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Properties

Label 16.0.400...969.16
Degree $16$
Signature $[0, 8]$
Discriminant $4.009\times 10^{30}$
Root discriminant \(81.79\)
Ramified primes $17,47$
Class number $656$ (GRH)
Class group [2, 2, 2, 82] (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 20*x^14 - 70*x^13 + 359*x^12 - 902*x^11 + 2062*x^10 + 7163*x^9 - 33118*x^8 + 13874*x^7 + 167137*x^6 - 248076*x^5 - 123004*x^4 + 2291467*x^3 - 3802366*x^2 - 9958815*x + 21067657)
 
gp: K = bnfinit(y^16 - 2*y^15 + 20*y^14 - 70*y^13 + 359*y^12 - 902*y^11 + 2062*y^10 + 7163*y^9 - 33118*y^8 + 13874*y^7 + 167137*y^6 - 248076*y^5 - 123004*y^4 + 2291467*y^3 - 3802366*y^2 - 9958815*y + 21067657, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 20*x^14 - 70*x^13 + 359*x^12 - 902*x^11 + 2062*x^10 + 7163*x^9 - 33118*x^8 + 13874*x^7 + 167137*x^6 - 248076*x^5 - 123004*x^4 + 2291467*x^3 - 3802366*x^2 - 9958815*x + 21067657);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 + 20*x^14 - 70*x^13 + 359*x^12 - 902*x^11 + 2062*x^10 + 7163*x^9 - 33118*x^8 + 13874*x^7 + 167137*x^6 - 248076*x^5 - 123004*x^4 + 2291467*x^3 - 3802366*x^2 - 9958815*x + 21067657)
 

\( x^{16} - 2 x^{15} + 20 x^{14} - 70 x^{13} + 359 x^{12} - 902 x^{11} + 2062 x^{10} + 7163 x^{9} + \cdots + 21067657 \) 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:   \(4009292695690170390860412175969\) \(\medspace = 17^{14}\cdot 47^{8}\) 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:  \(81.79\)
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:  $17^{7/8}47^{1/2}\approx 81.7884043055344$
Ramified primes:   \(17\), \(47\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{26}a^{13}+\frac{5}{26}a^{12}-\frac{4}{13}a^{11}-\frac{7}{26}a^{10}+\frac{7}{26}a^{9}-\frac{5}{13}a^{8}-\frac{5}{26}a^{7}-\frac{7}{26}a^{6}-\frac{6}{13}a^{5}-\frac{3}{26}a^{4}+\frac{7}{26}a^{3}+\frac{6}{13}a^{2}+\frac{7}{26}a-\frac{1}{2}$, $\frac{1}{15938}a^{14}-\frac{209}{15938}a^{13}+\frac{235}{15938}a^{12}+\frac{3421}{15938}a^{11}+\frac{3351}{15938}a^{10}+\frac{321}{1226}a^{9}-\frac{6237}{15938}a^{8}+\frac{3299}{15938}a^{7}+\frac{4931}{15938}a^{6}+\frac{7869}{15938}a^{5}-\frac{1015}{15938}a^{4}+\frac{7705}{15938}a^{3}+\frac{471}{1226}a^{2}+\frac{3013}{15938}a-\frac{601}{1226}$, $\frac{1}{33\!\cdots\!22}a^{15}+\frac{39\!\cdots\!52}{16\!\cdots\!11}a^{14}+\frac{57\!\cdots\!80}{16\!\cdots\!11}a^{13}+\frac{63\!\cdots\!01}{33\!\cdots\!22}a^{12}-\frac{29\!\cdots\!35}{16\!\cdots\!11}a^{11}+\frac{41\!\cdots\!91}{16\!\cdots\!11}a^{10}+\frac{63\!\cdots\!49}{33\!\cdots\!22}a^{9}-\frac{32\!\cdots\!50}{16\!\cdots\!11}a^{8}-\frac{71\!\cdots\!11}{16\!\cdots\!11}a^{7}-\frac{37\!\cdots\!39}{33\!\cdots\!22}a^{6}-\frac{38\!\cdots\!93}{16\!\cdots\!11}a^{5}+\frac{46\!\cdots\!49}{16\!\cdots\!11}a^{4}+\frac{93\!\cdots\!17}{33\!\cdots\!22}a^{3}+\frac{56\!\cdots\!89}{16\!\cdots\!11}a^{2}+\frac{72\!\cdots\!24}{16\!\cdots\!11}a-\frac{27\!\cdots\!23}{12\!\cdots\!47}$ 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

