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Properties

Label 16.8.281...000.3
Degree $16$
Signature $[8, 4]$
Discriminant $2.818\times 10^{21}$
Root discriminant \(21.91\)
Ramified primes $2,3,5$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $Q_8 : C_2^2$ (as 16T23)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^14 - 30*x^12 - 64*x^11 - 16*x^10 + 384*x^9 + 154*x^8 - 768*x^7 - 128*x^6 + 552*x^5 + 100*x^4 - 160*x^3 - 48*x^2 + 16*x + 4)
 
gp: K = bnfinit(y^16 + 4*y^14 - 30*y^12 - 64*y^11 - 16*y^10 + 384*y^9 + 154*y^8 - 768*y^7 - 128*y^6 + 552*y^5 + 100*y^4 - 160*y^3 - 48*y^2 + 16*y + 4, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 4*x^14 - 30*x^12 - 64*x^11 - 16*x^10 + 384*x^9 + 154*x^8 - 768*x^7 - 128*x^6 + 552*x^5 + 100*x^4 - 160*x^3 - 48*x^2 + 16*x + 4);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 4*x^14 - 30*x^12 - 64*x^11 - 16*x^10 + 384*x^9 + 154*x^8 - 768*x^7 - 128*x^6 + 552*x^5 + 100*x^4 - 160*x^3 - 48*x^2 + 16*x + 4)
 

\( x^{16} + 4 x^{14} - 30 x^{12} - 64 x^{11} - 16 x^{10} + 384 x^{9} + 154 x^{8} - 768 x^{7} - 128 x^{6} + \cdots + 4 \) 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:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(2817928042905600000000\) \(\medspace = 2^{40}\cdot 3^{8}\cdot 5^{8}\) 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:  \(21.91\)
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:  $2^{5/2}3^{1/2}5^{1/2}\approx 21.908902300206645$
Ramified primes:   \(2\), \(3\), \(5\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{6}a^{10}+\frac{1}{6}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{4}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{6}a^{11}+\frac{1}{6}a^{9}+\frac{1}{6}a^{8}+\frac{1}{3}a^{5}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{18}a^{12}-\frac{1}{18}a^{11}-\frac{1}{18}a^{10}+\frac{1}{6}a^{9}-\frac{1}{6}a^{8}+\frac{2}{9}a^{7}+\frac{1}{9}a^{6}-\frac{1}{9}a^{5}-\frac{2}{9}a^{4}+\frac{2}{9}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{9}$, $\frac{1}{18}a^{13}+\frac{1}{18}a^{11}-\frac{1}{18}a^{10}+\frac{1}{6}a^{9}+\frac{1}{18}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{4}-\frac{1}{9}a^{3}-\frac{1}{3}a^{2}+\frac{2}{9}a+\frac{2}{9}$, $\frac{1}{270}a^{14}+\frac{2}{135}a^{13}+\frac{1}{270}a^{12}+\frac{1}{15}a^{11}-\frac{11}{135}a^{10}+\frac{23}{135}a^{9}+\frac{1}{270}a^{8}+\frac{1}{5}a^{7}+\frac{2}{15}a^{6}+\frac{2}{9}a^{5}-\frac{61}{135}a^{4}+\frac{47}{135}a^{3}+\frac{2}{135}a^{2}-\frac{13}{27}a-\frac{52}{135}$, $\frac{1}{1388233890}a^{15}+\frac{110189}{1388233890}a^{14}+\frac{11424013}{694116945}a^{13}-\frac{2155457}{1388233890}a^{12}-\frac{26407661}{694116945}a^{11}+\frac{3157877}{462744630}a^{10}+\frac{10023461}{1388233890}a^{9}+\frac{139525007}{694116945}a^{8}-\frac{12414703}{77124105}a^{7}-\frac{6018472}{46274463}a^{6}+\frac{49293284}{694116945}a^{5}+\frac{112576327}{694116945}a^{4}+\frac{42130867}{694116945}a^{3}-\frac{4722601}{46274463}a^{2}+\frac{61374991}{231372315}a-\frac{41772812}{138823389}$ 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

