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Properties

Label 16.0.155...209.6
Degree $16$
Signature $[0, 8]$
Discriminant $1.554\times 10^{41}$
Root discriminant \(375.38\)
Ramified primes $47,97$
Class number $46080$ (GRH)
Class group [4, 16, 720] (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 - 122*x^14 + 7991*x^12 - 330492*x^10 + 11771701*x^8 - 185395116*x^6 + 1342090511*x^4 - 1443329186*x^2 + 8882874001)
 
gp: K = bnfinit(y^16 - 122*y^14 + 7991*y^12 - 330492*y^10 + 11771701*y^8 - 185395116*y^6 + 1342090511*y^4 - 1443329186*y^2 + 8882874001, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 122*x^14 + 7991*x^12 - 330492*x^10 + 11771701*x^8 - 185395116*x^6 + 1342090511*x^4 - 1443329186*x^2 + 8882874001);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 122*x^14 + 7991*x^12 - 330492*x^10 + 11771701*x^8 - 185395116*x^6 + 1342090511*x^4 - 1443329186*x^2 + 8882874001)
 

\( x^{16} - 122 x^{14} + 7991 x^{12} - 330492 x^{10} + 11771701 x^{8} - 185395116 x^{6} + 1342090511 x^{4} + \cdots + 8882874001 \) 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:   \(155448717458114694507131268341456449334209\) \(\medspace = 47^{8}\cdot 97^{14}\) 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:  \(375.38\)
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:  $47^{1/2}97^{7/8}\approx 375.3826504989875$
Ramified primes:   \(47\), \(97\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{12}a^{10}+\frac{1}{12}a^{8}-\frac{1}{12}a^{6}+\frac{1}{12}a^{4}+\frac{5}{12}a^{2}-\frac{1}{2}a-\frac{1}{3}$, $\frac{1}{12}a^{11}+\frac{1}{12}a^{9}-\frac{1}{12}a^{7}+\frac{1}{12}a^{5}+\frac{5}{12}a^{3}-\frac{1}{2}a^{2}-\frac{1}{3}a$, $\frac{1}{792}a^{12}-\frac{1}{24}a^{11}-\frac{1}{33}a^{10}+\frac{1}{12}a^{9}-\frac{83}{792}a^{8}-\frac{1}{12}a^{7}-\frac{31}{792}a^{6}+\frac{1}{12}a^{5}+\frac{61}{792}a^{4}+\frac{5}{12}a^{3}+\frac{3}{11}a^{2}+\frac{7}{24}a-\frac{203}{792}$, $\frac{1}{243144}a^{13}+\frac{771}{27016}a^{11}+\frac{1237}{243144}a^{9}-\frac{1}{8}a^{8}-\frac{39763}{243144}a^{7}-\frac{1}{8}a^{6}-\frac{57623}{243144}a^{5}-\frac{1}{8}a^{4}-\frac{125}{20262}a^{3}-\frac{1}{8}a^{2}+\frac{14633}{121572}a+\frac{3}{8}$, $\frac{1}{17\!\cdots\!04}a^{14}-\frac{23\!\cdots\!43}{17\!\cdots\!04}a^{12}+\frac{70\!\cdots\!85}{17\!\cdots\!04}a^{10}-\frac{1}{8}a^{9}+\frac{78\!\cdots\!57}{16\!\cdots\!64}a^{8}-\frac{1}{8}a^{7}-\frac{57\!\cdots\!97}{17\!\cdots\!04}a^{6}-\frac{1}{8}a^{5}-\frac{13\!\cdots\!59}{88\!\cdots\!52}a^{4}-\frac{1}{8}a^{3}-\frac{12\!\cdots\!35}{88\!\cdots\!52}a^{2}-\frac{1}{8}a+\frac{26\!\cdots\!61}{94\!\cdots\!48}$, $\frac{1}{54\!\cdots\!28}a^{15}-\frac{23\!\cdots\!43}{54\!\cdots\!28}a^{13}+\frac{12\!\cdots\!71}{54\!\cdots\!28}a^{11}-\frac{1}{24}a^{10}-\frac{53\!\cdots\!43}{49\!\cdots\!48}a^{9}-\frac{1}{24}a^{8}+\frac{90\!\cdots\!23}{54\!\cdots\!28}a^{7}-\frac{5}{24}a^{6}+\frac{19\!\cdots\!01}{27\!\cdots\!64}a^{5}-\frac{1}{24}a^{4}+\frac{56\!\cdots\!25}{27\!\cdots\!64}a^{3}-\frac{5}{24}a^{2}+\frac{93\!\cdots\!59}{72\!\cdots\!09}a-\frac{1}{3}$ 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

