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Properties

Label 16.0.102...625.6
Degree $16$
Signature $[0, 8]$
Discriminant $1.027\times 10^{23}$
Root discriminant \(27.43\)
Ramified primes $5,29$
Class number $4$ (GRH)
Class group [4] (GRH)
Galois group $C_2^3:C_4$ (as 16T33)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 23*x^14 - 50*x^13 + 115*x^12 - 263*x^11 + 384*x^10 - 245*x^9 + 289*x^8 - 890*x^7 + 1101*x^6 - 1581*x^5 + 3875*x^4 - 7140*x^3 + 8532*x^2 - 5184*x + 1296)
 
gp: K = bnfinit(y^16 - 7*y^15 + 23*y^14 - 50*y^13 + 115*y^12 - 263*y^11 + 384*y^10 - 245*y^9 + 289*y^8 - 890*y^7 + 1101*y^6 - 1581*y^5 + 3875*y^4 - 7140*y^3 + 8532*y^2 - 5184*y + 1296, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 7*x^15 + 23*x^14 - 50*x^13 + 115*x^12 - 263*x^11 + 384*x^10 - 245*x^9 + 289*x^8 - 890*x^7 + 1101*x^6 - 1581*x^5 + 3875*x^4 - 7140*x^3 + 8532*x^2 - 5184*x + 1296);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 7*x^15 + 23*x^14 - 50*x^13 + 115*x^12 - 263*x^11 + 384*x^10 - 245*x^9 + 289*x^8 - 890*x^7 + 1101*x^6 - 1581*x^5 + 3875*x^4 - 7140*x^3 + 8532*x^2 - 5184*x + 1296)
 

\( x^{16} - 7 x^{15} + 23 x^{14} - 50 x^{13} + 115 x^{12} - 263 x^{11} + 384 x^{10} - 245 x^{9} + \cdots + 1296 \) 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:   \(102711726879931884765625\) \(\medspace = 5^{12}\cdot 29^{10}\) 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:  \(27.43\)
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:  $5^{3/4}29^{3/4}\approx 41.78553833475025$
Ramified primes:   \(5\), \(29\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{8}a^{3}+\frac{1}{8}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{8}a^{4}+\frac{1}{4}a^{3}-\frac{1}{8}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{8}a^{5}-\frac{1}{2}a^{3}+\frac{1}{8}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{8}+\frac{1}{8}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{8}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{48}a^{13}-\frac{1}{48}a^{12}+\frac{1}{24}a^{11}+\frac{1}{48}a^{10}+\frac{1}{48}a^{9}+\frac{1}{12}a^{8}-\frac{3}{16}a^{7}+\frac{1}{48}a^{6}-\frac{1}{24}a^{5}+\frac{7}{48}a^{4}-\frac{3}{16}a^{3}+\frac{1}{4}a^{2}-\frac{1}{3}a-\frac{1}{2}$, $\frac{1}{8352}a^{14}-\frac{61}{8352}a^{13}-\frac{11}{4176}a^{12}-\frac{329}{8352}a^{11}-\frac{11}{8352}a^{10}-\frac{235}{4176}a^{9}-\frac{223}{2784}a^{8}-\frac{983}{8352}a^{7}+\frac{401}{4176}a^{6}-\frac{2015}{8352}a^{5}-\frac{83}{2784}a^{4}-\frac{541}{1392}a^{3}-\frac{919}{2088}a^{2}-\frac{13}{174}a+\frac{12}{29}$, $\frac{1}{36\!\cdots\!56}a^{15}+\frac{166800029227}{18\!\cdots\!28}a^{14}-\frac{250579536593839}{36\!\cdots\!56}a^{13}-\frac{49042979001697}{12\!\cdots\!64}a^{12}+\frac{260369428657667}{18\!\cdots\!28}a^{11}+\frac{299383558289731}{36\!\cdots\!56}a^{10}+\frac{6817567731319}{237931881730752}a^{9}+\frac{22\!\cdots\!97}{18\!\cdots\!28}a^{8}+\frac{64\!\cdots\!91}{36\!\cdots\!56}a^{7}-\frac{831574453427015}{36\!\cdots\!56}a^{6}-\frac{13\!\cdots\!07}{60\!\cdots\!76}a^{5}-\frac{21\!\cdots\!77}{12\!\cdots\!52}a^{4}-\frac{34\!\cdots\!93}{91\!\cdots\!64}a^{3}-\frac{12\!\cdots\!59}{30\!\cdots\!88}a^{2}-\frac{5559138057529}{252802624338924}a+\frac{5795756044763}{84267541446308}$ 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