$C_{2}\times C_{2}\times C_{2}\times C_{82}$, which has order $656$ (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:   $\frac{17\!\cdots\!88}{23\!\cdots\!11}a^{15}-\frac{76\!\cdots\!70}{23\!\cdots\!11}a^{14}+\frac{46\!\cdots\!65}{46\!\cdots\!22}a^{13}-\frac{73\!\cdots\!08}{23\!\cdots\!11}a^{12}+\frac{33\!\cdots\!65}{23\!\cdots\!11}a^{11}-\frac{82\!\cdots\!19}{46\!\cdots\!22}a^{10}-\frac{24\!\cdots\!75}{23\!\cdots\!11}a^{9}+\frac{19\!\cdots\!86}{23\!\cdots\!11}a^{8}-\frac{82\!\cdots\!09}{46\!\cdots\!22}a^{7}-\frac{12\!\cdots\!38}{23\!\cdots\!11}a^{6}+\frac{37\!\cdots\!82}{23\!\cdots\!11}a^{5}+\frac{57\!\cdots\!99}{46\!\cdots\!22}a^{4}-\frac{14\!\cdots\!45}{23\!\cdots\!11}a^{3}+\frac{75\!\cdots\!28}{23\!\cdots\!11}a^{2}+\frac{59\!\cdots\!09}{46\!\cdots\!22}a-\frac{75\!\cdots\!82}{17\!\cdots\!47}$, $\frac{80\!\cdots\!70}{23\!\cdots\!11}a^{15}+\frac{25\!\cdots\!08}{23\!\cdots\!11}a^{14}+\frac{17\!\cdots\!85}{23\!\cdots\!11}a^{13}+\frac{35\!\cdots\!41}{23\!\cdots\!11}a^{12}+\frac{84\!\cdots\!52}{23\!\cdots\!11}a^{11}+\frac{46\!\cdots\!35}{23\!\cdots\!11}a^{10}-\frac{18\!\cdots\!85}{23\!\cdots\!11}a^{9}+\frac{12\!\cdots\!66}{23\!\cdots\!11}a^{8}-\frac{16\!\cdots\!21}{23\!\cdots\!11}a^{7}-\frac{37\!\cdots\!77}{23\!\cdots\!11}a^{6}+\frac{76\!\cdots\!00}{23\!\cdots\!11}a^{5}-\frac{10\!\cdots\!30}{23\!\cdots\!11}a^{4}+\frac{82\!\cdots\!27}{23\!\cdots\!11}a^{3}+\frac{56\!\cdots\!18}{23\!\cdots\!11}a^{2}-\frac{46\!\cdots\!74}{23\!\cdots\!11}a+\frac{44\!\cdots\!39}{17\!\cdots\!47}$, $\frac{20\!\cdots\!53}{33\!\cdots\!22}a^{15}-\frac{53\!\cdots\!93}{33\!\cdots\!22}a^{14}+\frac{57\!\cdots\!17}{33\!\cdots\!22}a^{13}-\frac{18\!\cdots\!93}{33\!\cdots\!22}a^{12}+\frac{11\!\cdots\!71}{33\!\cdots\!22}a^{11}-\frac{32\!\cdots\!85}{33\!\cdots\!22}a^{10}+\frac{11\!\cdots\!19}{33\!\cdots\!22}a^{9}-\frac{27\!\cdots\!67}{33\!\cdots\!22}a^{8}-\frac{30\!\cdots\!11}{33\!\cdots\!22}a^{7}+\frac{10\!\cdots\!13}{33\!\cdots\!22}a^{6}+\frac{15\!\cdots\!63}{33\!\cdots\!22}a^{5}+\frac{27\!\cdots\!27}{33\!\cdots\!22}a^{4}-\frac{19\!\cdots\!69}{33\!\cdots\!22}a^{3}+\frac{44\!\cdots\!01}{33\!\cdots\!22}a^{2}+\frac{15\!\cdots\!17}{33\!\cdots\!22}a-\frac{15\!\cdots\!40}{12\!\cdots\!47}$, $\frac{55\!\cdots\!10}{23\!\cdots\!11}a^{15}+\frac{44\!\cdots\!82}{23\!\cdots\!11}a^{14}+\frac{17\!\cdots\!87}{46\!\cdots\!22}a^{13}-\frac{65\!\cdots\!78}{23\!\cdots\!11}a^{12}+\frac{87\!\cdots\!27}{23\!\cdots\!11}a^{11}+\frac{43\!\cdots\!71}{46\!\cdots\!22}a^{10}-\frac{47\!\cdots\!59}{23\!\cdots\!11}a^{9}+\frac{70\!\cdots\!65}{23\!\cdots\!11}a^{8}-\frac{24\!\cdots\!07}{46\!\cdots\!22}a^{7}-\frac{35\!\cdots\!24}{23\!\cdots\!11}a^{6}+\frac{93\!\cdots\!80}{23\!\cdots\!11}a^{5}-\frac{36\!\cdots\!07}{46\!