Trivial group, which has order $1$ (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:  $11$
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:   $\frac{4408}{96795}a^{15}-\frac{6349}{96795}a^{14}+\frac{11024}{96795}a^{13}-\frac{73729}{193590}a^{12}-\frac{171304}{96795}a^{11}-\frac{10196}{6453}a^{10}+\frac{465032}{96795}a^{9}+\frac{4960787}{193590}a^{8}-\frac{71024}{10755}a^{7}-\frac{1848112}{32265}a^{6}+\frac{502054}{96795}a^{5}+\frac{3600359}{96795}a^{4}+\frac{800408}{96795}a^{3}-\frac{281788}{32265}a^{2}-\frac{136084}{32265}a+\frac{12529}{96795}$, $\frac{96686101}{1388233890}a^{15}+\frac{3259891}{277646778}a^{14}+\frac{45949112}{138823389}a^{13}+\frac{213482179}{1388233890}a^{12}-\frac{1220955152}{694116945}a^{11}-\frac{1946882927}{462744630}a^{10}-\frac{1923701174}{694116945}a^{9}+\frac{2909451415}{138823389}a^{8}+\frac{75701054}{15424821}a^{7}-\frac{10374980384}{231372315}a^{6}+\frac{9538295279}{694116945}a^{5}+\frac{27260409271}{694116945}a^{4}-\frac{10319981246}{694116945}a^{3}-\frac{151686658}{8569345}a^{2}+\frac{300820856}{231372315}a+\frac{2200054733}{694116945}$, $\frac{464359}{3910518}a^{15}-\frac{86968}{9776295}a^{14}+\frac{10422751}{19552590}a^{13}+\frac{41152}{9776295}a^{12}-\frac{31895389}{9776295}a^{11}-\frac{7703476}{1086255}a^{10}-\frac{26226553}{9776295}a^{9}+\frac{403788202}{9776295}a^{8}+\frac{3644014}{362085}a^{7}-\frac{250477486}{3258765}a^{6}+\frac{13965032}{1955259}a^{5}+\frac{411946346}{9776295}a^{4}-\frac{46910452}{9776295}a^{3}-\frac{24804479}{3258765}a^{2}+\frac{776}{217251}a+\frac{10241407}{9776295}$, $\frac{91349963}{462744630}a^{15}-\frac{3664726}{46274463}a^{14}+\frac{39347849}{46274463}a^{13}-\frac{15887707}{51416070}a^{12}-\frac{1297471141}{231372315}a^{11}-\frac{2345394194}{231372315}a^{10}+\frac{126091708}{231372315}a^{9}+\frac{2268250421}{30849642}a^{8}-\frac{40386899}{15424821}a^{7}-\frac{1257234368}{8569345}a^{6}+\frac{8771712997}{231372315}a^{5}+\frac{19662356173}{231372315}a^{4}-\frac{3425511518}{231372315}a^{3}-\frac{4003324384}{231372315}a^{2}-\frac{350949116}{231372315}a+\frac{97422448}{77124105}$, $\frac{6853811}{694116945}a^{15}+\frac{20485726}{694116945}a^{14}+\frac{178102313}{1388233890}a^{13}+\frac{21936367}{138823389}a^{12}+\frac{86300059}{694116945}a^{11}-\frac{614709197}{462744630}a^{10}-\frac{6095044519}{1388233890}a^{9}-\frac{4787605103}{1388233890}a^{8}+\frac{61970474}{8569345}a^{7}+\frac{5852214874}{231372315}a^{6}+\frac{628157678}{694116945}a^{5}-\frac{5531575771}{138823389}a^{4}-\frac{4003782623}{694116945}a^{3}+\frac{1338768622}{77124105}a^{2}+\frac{870803672}{231372315}a-\frac{171243073}{694116945}$, $\frac{49691350}{138823389}a^{15}+\frac{195051943}{694116945}a^{14}+\frac{2461863299}{1388233890}a^{13}+\frac{2000449501}{1388233890}a^{12}-\frac{6253318076}{694116945}a^{11}-\frac{6872889002}{231372315}a^{10}-\frac{22280542772}{694116945}a^{9}+\frac{143632630621}{1388233890}a^{8}+\frac{3303613466}{25708035}a^{7}-\frac{31838051159}{231372315}a^{6}-\frac{17524407473}{138823389}a^{5}+\frac{31566651649}{694116945}a^{4}+\frac{38795205577}{694116945}a^{3}+\frac{1051622164}{231372315}a^{2}-\frac{420951202}{46274463}a-\frac{892377052}{694116945}$, $\frac{2354377}{77124105}a^{15}+\frac{1003067}{10283214}a^{14}+\frac{4437205}{30849642}a^{13}+\frac{68577361}{154248210}a^{12}-\frac{21236956}{25708035}a^{11}-\frac{359523752}{77124105}a^{10}-\frac{190253242}{25708035}a^{9}+\frac{72799723}{10283214}a^{8}+\frac{195737186}{5141607}a^{7}-\frac{40047331}{25708035}a^{6}-\frac{4172141789}{77124105}a^{5}-\frac{1081297}{25708035}a^{4}+\frac{1472836621}{77124105}a^{3}+\frac{318958228}{77124105}a^{2}-\frac{12506632}{8569345}a+\frac{20658349}{25708035}$, $\frac{17934487}{1388233890}a^{15}+\frac{26406028}{694116945}a^{14}+\frac{10883399}{1388233890}a^{13}+\frac{74418229}{1388233890}a^{12}-\frac{181870511}{277646778}a^{11}-\frac{192227813}{77124105}a^{10}-\frac{253915804}{138823389}a^{9}+\frac{13098186481}{1388233890}a^{8}+\frac{1978541083}{77124105}a^{7}-\frac{2394146102}{231372315}a^{6}-\frac{43444052362}{694116945}a^{5}-\frac{128907029}{694116945}a^{4}+\frac{7779676103}{138823389}a^{3}+\frac{1882931252}{231372315}a^{2}-\frac{1177571416}{77124105}a-\frac{3651854356}{694116945}$, $\frac{8100358}{694116945}a^{15}-\frac{8695436}{138823389}a^{14}-\frac{10386799}{277646778}a^{13}-\frac{501677611}{1388233890}a^{12}-\frac{581874622}{694116945}a^{11}+\frac{26993837}{51416070}a^{10}+\frac{7650368467}{1388233890}a^{9}+\frac{3692235013}{277646778}a^{8}-\frac{162143038}{15424821}a^{7}-\frac{8274093019}{231372315}a^{6}+\frac{6183554524}{694116945}a^{5}+\frac{18155256746}{694116945}a^{4}+\frac{1578495959}{694116945}a^{3}-\frac{585322136}{231372315}a^{2}-\frac{219064478}{77124105}a-\frac{1461738647}{694116945}$, $\frac{108038}{2904255}a^{15}+\frac{574249}{2904255}a^{14}+\frac{1050346}{2904255}a^{13}+\frac{5812867}{5808510}a^{12}-\frac{12658}{580851}a^{11}-\frac{6985007}{968085}a^{10}-\frac{10683841}{580851}a^{9}-\frac{21630526}{2904255}a^{8}+\frac{19013704}{322695}a^{7}+\frac{56778679}{968085}a^{6}-\frac{204075376}{2904255}a^{5}-\frac{207877022}{2904255}a^{4}+\frac{16668050}{580851}a^{3}+\frac{27700486}{968085}a^{2}-\frac{48824}{968085}a-\frac{3791503}{2904255}$, $\frac{27226861}{694116945}a^{15}-\frac{69620363}{1388233890}a^{14}+\frac{51210944}{694116945}a^{13}-\frac{85400453}{277646778}a^{12}-\frac{1129917526}{694116945}a^{11}-\frac{360044326}{231372315}a^{10}+\frac{6218276341}{1388233890}a^{9}+\frac{33031015637}{1388233890}a^{8}-\frac{160064554}{77124105}a^{7}-\frac{13261135996}{231372315}a^{6}-\frac{5208866597}{694116945}a^{5}+\frac{5929132030}{138823389}a^{4}+\frac{10993121777}{694116945}a^{3}-\frac{1681884109}{231372315}a^{2}-\frac{1170798088}{231372315}a-\frac{607554818}{694116945}$ 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:  \( 85695.3684811 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 85695.3684811 \cdot 1}{2\cdot\sqrt{2817928042905600000000}}\cr\approx \mathstrut & 0.322048917807 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^14 - 30*x^12 - 64*x^11 - 16*x^10 + 384*x^9 + 154*x^8 - 768*x^7 - 128*x^6 + 552*x^5 + 100*x^4 - 160*x^3 - 48*x^2 + 16*x + 4)
 