$C_{4}\times C_{16}\times C_{720}$, which has order $46080$ (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:   $\frac{10\!\cdots\!19}{29\!\cdots\!16}a^{15}-\frac{30199098143}{52\!\cdots\!76}a^{14}-\frac{11\!\cdots\!43}{24\!\cdots\!18}a^{13}+\frac{3477209417261}{52\!\cdots\!76}a^{12}+\frac{11\!\cdots\!86}{37\!\cdots\!27}a^{11}-\frac{430783812279819}{10\!\cdots\!52}a^{10}-\frac{50\!\cdots\!56}{37\!\cdots\!27}a^{9}+\frac{82\!\cdots\!33}{52\!\cdots\!76}a^{8}+\frac{18\!\cdots\!86}{37\!\cdots\!27}a^{7}-\frac{28\!\cdots\!23}{52\!\cdots\!76}a^{6}-\frac{88\!\cdots\!57}{99\!\cdots\!72}a^{5}+\frac{80\!\cdots\!65}{13\!\cdots\!69}a^{4}+\frac{60\!\cdots\!05}{74\!\cdots\!54}a^{3}-\frac{54\!\cdots\!47}{11\!\cdots\!79}a^{2}-\frac{12\!\cdots\!84}{43\!\cdots\!47}a-\frac{44\!\cdots\!99}{10\!\cdots\!52}$, $\frac{30\!\cdots\!15}{29\!\cdots\!16}a^{15}+\frac{30199098143}{52\!\cdots\!76}a^{14}-\frac{12\!\cdots\!65}{12\!\cdots\!09}a^{13}-\frac{3477209417261}{52\!\cdots\!76}a^{12}+\frac{20\!\cdots\!32}{37\!\cdots\!27}a^{11}+\frac{430783812279819}{10\!\cdots\!52}a^{10}-\frac{26\!\cdots\!49}{14\!\cdots\!08}a^{9}-\frac{82\!\cdots\!33}{52\!\cdots\!76}a^{8}+\frac{82\!\cdots\!11}{14\!\cdots\!08}a^{7}+\frac{28\!\cdots\!23}{52\!\cdots\!76}a^{6}+\frac{31\!\cdots\!61}{99\!\cdots\!72}a^{5}-\frac{80\!\cdots\!65}{13\!\cdots\!69}a^{4}-\frac{23\!\cdots\!65}{14\!\cdots\!08}a^{3}+\frac{54\!\cdots\!47}{11\!\cdots\!79}a^{2}+\frac{84\!\cdots\!75}{17\!\cdots\!88}a-\frac{22\!\cdots\!85}{10\!\cdots\!52}$, $\frac{21\!\cdots\!91}{37\!\cdots\!27}a^{15}+\frac{392588275859}{26\!\cdots\!38}a^{14}-\frac{16\!\cdots\!91}{24\!\cdots\!18}a^{13}-\frac{45203722424393}{26\!\cdots\!38}a^{12}+\frac{15\!\cdots\!58}{37\!\cdots\!27}a^{11}+\frac{56\!\cdots\!47}{52\!\cdots\!76}a^{10}-\frac{22\!\cdots\!85}{14\!\cdots\!08}a^{9}-\frac{10\!\cdots\!29}{26\!\cdots\!38}a^{8}+\frac{78\!\cdots\!71}{14\!\cdots\!08}a^{7}+\frac{37\!\cdots\!99}{26\!\cdots\!38}a^{6}-\frac{25\!\cdots\!93}{49\!\cdots\!36}a^{5}-\frac{20\!\cdots\!90}{13\!\cdots\!69}a^{4}+\frac{63\!\cdots\!25}{14\!\cdots\!08}a^{3}+\frac{14\!\cdots\!22}{11\!\cdots\!79}a^{2}-\frac{62\!\cdots\!77}{17\!\cdots\!88}a-\frac{57\!\cdots\!23}{52\!\cdots\!76}$, $\frac{42\!\cdots\!63}{27\!\cdots\!64}a^{15}+\frac{74\!\cdots\!01}{29\!\cdots\!84}a^{14}-\frac{92\!\cdots\!67}{49\!\cdots\!48}a^{13}-\frac{47\!\cdots\!39}{14\!\cdots\!42}a^{12}+\frac{65\!\cdots\!63}{54\!\cdots\!28}a^{11}+\frac{65\!\cdots\!49}{29\!\cdots\!84}a^{10}-\frac{26\!\cdots\!09}{54\!\cdots\!28}a^{9}-\frac{57\!\cdots\!73}{59\!\cdots\!68}a^{8}+\frac{94\!\cdots\!75}{54\!\cdots\!28}a^{7}+\frac{19\!\cdots\!95}{59\!\cdots\!68}a^{6}-\frac{14\!\cdots\!15}{54\!\cdots\!28}a^{5}-\frac{31\!