\cdots\!22}a^{4}-\frac{11\!\cdots\!85}{23\!\cdots\!11}a^{3}+\frac{13\!\cdots\!31}{23\!\cdots\!11}a^{2}-\frac{96\!\cdots\!85}{46\!\cdots\!22}a-\frac{20\!\cdots\!41}{17\!\cdots\!47}$, $\frac{50\!\cdots\!57}{33\!\cdots\!22}a^{15}-\frac{46\!\cdots\!95}{16\!\cdots\!11}a^{14}+\frac{59\!\cdots\!99}{16\!\cdots\!11}a^{13}-\frac{35\!\cdots\!11}{33\!\cdots\!22}a^{12}+\frac{10\!\cdots\!12}{16\!\cdots\!11}a^{11}-\frac{27\!\cdots\!97}{16\!\cdots\!11}a^{10}+\frac{13\!\cdots\!83}{33\!\cdots\!22}a^{9}+\frac{84\!\cdots\!91}{16\!\cdots\!11}a^{8}-\frac{77\!\cdots\!18}{16\!\cdots\!11}a^{7}+\frac{13\!\cdots\!27}{33\!\cdots\!22}a^{6}+\frac{18\!\cdots\!91}{16\!\cdots\!11}a^{5}-\frac{22\!\cdots\!19}{16\!\cdots\!11}a^{4}-\frac{69\!\cdots\!87}{33\!\cdots\!22}a^{3}+\frac{37\!\cdots\!69}{16\!\cdots\!11}a^{2}-\frac{13\!\cdots\!22}{16\!\cdots\!11}a-\frac{16\!\cdots\!82}{12\!\cdots\!47}$, $\frac{24\!\cdots\!67}{33\!\cdots\!22}a^{15}-\frac{97\!\cdots\!11}{33\!\cdots\!22}a^{14}+\frac{65\!\cdots\!09}{33\!\cdots\!22}a^{13}-\frac{29\!\cdots\!97}{33\!\cdots\!22}a^{12}+\frac{13\!\cdots\!37}{33\!\cdots\!22}a^{11}-\frac{48\!\cdots\!07}{33\!\cdots\!22}a^{10}+\frac{13\!\cdots\!77}{33\!\cdots\!22}a^{9}-\frac{53\!\cdots\!87}{33\!\cdots\!22}a^{8}-\frac{85\!\cdots\!71}{33\!\cdots\!22}a^{7}+\frac{21\!\cdots\!73}{33\!\cdots\!22}a^{6}-\frac{71\!\cdots\!61}{33\!\cdots\!22}a^{5}-\frac{32\!\cdots\!57}{33\!\cdots\!22}a^{4}+\frac{23\!\cdots\!09}{33\!\cdots\!22}a^{3}+\frac{57\!\cdots\!49}{33\!\cdots\!22}a^{2}-\frac{22\!\cdots\!63}{33\!\cdots\!22}a+\frac{87\!\cdots\!75}{12\!\cdots\!47}$, $\frac{67\!\cdots\!90}{16\!\cdots\!11}a^{15}+\frac{59\!\cdots\!24}{16\!\cdots\!11}a^{14}-\frac{64\!\cdots\!77}{33\!\cdots\!22}a^{13}+\frac{23\!\cdots\!99}{16\!\cdots\!11}a^{12}-\frac{11\!\cdots\!41}{16\!\cdots\!11}a^{11}+\frac{11\!\cdots\!31}{33\!\cdots\!22}a^{10}-\frac{22\!\cdots\!48}{16\!\cdots\!11}a^{9}+\frac{77\!\cdots\!02}{16\!\cdots\!11}a^{8}-\frac{23\!\cdots\!75}{33\!\cdots\!22}a^{7}-\frac{29\!\cdots\!67}{16\!\cdots\!11}a^{6}+\frac{19\!\cdots\!86}{16\!\cdots\!11}a^{5}-\frac{82\!\cdots\!09}{33\!\cdots\!22}a^{4}+\frac{22\!\cdots\!36}{16\!\cdots\!11}a^{3}+\frac{13\!\cdots\!12}{16\!\cdots\!11}a^{2}-\frac{86\!\cdots\!55}{33\!\cdots\!22}a+\frac{30\!\cdots\!92}{12\!\cdots\!47}$ 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:  \( 842170.663492 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 842170.663492 \cdot 656}{2\cdot\sqrt{4009292695690170390860412175969}}\cr\approx \mathstrut & 0.335103541886 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 20*x^14 - 70*x^13 + 359*x^12 - 902*x^11 + 2062*x^10 + 7163*x^9 - 33118*x^8 + 13874*x^7 + 167137*x^6 - 248076*x^5 - 123004*x^4 + 2291467*x^3 - 3802366*x^2 - 9958815*x + 21067657)
 
