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Properties

Label 16.0.556...681.25
Degree $16$
Signature $[0, 8]$
Discriminant $5.569\times 10^{36}$
Root discriminant \(197.98\)
Ramified primes $41,59$
Class number $1728$ (GRH)
Class group [12, 12, 12] (GRH)
Galois group $\OD_{16}:C_2$ (as 16T36)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 32*x^14 + 366*x^12 - 35236*x^10 + 388601*x^8 - 4143076*x^6 + 647985536*x^4 - 4660210752*x^2 + 16429086976)
 
gp: K = bnfinit(y^16 - 32*y^14 + 366*y^12 - 35236*y^10 + 388601*y^8 - 4143076*y^6 + 647985536*y^4 - 4660210752*y^2 + 16429086976, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 32*x^14 + 366*x^12 - 35236*x^10 + 388601*x^8 - 4143076*x^6 + 647985536*x^4 - 4660210752*x^2 + 16429086976);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 32*x^14 + 366*x^12 - 35236*x^10 + 388601*x^8 - 4143076*x^6 + 647985536*x^4 - 4660210752*x^2 + 16429086976)
 

\( x^{16} - 32 x^{14} + 366 x^{12} - 35236 x^{10} + 388601 x^{8} - 4143076 x^{6} + 647985536 x^{4} + \cdots + 16429086976 \) 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:   \(5569165027023164184679061585237737681\) \(\medspace = 41^{14}\cdot 59^{8}\) 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:  \(197.98\)
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:  $41^{7/8}59^{1/2}\approx 197.97569691730635$
Ramified primes:   \(41\), \(59\) 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 \(a\))

$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{3}$, $\frac{1}{8}a^{6}-\frac{1}{8}a^{5}-\frac{1}{8}a^{4}+\frac{1}{8}a^{3}$, $\frac{1}{16}a^{7}-\frac{1}{8}a^{5}+\frac{1}{16}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{32}a^{8}-\frac{1}{16}a^{6}-\frac{1}{8}a^{5}+\frac{1}{32}a^{4}-\frac{1}{8}a^{3}+\frac{1}{4}a$, $\frac{1}{32}a^{9}+\frac{1}{32}a^{5}-\frac{1}{16}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{64}a^{10}-\frac{1}{64}a^{9}-\frac{1}{32}a^{7}+\frac{1}{64}a^{6}+\frac{3}{64}a^{5}+\frac{3}{32}a^{4}-\frac{1}{4}a^{3}-\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{64}a^{11}-\frac{1}{64}a^{9}-\frac{1}{64}a^{7}+\frac{1}{64}a^{5}-\frac{1}{8}a^{4}+\frac{1}{8}a^{2}$, $\frac{1}{7552}a^{12}-\frac{37}{7552}a^{10}-\frac{1}{64}a^{9}-\frac{1}{7552}a^{8}-\frac{1}{32}a^{7}-\frac{411}{7552}a^{6}+\frac{3}{64}a^{5}-\frac{5}{472}a^{4}+\frac{9}{472}a^{2}-\frac{1}{2}a+\frac{35}{118}$, $\frac{1}{15104}a^{13}-\frac{1}{15104}a^{12}+\frac{81}{15104}a^{11}-\frac{81}{15104}a^{10}+\frac{235}{15104}a^{9}-\frac{235}{15104}a^{8}+\frac{179}{15104}a^{7}-\frac{179}{15104}a^{6}+\frac{113}{944}a^{5}-\frac{113}{944}a^{4}+\frac{17}{236}a^{3}-\frac{17}{236}a^{2}-\frac{83}{236}a+\frac{83}{236}$, $\frac{1}{74\!\cdots\!16}a^{14}-\frac{75\!\cdots\!67}{18\!\cdots\!04}a^{12}-\frac{22\!\cdots\!25}{37\!\cdots\!08}a^{10}+\frac{25\!\cdots\!21}{18\!\cdots\!04}a^{8}-\frac{41\!\cdots\!35}{74\!\cdots\!16}a^{6}-\frac{1}{8}a^{5}-\frac{25\!\cdots\!05}{46\!\cdots\!76}a^{4}-\frac{1}{8}a^{3}-\frac{40\!\cdots\!31}{11\!\cdots\!44}a^{2}+\frac{1}{4}a-\frac{34\!\cdots\!57}{11\!\cdots\!44}$, $\frac{1}{23\!\cdots\!04}a^{15}-\frac{1}{14\!\cdots\!32}a^{14}-\frac{82\!\cdots\!19}{59\!\cdots\!76}a^{13}-\frac{17\!\cdots\!85}{37\!\cdots\!08}a^{12}+\frac{19\!\cdots\!11}{11\!\cdots\!52}a^{11}-\frac{16\!\cdots\!99}{74\!\cdots\!16}a^{10}+\frac{68\!\cdots\!53}{59\!\cdots\!76}a^{9}-\frac{25\!\cdots\!69}{37\!\cdots\!08}a^{8}+\frac{34\!\cdots\!85}{23\!\cdots\!04}a^{7}-\frac{59\!\cdots\!73}{14\!\cdots\!32}a^{6}-\frac{59\!\cdots\!21}{14\!\cdots\!44}a^{5}-\frac{71\!\cdots\!31}{92\!\cdots\!52}a^{4}-\frac{17\!\cdots\!99}{37\!\cdots\!36}a^{3}-\frac{12\!\cdots\!05}{23\!\cdots\!88}a^{2}+\frac{14\!\cdots\!99}{37\!\cdots\!36}a-\frac{90\!\cdots\!23}{23\!\cdots\!88}$ 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