$C_{4}$, which has order $4$ (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:   $\frac{788927587}{3003099975648}a^{15}-\frac{3683153845}{3003099975648}a^{14}+\frac{2964142303}{1501549987824}a^{13}-\frac{4064246885}{3003099975648}a^{12}+\frac{25746706033}{3003099975648}a^{11}-\frac{27201264175}{1501549987824}a^{10}-\frac{446604457}{19628104416}a^{9}+\frac{226532985277}{3003099975648}a^{8}+\frac{139182455615}{1501549987824}a^{7}-\frac{413855948339}{3003099975648}a^{6}-\frac{287753289583}{1001033325216}a^{5}-\frac{61422406285}{500516662608}a^{4}+\frac{1068824739595}{1501549987824}a^{3}+\frac{90825082477}{250258331304}a^{2}-\frac{12520467289}{13903240628}a+\frac{3386213077}{3475810157}$, $\frac{5145212429281}{16\!\cdots\!48}a^{15}-\frac{32791152448477}{16\!\cdots\!48}a^{14}+\frac{23853668502467}{413677021645512}a^{13}-\frac{182075063006927}{16\!\cdots\!48}a^{12}+\frac{435031088363221}{16\!\cdots\!48}a^{11}-\frac{63028610522291}{103419255411378}a^{10}+\frac{22968208123811}{32445256599648}a^{9}-\frac{96665931022151}{16\!\cdots\!48}a^{8}+\frac{63711163714069}{103419255411378}a^{7}-\frac{42\!\cdots\!21}{16\!\cdots\!48}a^{6}+\frac{304678839159607}{183856454064672}a^{5}-\frac{338949444022607}{137892340548504}a^{4}+\frac{86\!\cdots\!47}{827354043291024}a^{3}-\frac{355387188336319}{22982056758084}a^{2}+\frac{132457834430341}{11491028379042}a-\frac{12492754940419}{3830342793014}$, $\frac{483228355153}{478994446115856}a^{15}-\frac{5889179719811}{957988892231712}a^{14}+\frac{16701891817063}{957988892231712}a^{13}-\frac{16636144125665}{478994446115856}a^{12}+\frac{83725217322023}{957988892231712}a^{11}-\frac{183816052817191}{957988892231712}a^{10}+\frac{2004452856961}{9392047963056}a^{9}-\frac{79250777571769}{957988892231712}a^{8}+\frac{304733526195437}{957988892231712}a^{7}-\frac{303129371259005}{478994446115856}a^{6}+\frac{115992018527003}{319329630743904}a^{5}-\frac{576550733134567}{319329630743904}a^{4}+\frac{742920702915583}{239497223057928}a^{3}-\frac{293163577144853}{79832407685976}a^{2}+\frac{12695089301987}{2217566880166}a-\frac{6342554304595}{2217566880166}$, $\frac{104988950749}{758407873016772}a^{15}-\frac{2735799853543}{60\!\cdots\!76}a^{14}+\frac{1477333203293}{20\!\cdots\!92}a^{13}-\frac{4534330905439}{30\!\cdots\!88}a^{12}+\frac{18106980950357}{20\!\cdots\!92}a^{11}-\frac{34185829420009}{20\!\cdots\!92}a^{10}+\frac{3317702262733}{178448911298064}a^{9}-\frac{266927149762441}{60\!\cdots\!76}a^{8}+\frac{121705430482837}{674140331570464}a^{7}-\frac{140158210453541}{10\!\cdots\!96}a^{6}+\frac{404383606567825}{60\!\cdots\!76}a^{5}-\frac{288765244364675}{20\!\cdots\!92}a^{4}+\frac{11\!\cdots\!89}{30\!\cdots\!88}a^{3}+\frac{2952841007351}{379203936508386}a^{2}-\frac{73846612180609}{252802624338924}a+\frac{2300128698797}{42133770723154}$, $\frac{18903106310563}{18\!\cdots\!28}a^{15}-\frac{127848266489641}{18\!\cdots\!28}a^{14}+\frac{51822265518457}{22\!\cdots\!16}a^{13}-\frac{915274507109399}{18\!\cdots\!28}a^{12}+\frac{21\!\cdots\!65}{18\!\cdots\!28}a^{11}-\frac{604569429459217}{22\!\cdots\!16}a^{10}+\frac{43889283029017}{118965940865376}a^{9}-\frac{42\!\cdots\!99}{18\!\cdots\!28}a^{8}+\frac{14\!\cdots\!47}{45\!\cdots\!32}a^{7}-\frac{13\!\cdots\!93}{18\!\cdots\!28}a^{6}+\frac{38\!\cdots\!45}{60\!\cdots\!76}a^{5}-\frac{25\!\cdots\!11}{15\!\cdots\!44}a^{4}+\frac{22\!\cdots\!15}{45\!\cdots\!32}a^{3}-\frac{198307279882307}{26151995621268}a^{2}+\frac{899945829633629}{126401312169462}a-\frac{58358321818563}{21066885361577}$, $\frac{15246891194831}{45\!\cdots\!32}a^{15}-\frac{397192103466071}{18\!\cdots\!28}a^{14}+\frac{11\!\cdots\!17}{18\!\cdots\!28}a^{13}-\frac{577892678521759}{45\!\cdots\!32}a^{12}+\frac{54\!\cdots\!23}{18\!\cdots\!28}a^{11}-\frac{12\!\cdots\!57}{18\!\cdots\!28}a^{10}+\frac{75870551667415}{89224455649032}a^{9}-\frac{33\!\cdots\!61}{18\!\cdots\!28}a^{8}+\frac{12\!\cdots\!67}{18\!\cdots\!28}a^{7}-\frac{16\!\cdots\!61}{568805904762579}a^{6}+\frac{46\!\cdots\!49}{20\!\cdots\!92}a^{5}-\frac{20\!\cdots\!81}{60\!\cdots\!76}a^{4}+\frac{48\!\cdots\!81}{45\!\cdots\!32}a^{3}-\frac{299053238346523}{17434663747512}a^{2}+\frac{10\!\cdots\!58}{63200656084731}a-\frac{289296773871601}{42133770723154}$, $\frac{118510669}{39081521952}a^{15}-\frac{642147115}{39081521952}a^{14}+\frac{795409525}{19540760976}a^{13}-\frac{2718446255}{39081521952}a^{12}+\frac{7504330555}{39081521952}a^{11}-\frac{8055151321}{19540760976}a^{10}+\frac{78108665}{255434784}a^{9}+\frac{7616700115}{39081521952}a^{8}+\frac{15692719385}{19540760976}a^{7}-\frac{66129075905}{39081521952}a^{6}+\frac{2436663179}{13027173984}a^{5}-\frac{18137508055}{6513586992}a^{4}+\frac{130652372545}{19540760976}a^{3}-\frac{887057725}{112303224}a^{2}+\frac{859025525}{180932972}a-\frac{48247977}{45233243}$ 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:  \( 119871.996246 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 119871.996246 \cdot 4}{2\cdot\sqrt{102711726879931884765625}}\cr\approx \mathstrut & 1.81709094743 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 23*x^14 - 50*x^13 + 115*x^12 - 263*x^11 + 384*x^10 - 245*x^9 + 289*x^8 - 890*x^7 + 1101*x^6 - 1581*x^5 + 3875*x^4 - 7140*x^3 + 8532*x^2 - 5184*x + 1296)
 
