, $\chi_{285}(94,·)$, $\chi_{285}(37,·)$, $\chi_{285}(227,·)$, $\chi_{285}(229,·)$, $\chi_{285}(172,·)$, $\chi_{285}(113,·)$, $\chi_{285}(56,·)$, $\chi_{285}(58,·)$, $\chi_{285}(191,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5}a^{9}-\frac{1}{5}a^{8}+\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{4}-\frac{1}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{275}a^{10}-\frac{1}{25}a^{9}-\frac{2}{5}a^{8}+\frac{1}{25}a^{7}-\frac{11}{25}a^{6}-\frac{109}{275}a^{5}+\frac{2}{5}a^{4}-\frac{6}{25}a^{3}-\frac{9}{25}a^{2}+\frac{2}{5}a+\frac{4}{11}$, $\frac{1}{1375}a^{11}-\frac{1}{1375}a^{10}+\frac{1}{25}a^{9}+\frac{51}{125}a^{8}+\frac{24}{125}a^{7}+\frac{331}{1375}a^{6}+\frac{134}{275}a^{5}+\frac{44}{125}a^{4}+\frac{6}{125}a^{3}+\frac{9}{25}a^{2}-\frac{18}{55}a-\frac{3}{11}$, $\frac{1}{27500}a^{12}+\frac{1}{6875}a^{11}+\frac{1}{2500}a^{9}+\frac{226}{625}a^{8}-\frac{756}{6875}a^{7}+\frac{1}{220}a^{6}-\frac{164}{625}a^{5}-\frac{131}{625}a^{4}+\frac{1}{4}a^{3}+\frac{1}{275}a^{2}+\frac{14}{55}a-\frac{1}{4}$, $\frac{1}{243237500}a^{13}+\frac{629}{243237500}a^{12}-\frac{72}{486475}a^{11}+\frac{3501}{22112500}a^{10}-\frac{820971}{22112500}a^{9}+\frac{14323994}{60809375}a^{8}+\frac{327069}{1945900}a^{7}+\frac{48514909}{243237500}a^{6}+\frac{1234244}{5528125}a^{5}-\frac{33807}{176900}a^{4}-\frac{2004361}{9729500}a^{3}+\frac{217994}{486475}a^{2}+\frac{134413}{389180}a+\frac{1803}{7076}$, $\frac{1}{1216187500}a^{14}-\frac{1}{1216187500}a^{13}+\frac{499}{60809375}a^{12}-\frac{278489}{1216187500}a^{11}+\frac{318389}{1216187500}a^{10}-\frac{21552561}{304046875}a^{9}+\frac{84062849}{243237500}a^{8}+\frac{323550659}{1216187500}a^{7}-\frac{18720796}{304046875}a^{6}+\frac{41316639}{243237500}a^{5}+\frac{13932769}{48647500}a^{4}+\frac{784951}{2432375}a^{3}-\frac{304223}{1945900}a^{2}-\frac{61173}{389180}a-\frac{4705}{19459}$, $\frac{1}{6080937500}a^{15}-\frac{1}{6080937500}a^{14}+\frac{1}{1216187500}a^{13}+\frac{40493}{3040468750}a^{12}+\frac{1195889}{6080937500}a^{11}+\frac{2850031}{6080937500}a^{10}+\frac{33321597}{608093750}a^{9}+\frac{1328617559}{6080937500}a^{8}+\frac{2921404941}{6080937500}a^{7}+\frac{173040217}{608093750}a^{6}+\frac{15754301}{48647500}a^{5}+\frac{224859}{48647500}a^{4}+\frac{633277}{4864750}a^{3}-\frac{930101}{1945900}a^{2}+\frac{35505}{77836}a+\frac{1139}{77836}$ 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