\cdots\!55}{59\!\cdots\!68}a^{4}+\frac{21\!\cdots\!79}{13\!\cdots\!82}a^{3}+\frac{20\!\cdots\!21}{53\!\cdots\!88}a^{2}+\frac{18\!\cdots\!12}{72\!\cdots\!09}a-\frac{19\!\cdots\!95}{62\!\cdots\!32}$, $\frac{80\!\cdots\!57}{54\!\cdots\!28}a^{15}-\frac{27\!\cdots\!52}{73\!\cdots\!21}a^{14}-\frac{10\!\cdots\!21}{54\!\cdots\!28}a^{13}+\frac{12\!\cdots\!67}{29\!\cdots\!84}a^{12}+\frac{72\!\cdots\!83}{54\!\cdots\!28}a^{11}-\frac{16\!\cdots\!29}{59\!\cdots\!68}a^{10}-\frac{32\!\cdots\!05}{54\!\cdots\!28}a^{9}+\frac{62\!\cdots\!07}{59\!\cdots\!68}a^{8}+\frac{11\!\cdots\!23}{54\!\cdots\!28}a^{7}-\frac{21\!\cdots\!49}{59\!\cdots\!68}a^{6}-\frac{11\!\cdots\!25}{27\!\cdots\!64}a^{5}+\frac{25\!\cdots\!35}{59\!\cdots\!68}a^{4}+\frac{27\!\cdots\!82}{68\!\cdots\!41}a^{3}-\frac{35\!\cdots\!45}{53\!\cdots\!88}a^{2}-\frac{84\!\cdots\!82}{65\!\cdots\!19}a-\frac{31\!\cdots\!29}{15\!\cdots\!58}$, $\frac{30\!\cdots\!91}{54\!\cdots\!28}a^{15}+\frac{17\!\cdots\!79}{17\!\cdots\!04}a^{14}-\frac{39\!\cdots\!07}{54\!\cdots\!28}a^{13}-\frac{19\!\cdots\!55}{17\!\cdots\!04}a^{12}+\frac{26\!\cdots\!81}{54\!\cdots\!28}a^{11}+\frac{61\!\cdots\!05}{88\!\cdots\!52}a^{10}-\frac{58\!\cdots\!51}{27\!\cdots\!64}a^{9}-\frac{23\!\cdots\!33}{88\!\cdots\!52}a^{8}+\frac{21\!\cdots\!19}{27\!\cdots\!64}a^{7}+\frac{81\!\cdots\!13}{88\!\cdots\!52}a^{6}-\frac{79\!\cdots\!57}{54\!\cdots\!28}a^{5}-\frac{18\!\cdots\!69}{17\!\cdots\!04}a^{4}+\frac{75\!\cdots\!33}{54\!\cdots\!28}a^{3}+\frac{13\!\cdots\!77}{17\!\cdots\!04}a^{2}-\frac{27\!\cdots\!31}{57\!\cdots\!72}a+\frac{35\!\cdots\!21}{47\!\cdots\!74}$, $\frac{55\!\cdots\!89}{27\!\cdots\!64}a^{15}+\frac{11\!\cdots\!85}{17\!\cdots\!04}a^{14}-\frac{12\!\cdots\!09}{54\!\cdots\!28}a^{13}-\frac{13\!\cdots\!17}{17\!\cdots\!04}a^{12}+\frac{75\!\cdots\!71}{54\!\cdots\!28}a^{11}+\frac{84\!\cdots\!09}{17\!\cdots\!04}a^{10}-\frac{14\!\cdots\!99}{27\!\cdots\!64}a^{9}-\frac{81\!\cdots\!13}{44\!\cdots\!26}a^{8}+\frac{49\!\cdots\!19}{27\!\cdots\!64}a^{7}+\frac{14\!\cdots\!49}{22\!\cdots\!63}a^{6}-\frac{24\!\cdots\!53}{13\!\cdots\!82}a^{5}-\frac{12\!\cdots\!89}{17\!\cdots\!04}a^{4}+\frac{80\!\cdots\!67}{54\!\cdots\!28}a^{3}+\frac{93\!\cdots\!33}{17\!\cdots\!04}a^{2}-\frac{71\!\cdots\!97}{57\!\cdots\!72}a-\frac{93\!\cdots\!81}{18\!\cdots\!96}$ 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:  \( 1817406770.17 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1817406770.17 \cdot 46080}{2\cdot\sqrt{155448717458114694507131268341456449334209}}\cr\approx \mathstrut & 0.257976408061 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 122*x^14 + 7991*x^12 - 330492*x^10 + 11771701*x^8 - 185395116*x^6 + 1342090511*x^4 - 1443329186*x^2 + 8882874001)
 
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 - 122*x^14 + 7991*x^12 - 330492*x^10 + 11771701*x^8 - 185395116*x^6 + 1342090511*x^4 - 1443329186*x^2 + 8882874001, 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 - 122*x^14 + 7991*x^12 - 330492*x^10 + 11771701*x^8 - 185395116*x^6 + 1342090511*x^4 - 1443329186*x^2 + 8882874001);
 
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 - 122*x^14 + 7991*x^12 - 330492*x^10 + 11771701*x^8 - 185395116*x^6 + 1342090511*x^4 - 1443329186*x^2 + 8882874001);
 
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

\(\Q(\sqrt{97}) \), \(\Q(\sqrt{-47}) \), \(\Q(\sqrt{-4559}) \), 4.0.2016094657.2, 4.4.912673.1, \(\Q(\sqrt{-47}, \sqrt{97})\), 8.0.394269853600442921953.3 x2, 8.4.178483410412151617.2 x2, 8.0.4064637665983947649.3

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: 8.4.178483410412151617.2, 8.0.394269853600442921953.3
Degree 16 siblings: 16.4.70370628093306787916311121929133748001.7, 16.0.155448717458114694507131268341456449334209.11
Minimal sibling: 8.4.178483410412151617.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{4}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ R ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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
\(47\) Copy content Toggle raw display 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}$
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}$
\(97\) Copy content Toggle raw display 97.8.7.1$x^{8} + 97$$8$$1$$7$$C_8$$[\ ]_{8}$
97.8.7.1$x^{8} + 97$$8$$1$$7$$C_8$$[\ ]_{8}$