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Properties

Label 16.0.876...713.5
Degree $16$
Signature $[0, 8]$
Discriminant $8.768\times 10^{36}$
Root discriminant \(203.67\)
Ramified primes $61,97$
Class number $253504$ (GRH)
Class group [2, 2, 4, 15844] (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 177*x^14 - 652*x^13 + 12492*x^12 - 48379*x^11 + 504319*x^10 - 2195433*x^9 + 13672084*x^8 - 78136205*x^7 + 244603409*x^6 - 998379714*x^5 + 4521909355*x^4 - 12093973889*x^3 + 20305914608*x^2 - 24281793064*x + 15386186173)
 
gp: K = bnfinit(y^16 - 3*y^15 + 177*y^14 - 652*y^13 + 12492*y^12 - 48379*y^11 + 504319*y^10 - 2195433*y^9 + 13672084*y^8 - 78136205*y^7 + 244603409*y^6 - 998379714*y^5 + 4521909355*y^4 - 12093973889*y^3 + 20305914608*y^2 - 24281793064*y + 15386186173, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 177*x^14 - 652*x^13 + 12492*x^12 - 48379*x^11 + 504319*x^10 - 2195433*x^9 + 13672084*x^8 - 78136205*x^7 + 244603409*x^6 - 998379714*x^5 + 4521909355*x^4 - 12093973889*x^3 + 20305914608*x^2 - 24281793064*x + 15386186173);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 + 177*x^14 - 652*x^13 + 12492*x^12 - 48379*x^11 + 504319*x^10 - 2195433*x^9 + 13672084*x^8 - 78136205*x^7 + 244603409*x^6 - 998379714*x^5 + 4521909355*x^4 - 12093973889*x^3 + 20305914608*x^2 - 24281793064*x + 15386186173)
 