$C_{12}\times C_{12}\times C_{12}$, which has order $1728$ (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:   $\frac{18\!\cdots\!09}{18\!\cdots\!52}a^{15}+\frac{33775909}{75\!\cdots\!32}a^{14}-\frac{14\!\cdots\!31}{46\!\cdots\!88}a^{13}+\frac{381165529}{18\!\cdots\!08}a^{12}+\frac{27\!\cdots\!55}{93\!\cdots\!76}a^{11}-\frac{4510557413}{37\!\cdots\!16}a^{10}-\frac{14\!\cdots\!19}{46\!\cdots\!88}a^{9}-\frac{611412530543}{18\!\cdots\!08}a^{8}+\frac{78\!\cdots\!97}{18\!\cdots\!52}a^{7}-\frac{59455238930227}{75\!\cdots\!32}a^{6}-\frac{83\!\cdots\!77}{11\!\cdots\!72}a^{5}-\frac{27835240572141}{47\!\cdots\!52}a^{4}+\frac{15\!\cdots\!65}{29\!\cdots\!68}a^{3}+\frac{77008387520597}{11\!\cdots\!88}a^{2}-\frac{18\!\cdots\!37}{29\!\cdots\!68}a+\frac{37\!\cdots\!75}{11\!\cdots\!88}$, $\frac{36\!\cdots\!03}{18\!\cdots\!52}a^{15}+\frac{33775909}{75\!\cdots\!32}a^{14}-\frac{34\!\cdots\!89}{46\!\cdots\!88}a^{13}+\frac{381165529}{18\!\cdots\!08}a^{12}+\frac{86\!\cdots\!93}{93\!\cdots\!76}a^{11}-\frac{4510557413}{37\!\cdots\!16}a^{10}-\frac{33\!\cdots\!65}{46\!\cdots\!88}a^{9}-\frac{611412530543}{18\!\cdots\!08}a^{8}+\frac{22\!\cdots\!19}{18\!\cdots\!52}a^{7}-\frac{59455238930227}{75\!\cdots\!32}a^{6}-\frac{11\!\cdots\!47}{11\!\cdots\!72}a^{5}-\frac{27835240572141}{47\!\cdots\!52}a^{4}+\frac{42\!\cdots\!51}{29\!\cdots\!68}a^{3}+\frac{77008387520597}{11\!\cdots\!88}a^{2}-\frac{51\!\cdots\!23}{29\!\cdots\!68}a+\frac{33\!\cdots\!95}{11\!\cdots\!88}$, $\frac{57\!\cdots\!07}{58\!\cdots\!36}a^{15}+\frac{33775909}{37\!\cdots\!16}a^{14}-\frac{26\!\cdots\!97}{14\!\cdots\!84}a^{13}+\frac{381165529}{94\!\cdots\!04}a^{12}-\frac{10\!\cdots\!79}{29\!\cdots\!68}a^{11}-\frac{4510557413}{18\!\cdots\!08}a^{10}-\frac{32\!\cdots\!81}{14\!\cdots\!84}a^{9}-\frac{611412530543}{94\!\cdots\!04}a^{8}+\frac{18\!\cdots\!71}{58\!\cdots\!36}a^{7}-\frac{59455238930227}{37\!\cdots\!16}a^{6}-\frac{41\!\cdots\!87}{36\!\cdots\!96}a^{5}-\frac{27835240572141}{23\!\cdots\!76}a^{4}+\frac{14\!\cdots\!67}{91\!\cdots\!24}a^{3}+\frac{77008387520597}{58\!\cdots\!44}a^{2}-\frac{11\!\cdots\!59}{91\!\cdots\!24}a+\frac{797169563705903}{58\!\cdots\!44}$, $\frac{26\!\cdots\!27}{11\!\cdots\!52}a^{15}+\frac{84\!\cdots\!27}{74\!\cdots\!16}a^{14}-\frac{39\!\cdots\!49}{29\!\cdots\!88}a^{13}-\frac{14\!\cdots\!89}{18\!\cdots\!04}a^{12}+\frac{21\!\cdots\!49}{59\!\cdots\!76}a^{11}+\frac{58\!\cdots\!49}{37\!\cdots\!08}a^{10}-\frac{21\!\cdots\!05}{29\!\cdots\!88}a^{9}-\frac{70\!\cdots\!49}{18\!