$C_{8}$, which has order $8$ (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:   \( \frac{1}{389180} a^{15} - \frac{1}{77836} a^{14} + \frac{7}{194590} a^{13} - \frac{7}{194590} a^{12} - \frac{31}{389180} a^{11} + \frac{3}{38918} a^{10} - \frac{217}{194590} a^{9} + \frac{1059}{389180} a^{8} - \frac{217}{38918} a^{7} + \frac{75}{38918} a^{6} - \frac{775}{77836} a^{5} - \frac{875}{38918} a^{4} + \frac{4375}{38918} a^{3} - \frac{15625}{77836} a^{2} - \frac{1253}{194590} a - \frac{78125}{77836} \)  (order $30$) 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:   $\frac{3024}{1520234375}a^{15}-\frac{6804}{1520234375}a^{14}+\frac{35789}{1520234375}a^{12}+\frac{756}{1520234375}a^{11}+\frac{305424}{1520234375}a^{10}-\frac{756}{2432375}a^{9}+\frac{76356}{138203125}a^{8}-\frac{4275936}{1520234375}a^{7}+\frac{756}{486475}a^{6}+\frac{76356}{12161875}a^{5}-\frac{42336}{2432375}a^{4}+\frac{756}{44225}a^{3}-\frac{76356}{486475}a^{2}+\frac{3024}{19459}a-\frac{15679}{19459}$, $\frac{2601}{1216187500}a^{15}-\frac{8201}{243237500}a^{14}+\frac{63559}{1216187500}a^{13}-\frac{28546}{304046875}a^{12}+\frac{79309}{243237500}a^{11}-\frac{101}{884500}a^{10}-\frac{714826}{304046875}a^{9}+\frac{5121579}{1216187500}a^{8}-\frac{2289829}{243237500}a^{7}+\frac{8003464}{304046875}a^{6}-\frac{969701}{22112500}a^{5}-\frac{2023559}{48647500}a^{4}+\frac{12796}{97295}a^{3}-\frac{28405}{77836}a^{2}+\frac{459531}{389180}a-\frac{203611}{77836}$, $\frac{559}{6080937500}a^{15}+\frac{38783}{3040468750}a^{14}+\frac{559}{1216187500}a^{13}-\frac{3913}{3040468750}a^{12}+\frac{3913}{3040468750}a^{11}+\frac{17329}{6080937500}a^{10}-\frac{1677}{608093750}a^{9}+\frac{121303}{3040468750}a^{8}-\frac{591981}{6080937500}a^{7}+\frac{121303}{608093750}a^{6}-\frac{1677}{24323750}a^{5}+\frac{17329}{48647500}a^{4}+\frac{3913}{4864750}a^{3}-\frac{3913}{972950}a^{2}+\frac{559}{77836}a-\frac{78395}{77836}$, $\frac{29233}{3040468750}a^{15}-\frac{7533}{3040468750}a^{14}-\frac{35217}{608093750}a^{13}-\frac{1917}{276406250}a^{12}-\frac{1035163}{3040468750}a^{11}+\frac{1987723}{3040468750}a^{10}-\frac{129573}{608093750}a^{9}-\frac{9491303}{3040468750}a^{8}-\frac{295677}{276406250}a^{7}-\frac{8715053}{608093750}a^{6}+\frac{7620353}{121618750}a^{5}-\frac{103977}{24323750}a^{4}+\frac{17009}{972950}a^{3}-\frac{135}{3538}a^{2}-\frac{160463}{194590}a+\frac{32880}{19459}$, $\frac{2923}{6080937500}a^{15}+\frac{175937}{6080937500}a^{14}-\frac{26789}{1216187500}a^{13}+\frac{131039}{3040468750}a^{12}-\frac{3634993}{6080937500}a^{11}+\frac{940453}{6080937500}a^{10}+\frac{237267}{608093750}a^{9}+\frac{15134957}{6080937500}a^{8}+\frac{24804783}{6080937500}a^{7}-\frac{6597713}{608093750}a^{6}+\frac{584003}{48647500}a^{5}-\frac{127547}{48647500}a^{4}-\frac{256559}{4864750}a^{3}-\frac{69451}{1945900}a^{2}-\frac{184153}{389180}a+\frac{5767}{7076}$, $\frac{118367}{6080937500}a^{15}-\frac{249301}{3040468750}a^{14}+\frac{103727}{608093750}a^{13}-\frac{2047763}{6080937500}a^{12}+\frac{50141}{104843750}a^{11}+\frac{3987581}{3040468750}a^{10}-\frac{7515949}{1216187500}a^{9}+\frac{47140639}{3040468750}a^{8}-\frac{120655509}{3040468750}a^{7}+\frac{85126661}{1216187500}a^{6}-\frac{429351}{24323750}a^{5}-\frac{917319}{4864750}a^{4}+\frac{6364219}{9729500}a^{3}-\frac{1645189}{972950}a^{2}+\frac{111371}{38918}a-\frac{39731}{19459}$, $\frac{12217}{1216187500}a^{15}+\frac{53237}{1216187500}a^{14}+\frac{41231}{1216187500}a^{13}+\frac{189207}{1216187500}a^{12}-\frac{759893}{1216187500}a^{11}-\frac{784567}{1216187500}a^{10}+\frac{572789}{1216187500}a^{9}+\frac{4026333}{1216187500}a^{8}+\frac{37636683}{1216187500}a^{7}+\frac{6717379}{1216187500}a^{6}-\frac{2406409}{243237500}a^{5}-\frac{7392507}{48647500}a^{4}-\frac{2352231}{9729500}a^{3}+\frac{1243101}{1945900}a^{2}+\frac{315511}{389180}a+\frac{53553}{19459}$ 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:  \( 49344.1781525 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 49344.1781525 \cdot 8}{30\cdot\sqrt{27204384060547119140625}}\cr\approx \mathstrut & 0.193786781781 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 5*x^14 - 14*x^13 + 14*x^12 + 31*x^11 - 30*x^10 + 434*x^9 - 1059*x^8 + 2170*x^7 - 750*x^6 + 3875*x^5 + 8750*x^4 - 43750*x^3 + 78125*x^2 - 78125*x + 390625)
 
