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Properties

Label 18.18.391...336.1
Degree $18$
Signature $[18, 0]$
Discriminant $3.919\times 10^{45}$
Root discriminant \(341.16\)
Ramified primes $2,3,19$
Class number $3$ (GRH)
Class group [3] (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 342*x^16 + 48735*x^14 - 3745014*x^12 + 167723127*x^10 - 4412408418*x^8 + 65205591066*x^6 - 482690739060*x^4 + 1375668606321*x^2 - 1132751510659)
 
gp: K = bnfinit(y^18 - 342*y^16 + 48735*y^14 - 3745014*y^12 + 167723127*y^10 - 4412408418*y^8 + 65205591066*y^6 - 482690739060*y^4 + 1375668606321*y^2 - 1132751510659, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 342*x^16 + 48735*x^14 - 3745014*x^12 + 167723127*x^10 - 4412408418*x^8 + 65205591066*x^6 - 482690739060*x^4 + 1375668606321*x^2 - 1132751510659);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 342*x^16 + 48735*x^14 - 3745014*x^12 + 167723127*x^10 - 4412408418*x^8 + 65205591066*x^6 - 482690739060*x^4 + 1375668606321*x^2 - 1132751510659)
 

\( x^{18} - 342 x^{16} + 48735 x^{14} - 3745014 x^{12} + 167723127 x^{10} - 4412408418 x^{8} + \cdots - 1132751510659 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[18, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(3919035016511176446693259642863771102588174336\) \(\medspace = 2^{18}\cdot 3^{44}\cdot 19^{15}\) 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:  \(341.16\)
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:  $2\cdot 3^{22/9}19^{5/6}\approx 341.1570865222663$
Ramified primes:   \(2\), \(3\), \(19\) 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:  \(\Q(\sqrt{19}) \)
$\card{ \Gal(K/\Q) }$:  $18$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2052=2^{2}\cdot 3^{3}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{2052}(1,·)$, $\chi_{2052}(331,·)$, $\chi_{2052}(961,·)$, $\chi_{2052}(715,·)$, $\chi_{2052}(1645,·)$, $\chi_{2052}(1489,·)$, $\chi_{2052}(1747,·)$, $\chi_{2052}(277,·)$, $\chi_{2052}(1015,·)$, $\chi_{2052}(1369,·)$, $\chi_{2052}(31,·)$, $\chi_{2052}(1699,·)$, $\chi_{2052}(805,·)$, $\chi_{2052}(1063,·)$, $\chi_{2052}(685,·)$, $\chi_{2052}(1399,·)$, $\chi_{2052}(121,·)$, $\chi_{2052}(379,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{19}a^{6}$, $\frac{1}{19}a^{7}$, $\frac{1}{19}a^{8}$, $\frac{1}{244169}a^{9}-\frac{9}{12851}a^{7}+\frac{513}{12851}a^{5}+\frac{2021}{12851}a^{3}-\frac{2524}{12851}a$, $\frac{1}{676592299}a^{10}-\frac{1914970}{676592299}a^{8}+\frac{6178227}{676592299}a^{6}-\frac{15432030}{35610121}a^{4}-\frac{14806876}{35610121}a^{2}+\frac{1016}{2771}$, $\frac{1}{676592299}a^{11}-\frac{11}{35610121}a^{9}-\frac{754815}{676592299}a^{7}+\frac{5367096}{35610121}a^{5}+\frac{9032037}{35610121}a^{3}-\frac{12433813}{35610121}a$, $\frac{1}{12855253681}a^{12}+\frac{12631507}{676592299}a^{8}+\frac{29681}{9529469}a^{6}-\frac{2471656}{35610121}a^{4}+\frac{10772124}{35610121}a^{2}+\frac{92}{2771}$, $\frac{1}{12855253681}a^{13}+\frac{1289}{676592299}a^{9}-\frac{10342752}{676592299}a^{7}-\frac{731468}{35610121}a^{5}+\frac{17558303}{35610121}a^{3}+\frac{8794229}{35610121}a$, $\frac{1}{12855253681}a^{14}+\frac{955229}{676592299}a^{8}-\frac{965391}{676592299}a^{6}+\frac{3387334}{35610121}a^{4}+\frac{7832537}{35610121}a^{2}+\frac{1059}{2771}$, $\frac{1}{12855253681}a^{15}-\frac{766}{676592299}a^{9}-\frac{15540851}{676592299}a^{7}+\frac{11503593}{35610121}a^{5}-\frac{1286824}{35610121}a^{3}+\frac{5052361}{35610121}a$, $\frac{1}{12855253681}a^{16}+\frac{777483}{39799547}a^{8}+\frac{75490}{39799547}a^{6}+\frac{19904}{2094713}a^{4}-\frac{764481}{2094713}a^{2}-\frac{395}{2771}$, $\frac{1}{12855253681}a^{17}-\frac{27}{39799547}a^{9}-\frac{1031932}{39799547}a^{7}-\frac{847256}{2094713}a^{5}+\frac{1017272}{2094713}a^{3}-\frac{10360442}{35610121}a$ 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:  $17$