\cdots\!04}a^{8}-\frac{13\!\cdots\!85}{11\!\cdots\!52}a^{7}-\frac{44\!\cdots\!01}{74\!\cdots\!16}a^{6}-\frac{30\!\cdots\!39}{74\!\cdots\!72}a^{5}-\frac{98\!\cdots\!39}{46\!\cdots\!76}a^{4}+\frac{61\!\cdots\!31}{18\!\cdots\!68}a^{3}+\frac{20\!\cdots\!47}{11\!\cdots\!44}a^{2}-\frac{24\!\cdots\!63}{18\!\cdots\!68}a-\frac{81\!\cdots\!63}{11\!\cdots\!44}$, $\frac{25\!\cdots\!51}{18\!\cdots\!68}a^{15}+\frac{20\!\cdots\!01}{37\!\cdots\!08}a^{14}-\frac{20\!\cdots\!95}{46\!\cdots\!92}a^{13}-\frac{31\!\cdots\!07}{92\!\cdots\!52}a^{12}-\frac{65\!\cdots\!51}{93\!\cdots\!84}a^{11}+\frac{58\!\cdots\!87}{18\!\cdots\!04}a^{10}-\frac{20\!\cdots\!99}{46\!\cdots\!92}a^{9}+\frac{41\!\cdots\!13}{92\!\cdots\!52}a^{8}+\frac{18\!\cdots\!67}{18\!\cdots\!68}a^{7}+\frac{15\!\cdots\!37}{37\!\cdots\!08}a^{6}-\frac{22\!\cdots\!83}{58\!\cdots\!24}a^{5}-\frac{31\!\cdots\!57}{23\!\cdots\!88}a^{4}-\frac{36\!\cdots\!43}{36\!\cdots\!14}a^{3}+\frac{51\!\cdots\!61}{58\!\cdots\!72}a^{2}+\frac{27\!\cdots\!45}{29\!\cdots\!12}a-\frac{15\!\cdots\!81}{58\!\cdots\!72}$, $\frac{68\!\cdots\!59}{29\!\cdots\!88}a^{15}-\frac{13\!\cdots\!93}{74\!\cdots\!72}a^{13}+\frac{94\!\cdots\!57}{14\!\cdots\!44}a^{11}-\frac{14\!\cdots\!45}{74\!\cdots\!72}a^{9}+\frac{13\!\cdots\!55}{29\!\cdots\!88}a^{7}-\frac{87\!\cdots\!07}{18\!\cdots\!68}a^{5}+\frac{10\!\cdots\!55}{46\!\cdots\!92}a^{3}-\frac{23\!\cdots\!03}{46\!\cdots\!92}a$, $\frac{59\!\cdots\!09}{11\!\cdots\!72}a^{15}+\frac{58\!\cdots\!37}{37\!\cdots\!08}a^{14}-\frac{33\!\cdots\!55}{29\!\cdots\!68}a^{13}-\frac{35\!\cdots\!27}{92\!\cdots\!52}a^{12}+\frac{27\!\cdots\!11}{58\!\cdots\!36}a^{11}+\frac{36\!\cdots\!63}{18\!\cdots\!04}a^{10}-\frac{47\!\cdots\!07}{29\!\cdots\!68}a^{9}-\frac{49\!\cdots\!63}{92\!\cdots\!52}a^{8}+\frac{23\!\cdots\!93}{11\!\cdots\!72}a^{7}+\frac{95\!\cdots\!09}{37\!\cdots\!08}a^{6}-\frac{27\!\cdots\!81}{73\!\cdots\!92}a^{5}-\frac{11\!\cdots\!81}{23\!\cdots\!88}a^{4}+\frac{55\!\cdots\!73}{18\!\cdots\!48}a^{3}+\frac{62\!\cdots\!97}{58\!\cdots\!72}a^{2}+\frac{10\!\cdots\!55}{18\!\cdots\!48}a-\frac{74\!\cdots\!41}{58\!\cdots\!72}$ 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:  \( 1833256639.05 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1833256639.05 \cdot 1728}{2\cdot\sqrt{5569165027023164184679061585237737681}}\cr\approx \mathstrut & 1.63035024166 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 32*x^14 + 366*x^12 - 35236*x^10 + 388601*x^8 - 4143076*x^6 + 647985536*x^4 - 4660210752*x^2 + 16429086976)
 