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 - 2*x^15 + 20*x^14 - 70*x^13 + 359*x^12 - 902*x^11 + 2062*x^10 + 7163*x^9 - 33118*x^8 + 13874*x^7 + 167137*x^6 - 248076*x^5 - 123004*x^4 + 2291467*x^3 - 3802366*x^2 - 9958815*x + 21067657, 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 - 2*x^15 + 20*x^14 - 70*x^13 + 359*x^12 - 902*x^11 + 2062*x^10 + 7163*x^9 - 33118*x^8 + 13874*x^7 + 167137*x^6 - 248076*x^5 - 123004*x^4 + 2291467*x^3 - 3802366*x^2 - 9958815*x + 21067657);
 
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 - 2*x^15 + 20*x^14 - 70*x^13 + 359*x^12 - 902*x^11 + 2062*x^10 + 7163*x^9 - 33118*x^8 + 13874*x^7 + 167137*x^6 - 248076*x^5 - 123004*x^4 + 2291467*x^3 - 3802366*x^2 - 9958815*x + 21067657);
 
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

$C_2^2:C_8$ (as 16T24):

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 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.2.230911.1, 4.2.13583.1, 8.4.906438128657.1, 8.0.2002321826203313.5, 8.4.53319889921.1

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 16 sibling: 16.8.821630081083204084623649.2
Minimal sibling: 16.8.821630081083204084623649.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{4}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{8}$ R ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ R ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(17\) Copy content Toggle raw display 17.8.7.3$x^{8} + 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} + 17$$8$$1$$7$$C_8$$[\ ]_{8}$
\(47\) Copy content Toggle raw display 47.2.1.1$x^{2} + 235$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.1$x^{2} + 235$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.1$x^{2} + 235$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.1$x^{2} + 235$$2$$1$$1$$C_2$$[\ ]_{2}$
47.4.2.1$x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$