Class group and class number

$C_{3}$, which has order $3$ (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:  $17$
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:   $\frac{231}{12855253681}a^{12}-\frac{2772}{676592299}a^{10}+\frac{12474}{35610121}a^{8}-\frac{9320656}{676592299}a^{6}+\frac{8641239}{35610121}a^{4}-\frac{53767188}{35610121}a^{2}+\frac{6381}{2771}$, $\frac{185}{12855253681}a^{12}-\frac{2220}{676592299}a^{10}+\frac{9990}{35610121}a^{8}-\frac{7618751}{676592299}a^{6}+\frac{7845411}{35610121}a^{4}-\frac{69421041}{35610121}a^{2}+\frac{11552}{2771}$, $\frac{12}{12855253681}a^{15}-\frac{180}{676592299}a^{13}-\frac{1754}{12855253681}a^{12}+\frac{1080}{35610121}a^{11}+\frac{21048}{676592299}a^{10}-\frac{1180216}{676592299}a^{9}-\frac{94716}{35610121}a^{8}+\frac{1849644}{35610121}a^{7}+\frac{69539176}{676592299}a^{6}-\frac{25426332}{35610121}a^{5}-\frac{58214058}{35610121}a^{4}+\frac{107582866}{35610121}a^{3}+\frac{197372304}{35610121}a^{2}-\frac{256419858}{35610121}a-\frac{27399}{2771}$, $\frac{1}{12855253681}a^{16}-\frac{265}{12855253681}a^{14}+\frac{27169}{12855253681}a^{12}-\frac{68385}{676592299}a^{10}+\frac{1228391}{676592299}a^{8}+\frac{17418119}{676592299}a^{6}-\frac{56570302}{35610121}a^{4}+\frac{814088522}{35610121}a^{2}-\frac{296656}{2771}$, $\frac{3}{12855253681}a^{16}-\frac{659}{12855253681}a^{14}+\frac{48154}{12855253681}a^{12}-\frac{43578}{676592299}a^{10}-\frac{2921908}{676592299}a^{8}+\frac{143384844}{676592299}a^{6}-\frac{105861129}{35610121}a^{4}+\frac{341424759}{35610121}a^{2}-\frac{21042}{2771}$, $\frac{5}{12855253681}a^{16}-\frac{1391}{12855253681}a^{14}+\frac{8858}{756191393}a^{12}-\frac{416122}{676592299}a^{10}+\frac{10685361}{676592299}a^{8}-\frac{118460074}{676592299}a^{6}+\frac{24708224}{35610121}a^{4}-\frac{54451228}{35610121}a^{2}+\frac{6184}{2771}$, $\frac{12}{12855253681}a^{16}-\frac{3412}{12855253681}a^{14}+\frac{22809}{756191393}a^{12}-\frac{1179515}{676592299}a^{10}+\frac{36391541}{676592299}a^{8}-\frac{33588332}{39799547}a^{6}+\frac{200439938}{35610121}a^{4}-\frac{261444493}{35610121}a^{2}+\frac{4056}{2771}$, $\frac{9}{12855253681}a^{16}-\frac{2809}{12855253681}a^{14}+\frac{354495}{12855253681}a^{12}-\frac{63782}{35610121}a^{10}+\frac{42403102}{676592299}a^{8}-\frac{749229315}{676592299}a^{6}+\frac{268686794}{35610121}a^{4}+\frac{77309373}{35610121}a^{2}-\frac{167836}{2771}$, $\frac{13}{12855253681}a^{16}-\frac{4042}{12855253681}a^{14}+\frac{514831}{12855253681}a^{12}-\frac{1815630}{676592299}a^{10}+\frac{68141271}{676592299}a^{8}-\frac{1399170548}{676592299}a^{6}+\frac{736220756}{35610121}a^{4}-\frac{2670385838}{35610121}a^{2}+\frac{233196}{2771}$, $\frac{94}{12855253681}a^{15}-\frac{1410}{676592299}a^{13}-\frac{7672}{12855253681}a^{12}+\frac{8460}{35610121}a^{11}+\frac{92064}{676592299}a^{10}-\frac{9320766}{676592299}a^{9}-\frac{414288}{35610121}a^{8}+\frac{15170544}{35610121}a^{7}+\frac{311716946}{676592299}a^{6}-\frac{238027896}{35610121}a^{5}-\frac{299943012}{35610121}a^{4}+\frac{1579913588}{35610121}a^{3}+\frac{2154772710}{35610121}a^{2}-\frac{1948023222}{35610121}a-\frac{171873}{2771}$, $\frac{3}{12855253681}a^{17}+\frac{5}{12855253681}a^{16}-\frac{932}{12855253681}a^{15}-\frac{1413}{12855253681}a^{14}+\frac{118147}{12855253681}a^{13}+\frac{159951}{12855253681}a^{12}-\frac{413242}{676592299}a^{11}-\frac{490092}{676592299}a^{10}+\frac{15380582}{676592299}a^{9}+\frac{15791099}{676592299}a^{8}-\frac{316780334}{676592299}a^{7}-\frac{286705393}{676592299}a^{6}+\frac{174913582}{35610121}a^{5}+\frac{152588136}{35610121}a^{4}-\frac{773142973}{35610121}a^{3}-\frac{759324267}{35610121}a^{2}+\frac{1076272187}{35610121}a+\frac{89525}{2771}$, $\frac{1}{12855253681}a^{17}+\frac{3}{12855253681}a^{16}-\frac{360}{12855253681}a^{15}-\frac{968}{12855253681}a^{14}+\frac{53735}{12855253681}a^{13}+\frac{126835}{12855253681}a^{12}-\frac{225650}{676592299}a^{11}-\frac{459589}{676592299}a^{10}+\frac{10338502}{676592299}a^{9}+\frac{18096356}{676592299}a^{8}-\frac{271579481}{676592299}a^{7}-\frac{419490787}{676592299}a^{6}+\frac{202847884}{35610121}a^{5}+\frac{296760758}{35610121}a^{4}-\frac{1350444942}{35610121}a^{3}-\frac{2018049353}{35610121}a^{2}+\frac{2915752634}{35610121}a+\frac{299937}{2771}$, $\frac{8}{12855253681}a^{16}+\frac{11}{12855253681}a^{15}-\frac{2685}{12855253681}a^{14}-\frac{3412}{12855253681}a^{13}+\frac{368898}{12855253681}a^{12}+\frac{21995}{676592299}a^{11}-\frac{1400073}{676592299}a^{10}-\frac{1341634}{676592299}a^{9}+\frac{56334965}{676592299}a^{8}+\frac{34893}{560557}a^{7}-\frac{1236072649}{676592299}a^{6}-\frac{33327546}{35610121}a^{5}+\frac{699244559}{35610121}a^{4}+\frac{207323705}{35610121}a^{3}-\frac{2901091759}{35610121}a^{2}-\frac{41913944}{2094713}a+\frac{263951}{2771}$, $\frac{1}{12855253681}a^{17}-\frac{7}{12855253681}a^{16}-\frac{312}{12855253681}a^{15}+\frac{2257}{12855253681}a^{14}+\frac{39038}{12855253681}a^{13}-\frac{295874}{12855253681}a^{12}-\frac{130349}{676592299}a^{11}+\frac{1061730}{676592299}a^{10}+\frac{4317307}{676592299}a^{9}-\frac{39888294}{676592299}a^{8}-\frac{3891277}{39799547}a^{7}+\frac{801621966}{676592299}a^{6}+\frac{11957475}{35610121}a^{5}-\frac{400958944}{35610121}a^{4}+\frac{123284159}{35610121}a^{3}+\frac{1376479707}{35610121}a^{2}+\frac{71439205}{35610121}a-\frac{123383}{2771}$, $\frac{1}{12855253681}a^{17}-\frac{1}{12855253681}a^{16}-\frac{349}{12855253681}a^{15}+\frac{394}{12855253681}a^{14}+\frac{50138}{12855253681}a^{13}-\frac{62039}{12855253681}a^{12}-\frac{202036}{676592299}a^{11}+\frac{268545}{676592299}a^{10}+\frac{524473}{39799547}a^{9}-\frac{12559531}{676592299}a^{8}-\frac{228289591}{676592299}a^{7}+\frac{338182742}{676592299}a^{6}+\frac{168042104}{35610121}a^{5}-\frac{261743392}{35610121}a^{4}-\frac{6451515}{218467}a^{3}+\frac{1733986810}{35610121}a^{2}+\frac{1165457915}{35610121}a-\frac{148995}{2771}$, $\frac{1}{12855253681}a^{17}-\frac{12}{12855253681}a^{16}-\frac{297}{12855253681}a^{15}+\frac{3665}{12855253681}a^{14}+\frac{35133}{12855253681}a^{13}-\frac{26735}{756191393}a^{12}-\frac{110572}{676592299}a^{11}+\frac{1544786}{676592299}a^{10}+\frac{3507299}{676592299}a^{9}-\frac{55425158}{676592299}a^{8}-\frac{57354817}{676592299}a^{7}+\frac{1092684154}{676592299}a^{6}+\frac{25105456}{35610121}a^{5}-\frac{577150947}{35610121}a^{4}-\frac{156981180}{35610121}a^{3}+\frac{2414754146}{35610121}a^{2}+\frac{1003445521}{35610121}a-\frac{172827}{2771}$, $\frac{39}{244169}a^{9}-\frac{351}{12851}a^{7}+\frac{20007}{12851}a^{5}-\frac{422370}{12851}a^{3}+\frac{2407509}{12851}a+170$ 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:  \( 222183985349289660 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 222183985349289660 \cdot 3}{2\cdot\sqrt{3919035016511176446693259642863771102588174336}}\cr\approx \mathstrut & 1.39557879367550 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 342*x^16 + 48735*x^14 - 3745014*x^12 + 167723127*x^10 - 4412408418*x^8 + 65205591066*x^6 - 482690739060*x^4 + 1375668606321*x^2 - 1132751510659)
 