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 - 32*x^14 + 366*x^12 - 35236*x^10 + 388601*x^8 - 4143076*x^6 + 647985536*x^4 - 4660210752*x^2 + 16429086976, 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 - 32*x^14 + 366*x^12 - 35236*x^10 + 388601*x^8 - 4143076*x^6 + 647985536*x^4 - 4660210752*x^2 + 16429086976);
 
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 - 32*x^14 + 366*x^12 - 35236*x^10 + 388601*x^8 - 4143076*x^6 + 647985536*x^4 - 4660210752*x^2 + 16429086976);
 
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

$\OD_{16}:C_2$ (as 16T36):

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 11 conjugacy class representatives for $\OD_{16}:C_2$
Character table for $\OD_{16}:C_2$

Intermediate fields

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{-59}) \), \(\Q(\sqrt{-2419}) \), 4.0.239914001.2, 4.4.68921.1, \(\Q(\sqrt{41}, \sqrt{-59})\), 8.0.2359907842908948041.2 x2, 8.4.677939627379761.3 x2, 8.0.57558727875828001.4

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: 8.0.2359907842908948041.2, 8.4.677939627379761.3
Degree 16 siblings: 16.0.5569165027023164184679061585237737681.24, 16.4.1599875043672267792208865723998201.20
Minimal sibling: 8.4.677939627379761.3

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.2.0.1}{2} }^{8}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ R ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ R

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
\(41\) Copy content Toggle raw display 41.8.7.3$x^{8} + 41$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} + 41$$8$$1$$7$$C_8$$[\ ]_{8}$
\(59\) Copy content Toggle raw display 59.2.1.2$x^{2} + 59$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 59$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 59$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 59$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 59$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 59$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 59$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 59$$2$$1$$1$$C_2$$[\ ]_{2}$