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^14 - 30*x^12 - 64*x^11 - 16*x^10 + 384*x^9 + 154*x^8 - 768*x^7 - 128*x^6 + 552*x^5 + 100*x^4 - 160*x^3 - 48*x^2 + 16*x + 4, 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^14 - 30*x^12 - 64*x^11 - 16*x^10 + 384*x^9 + 154*x^8 - 768*x^7 - 128*x^6 + 552*x^5 + 100*x^4 - 160*x^3 - 48*x^2 + 16*x + 4);
 
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^14 - 30*x^12 - 64*x^11 - 16*x^10 + 384*x^9 + 154*x^8 - 768*x^7 - 128*x^6 + 552*x^5 + 100*x^4 - 160*x^3 - 48*x^2 + 16*x + 4);
 
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

$D_4:C_2^2$ (as 16T23):

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 17 conjugacy class representatives for $Q_8 : C_2^2$
Character table for $Q_8 : C_2^2$

Intermediate fields

\(\Q(\sqrt{15}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{15})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{5}, \sqrt{6})\), 8.8.3317760000.1, 8.4.53084160000.1 x2, 8.4.3317760000.3 x2

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: 8.0.2123366400.9, 8.4.3317760000.3, 8.0.2123366400.8, 8.4.53084160000.1, 8.0.5898240000.2, 8.0.368640000.2
Degree 16 siblings: 16.0.4508684868648960000.1, 16.0.34789235097600000000.1, 16.0.11007531417600000000.5, 16.0.2817928042905600000000.7, 16.0.2817928042905600000000.15, 16.0.2817928042905600000000.9, 16.0.2817928042905600000000.10, 16.0.2817928042905600000000.13
Minimal sibling: 8.0.368640000.2

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 ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 Deg $16$$8$$2$$40$
\(3\) 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}$