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^18 - 342*x^16 + 48735*x^14 - 3745014*x^12 + 167723127*x^10 - 4412408418*x^8 + 65205591066*x^6 - 482690739060*x^4 + 1375668606321*x^2 - 1132751510659, 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^18 - 342*x^16 + 48735*x^14 - 3745014*x^12 + 167723127*x^10 - 4412408418*x^8 + 65205591066*x^6 - 482690739060*x^4 + 1375668606321*x^2 - 1132751510659);
 
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^18 - 342*x^16 + 48735*x^14 - 3745014*x^12 + 167723127*x^10 - 4412408418*x^8 + 65205591066*x^6 - 482690739060*x^4 + 1375668606321*x^2 - 1132751510659);
 
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

$C_{18}$ (as 18T1):

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 cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{19}) \), \(\Q(\zeta_{9})^+\), 6.6.2880121536.1, 9.9.1476349596018920529.1

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.9.0.1}{9} }^{2}$ $18$ $18$ $18$ ${\href{/padicField/17.1.0.1}{1} }^{18}$ R $18$ $18$ ${\href{/padicField/31.9.0.1}{9} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{3}$ $18$ $18$ $18$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.9.0.1}{9} }^{2}$

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
\(2\) Copy content Toggle raw display 2.18.18.118$x^{18} + 18 x^{17} + 198 x^{16} + 1536 x^{15} + 9312 x^{14} + 45696 x^{13} + 187776 x^{12} + 655872 x^{11} + 2010400 x^{10} + 5500224 x^{9} + 14116288 x^{8} + 34058240 x^{7} + 79898624 x^{6} + 169960448 x^{5} + 335809536 x^{4} + 542121984 x^{3} + 798549248 x^{2} + 783239680 x + 807955968$$2$$9$$18$$C_{18}$$[2]^{9}$
\(3\) Copy content Toggle raw display 3.9.22.6$x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$$9$$1$$22$$C_9$$[2, 3]$
3.9.22.6$x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$$9$$1$$22$$C_9$$[2, 3]$
\(19\) Copy content Toggle raw display 19.6.5.1$x^{6} + 38$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.1$x^{6} + 38$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.1$x^{6} + 38$$6$$1$$5$$C_6$$[\ ]_{6}$