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 - 7*x^15 + 23*x^14 - 50*x^13 + 115*x^12 - 263*x^11 + 384*x^10 - 245*x^9 + 289*x^8 - 890*x^7 + 1101*x^6 - 1581*x^5 + 3875*x^4 - 7140*x^3 + 8532*x^2 - 5184*x + 1296, 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 - 7*x^15 + 23*x^14 - 50*x^13 + 115*x^12 - 263*x^11 + 384*x^10 - 245*x^9 + 289*x^8 - 890*x^7 + 1101*x^6 - 1581*x^5 + 3875*x^4 - 7140*x^3 + 8532*x^2 - 5184*x + 1296);
 
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 - 7*x^15 + 23*x^14 - 50*x^13 + 115*x^12 - 263*x^11 + 384*x^10 - 245*x^9 + 289*x^8 - 890*x^7 + 1101*x^6 - 1581*x^5 + 3875*x^4 - 7140*x^3 + 8532*x^2 - 5184*x + 1296);
 
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

$C_2^3:C_4$ (as 16T33):

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 $C_2^3:C_4$
Character table for $C_2^3:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), 4.0.3625.1 x2, 4.0.105125.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.4.64097340625.1 x2, 8.0.11051265625.4, 8.0.11051265625.3 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: 8.4.64097340625.1, 8.0.371764575625.2, 8.4.9294114390625.1, 8.0.11051265625.3
Degree 16 siblings: 16.8.86380562306022715087890625.2, 16.0.3455222492240908603515625.6, 16.0.86380562306022715087890625.8
Minimal sibling: 8.0.11051265625.3

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.2.0.1}{2} }^{8}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ R ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$

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
\(5\) Copy content Toggle raw display 5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
\(29\) Copy content Toggle raw display 29.2.1.1$x^{2} + 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} + 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} + 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} + 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.3.1$x^{4} + 116$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.1$x^{4} + 116$$4$$1$$3$$C_4$$[\ ]_{4}$