\( x^{16} - 3 x^{15} + 177 x^{14} - 652 x^{13} + 12492 x^{12} - 48379 x^{11} + 504319 x^{10} + \cdots + 15386186173 \) 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:   \(8767895277848913089642817986417897713\) \(\medspace = 61^{4}\cdot 97^{15}\) 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:  \(203.67\)
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:  $61^{1/2}97^{15/16}\approx 569.1980694008912$
Ramified primes:   \(61\), \(97\) 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:  \(\Q(\sqrt{97}) \)
$\card{ \Aut(K/\Q) }$:  $8$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{206}a^{13}+\frac{23}{206}a^{12}+\frac{42}{103}a^{11}-\frac{95}{206}a^{10}+\frac{39}{206}a^{9}-\frac{6}{103}a^{8}-\frac{17}{103}a^{7}-\frac{14}{103}a^{6}-\frac{18}{103}a^{5}-\frac{35}{206}a^{4}+\frac{95}{206}a^{3}-\frac{33}{206}a^{2}-\frac{39}{103}a+\frac{45}{206}$, $\frac{1}{206}a^{14}-\frac{33}{206}a^{12}+\frac{33}{206}a^{11}-\frac{21}{103}a^{10}-\frac{85}{206}a^{9}+\frac{18}{103}a^{8}-\frac{35}{103}a^{7}-\frac{5}{103}a^{6}-\frac{31}{206}a^{5}+\frac{38}{103}a^{4}+\frac{24}{103}a^{3}+\frac{63}{206}a^{2}-\frac{15}{206}a-\frac{5}{206}$, $\frac{1}{33\!\cdots\!78}a^{15}+\frac{12\!\cdots\!31}{33\!\cdots\!78}a^{14}-\frac{19\!\cdots\!65}{33\!\cdots\!78}a^{13}-\frac{31\!\cdots\!12}{16\!\cdots\!39}a^{12}+\frac{15\!\cdots\!69}{33\!\cdots\!78}a^{11}-\frac{78\!\cdots\!25}{33\!\cdots\!78}a^{10}+\frac{12\!\cdots\!17}{33\!\cdots\!78}a^{9}-\frac{17\!\cdots\!10}{16\!\cdots\!39}a^{8}+\frac{14\!\cdots\!99}{16\!\cdots\!13}a^{7}+\frac{22\!\cdots\!61}{33\!\cdots\!78}a^{6}-\frac{60\!\cdots\!21}{33\!\cdots\!78}a^{5}+\frac{47\!\cdots\!58}{16\!\cdots\!39}a^{4}-\frac{61\!\cdots\!61}{33\!\cdots\!78}a^{3}-\frac{28\!\cdots\!14}{16\!\cdots\!39}a^{2}+\frac{78\!\cdots\!94}{16\!\cdots\!39}a-\frac{15\!\cdots\!83}{33\!\cdots\!78}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{15844}$, which has order $253504$ (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:   $\frac{17\!\cdots\!99}{52\!\cdots\!21}a^{15}-\frac{19\!\cdots\!66}{52\!\cdots\!21}a^{14}+\frac{30\!\cdots\!85}{52\!\cdots\!21}a^{13}-\frac{52\!\cdots\!99}{52\!\cdots\!21}a^{12}+\frac{20\!\cdots\!62}{52\!\cdots\!21}a^{11}-\frac{40\!\cdots\!64}{52\!\cdots\!21}a^{10}+\frac{80\!\cdots\!74}{52\!\cdots\!21}a^{9}-\frac{20\!\cdots\!62}{52\!\cdots\!21}a^{8}+\frac{19\!\cdots\!10}{52\!\cdots\!21}a^{7}-\frac{90\!\cdots\!57}{52\!\cdots\!21}a^{6}+\frac{23\!\cdots\!83}{52\!\cdots\!21}a^{5}-\frac{11\!\cdots\!21}{52\!\cdots\!21}a^{4}+\frac{45\!\cdots\!99}{52\!\cdots\!21}a^{3}-\frac{97\!\cdots\!13}{52\!\cdots\!21}a^{2}+\frac{12\!\cdots\!62}{52\!\cdots\!21}a-\frac{70\!\cdots\!28}{52\!\cdots\!21}$, $\frac{22\!\cdots\!57}{16\!\cdots\!39}a^{15}+\frac{80\!\cdots\!73}{16\!\cdots\!39}a^{14}+\frac{40\!\cdots\!28}{16\!\cdots\!39}a^{13}+\frac{11\!\cdots\!16}{16\!\cdots\!39}a^{12}+\frac{27\!\cdots\!23}{16\!\cdots\!39}a^{11}+\frac{65\!\cdots\!18}{16\!\cdots\!39}a^{10}+\frac{99\!\cdots\!03}{16\!\cdots\!39}a^{9}+\frac{16\!\cdots\!28}{16\!\cdots\!39}a^{8}+\frac{20\!\cdots\!41}{16\!\cdots\!39}a^{7}-\frac{23\!\cdots\!58}{16\!\cdots\!39}a^{6}-\frac{85\!\cdots\!28}{16\!\cdots\!39}a^{5}-\frac{10\!\cdots\!85}{16\!\cdots\!39}a^{4}+\frac{83\!\cdots\!85}{16\!\cdots\!39}a^{3}+\frac{64\!\cdots\!37}{16\!\cdots\!39}a^{2}-\frac{46\!\cdots\!60}{16\!\cdots\!39}a+\frac{15\!\cdots\!53}{16\!\cdots\!39}$, $\frac{27\!\cdots\!08}{16\!\cdots\!39}a^{15}-\frac{68\!\cdots\!33}{16\!\cdots\!39}a^{14}+\frac{48\!\cdots\!40}{16\!\cdots\!39}a^{13}-\frac{15\!\cdots\!61}{16\!\cdots\!39}a^{12}+\frac{34\!\cdots\!36}{16\!\cdots\!39}a^{11}-\frac{12\!\cdots\!99}{16\!\cdots\!39}a^{10}+\frac{13\!\cdots\!68}{16\!\cdots\!39}a^{9}-\frac{59\!