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 - x^15 + 5*x^14 - 14*x^13 + 14*x^12 + 31*x^11 - 30*x^10 + 434*x^9 - 1059*x^8 + 2170*x^7 - 750*x^6 + 3875*x^5 + 8750*x^4 - 43750*x^3 + 78125*x^2 - 78125*x + 390625, 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 - x^15 + 5*x^14 - 14*x^13 + 14*x^12 + 31*x^11 - 30*x^10 + 434*x^9 - 1059*x^8 + 2170*x^7 - 750*x^6 + 3875*x^5 + 8750*x^4 - 43750*x^3 + 78125*x^2 - 78125*x + 390625);
 
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 - x^15 + 5*x^14 - 14*x^13 + 14*x^12 + 31*x^11 - 30*x^10 + 434*x^9 - 1059*x^8 + 2170*x^7 - 750*x^6 + 3875*x^5 + 8750*x^4 - 43750*x^3 + 78125*x^2 - 78125*x + 390625);
 
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

\(\Q(\sqrt{285}) \), \(\Q(\sqrt{-95}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{57}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{-95})\), \(\Q(\sqrt{5}, \sqrt{57})\), \(\Q(\sqrt{-15}, \sqrt{-19})\), \(\Q(\sqrt{5}, \sqrt{-19})\), \(\Q(\sqrt{-15}, \sqrt{57})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-19})\), 4.0.406125.2, \(\Q(\zeta_{5})\), \(\Q(\zeta_{15})^+\), 4.4.45125.1, 8.0.6597500625.1, 8.0.164937515625.3, 8.8.164937515625.1, 8.0.164937515625.1, 8.0.2036265625.1, 8.0.164937515625.2, \(\Q(\zeta_{15})\)

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 ${\href{/padicField/2.4.0.1}{4} }^{4}$ R R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ R ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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
\(3\) Copy content Toggle raw display 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(5\) Copy content Toggle raw display 5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
\(19\) Copy content Toggle raw display 19.4.2.1$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$