\cdots\!30}{16\!\cdots\!39}a^{8}+\frac{38\!\cdots\!66}{16\!\cdots\!39}a^{7}-\frac{21\!\cdots\!14}{16\!\cdots\!39}a^{6}+\frac{68\!\cdots\!68}{16\!\cdots\!39}a^{5}-\frac{29\!\cdots\!24}{16\!\cdots\!39}a^{4}+\frac{13\!\cdots\!98}{16\!\cdots\!39}a^{3}-\frac{36\!\cdots\!19}{16\!\cdots\!39}a^{2}+\frac{44\!\cdots\!96}{16\!\cdots\!39}a-\frac{20\!\cdots\!48}{16\!\cdots\!39}$, $\frac{14\!\cdots\!65}{16\!\cdots\!39}a^{15}-\frac{54\!\cdots\!16}{16\!\cdots\!39}a^{14}+\frac{24\!\cdots\!21}{16\!\cdots\!39}a^{13}-\frac{11\!\cdots\!35}{16\!\cdots\!39}a^{12}+\frac{17\!\cdots\!12}{16\!\cdots\!39}a^{11}-\frac{80\!\cdots\!48}{16\!\cdots\!39}a^{10}+\frac{69\!\cdots\!45}{16\!\cdots\!39}a^{9}-\frac{34\!\cdots\!87}{16\!\cdots\!39}a^{8}+\frac{19\!\cdots\!95}{16\!\cdots\!39}a^{7}-\frac{11\!\cdots\!67}{16\!\cdots\!39}a^{6}+\frac{36\!\cdots\!98}{16\!\cdots\!39}a^{5}-\frac{13\!\cdots\!44}{16\!\cdots\!39}a^{4}+\frac{65\!\cdots\!74}{16\!\cdots\!39}a^{3}-\frac{17\!\cdots\!92}{16\!\cdots\!39}a^{2}+\frac{21\!\cdots\!11}{16\!\cdots\!39}a-\frac{96\!\cdots\!69}{16\!\cdots\!39}$, $\frac{51\!\cdots\!37}{16\!\cdots\!39}a^{15}-\frac{45\!\cdots\!39}{16\!\cdots\!39}a^{14}+\frac{88\!\cdots\!98}{16\!\cdots\!39}a^{13}-\frac{89\!\cdots\!31}{16\!\cdots\!39}a^{12}+\frac{65\!\cdots\!14}{16\!\cdots\!39}a^{11}-\frac{67\!\cdots\!08}{16\!\cdots\!39}a^{10}+\frac{31\!\cdots\!49}{16\!\cdots\!39}a^{9}-\frac{30\!\cdots\!85}{16\!\cdots\!39}a^{8}+\frac{11\!\cdots\!01}{16\!\cdots\!39}a^{7}-\frac{90\!\cdots\!08}{16\!\cdots\!39}a^{6}+\frac{33\!\cdots\!38}{16\!\cdots\!39}a^{5}-\frac{10\!\cdots\!08}{16\!\cdots\!39}a^{4}+\frac{63\!\cdots\!85}{16\!\cdots\!39}a^{3}-\frac{18\!\cdots\!80}{16\!\cdots\!39}a^{2}+\frac{22\!\cdots\!03}{16\!\cdots\!39}a-\frac{47\!\cdots\!63}{16\!\cdots\!39}$, $\frac{52\!\cdots\!31}{16\!\cdots\!39}a^{15}+\frac{17\!\cdots\!88}{16\!\cdots\!39}a^{14}+\frac{92\!\cdots\!08}{16\!\cdots\!39}a^{13}-\frac{33\!\cdots\!34}{16\!\cdots\!39}a^{12}+\frac{63\!\cdots\!60}{16\!\cdots\!39}a^{11}-\frac{43\!\cdots\!06}{16\!\cdots\!39}a^{10}+\frac{24\!\cdots\!24}{16\!\cdots\!39}a^{9}-\frac{34\!\cdots\!40}{16\!\cdots\!39}a^{8}+\frac{57\!\cdots\!38}{16\!\cdots\!39}a^{7}-\frac{21\!\cdots\!78}{16\!\cdots\!39}a^{6}+\frac{48\!\cdots\!77}{16\!\cdots\!39}a^{5}-\frac{34\!\cdots\!68}{16\!\cdots\!39}a^{4}+\frac{11\!\cdots\!04}{16\!\cdots\!39}a^{3}-\frac{20\!\cdots\!05}{16\!\cdots\!39}a^{2}+\frac{28\!\cdots\!38}{16\!\cdots\!39}a-\frac{22\!\cdots\!32}{16\!\cdots\!39}$, $\frac{52\!\cdots\!62}{16\!\cdots\!39}a^{15}-\frac{45\!\cdots\!28}{16\!\cdots\!39}a^{14}+\frac{92\!\cdots\!37}{16\!\cdots\!39}a^{13}-\frac{14\!\cdots\!13}{16\!\cdots\!39}a^{12}+\frac{63\!\cdots\!71}{16\!\cdots\!39}a^{11}-\frac{11\!\cdots\!87}{16\!\cdots\!39}a^{10}+\frac{24\!\cdots\!87}{16\!\cdots\!39}a^{9}-\frac{61\!\cdots\!87}{16\!\cdots\!39}a^{8}+\frac{60\!\cdots\!13}{16\!\cdots\!39}a^{7}-\frac{27\!\cdots\!33}{16\!\cdots\!39}a^{6}+\frac{73\!\cdots\!37}{16\!\cdots\!39}a^{5}-\frac{38\!\cdots\!36}{16\!\cdots\!39}a^{4}+\frac{15\!\cdots\!42}{16\!\cdots\!39}a^{3}-\frac{33\!\cdots\!60}{16\!\cdots\!39}a^{2}+\frac{42\!\cdots\!34}{16\!\cdots\!39}a-\frac{25\!\cdots\!33}{16\!\cdots\!39}$ 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:  \( 1675810.87182 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1675810.87182 \cdot 253504}{2\cdot\sqrt{8767895277848913089642817986417897713}}\cr\approx \mathstrut & 0.174249320879 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 177*x^14 - 652*x^13 + 12492*x^12 - 48379*x^11 + 504319*x^10 - 2195433*x^9 + 13672084*x^8 - 78136205*x^7 + 244603409*x^6 - 998379714*x^5 + 4521909355*x^4 - 12093973889*x^3 + 20305914608*x^2 - 24281793064*x + 15386186173)
 
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 - 3*x^15 + 177*x^14 - 652*x^13 + 12492*x^12 - 48379*x^11 + 504319*x^10 - 2195433*x^9 + 13672084*x^8 - 78136205*x^7 + 244603409*x^6 - 998379714*x^5 + 4521909355*x^4 - 12093973889*x^3 + 20305914608*x^2 - 24281793064*x + 15386186173, 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 - 3*x^15 + 177*x^14 - 652*x^13 + 12492*x^12 - 48379*x^11 + 504319*x^10 - 2195433*x^9 + 13672084*x^8 - 78136205*x^7 + 244603409*x^6 - 998379714*x^5 + 4521909355*x^4 - 12093973889*x^3 + 20305914608*x^2 - 24281793064*x + 15386186173);
 
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 - 3*x^15 + 177*x^14 - 652*x^13 + 12492*x^12 - 48379*x^11 + 504319*x^10 - 2195433*x^9 + 13672084*x^8 - 78136205*x^7 + 244603409*x^6 - 998379714*x^5 + 4521909355*x^4 - 12093973889*x^3 + 20305914608*x^2 - 24281793064*x + 15386186173);
 
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

$\OD_{32}$ (as 16T22):

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 $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ $16$

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
\(61\) Copy content Toggle raw display 61.2.0.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
\(97\) Copy content Toggle raw display 97.16.15.1$x^{16} + 97$$16$$1$$15$$C_{16}$$[\ ]_{16}$