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Properties

Label 36.36.799...784.1
Degree $36$
Signature $[36, 0]$
Discriminant $7.996\times 10^{62}$
Root discriminant \(55.89\)
Ramified primes $2,3,19$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2\times C_{18}$ (as 36T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 35*x^34 + 560*x^32 - 5426*x^30 + 35554*x^28 - 166634*x^26 + 576201*x^24 - 1494747*x^22 + 2929464*x^20 - 4334718*x^18 + 4805781*x^16 - 3932287*x^14 + 2318239*x^12 - 949739*x^10 + 256105*x^8 - 41769*x^6 + 3570*x^4 - 120*x^2 + 1)
 
gp: K = bnfinit(y^36 - 35*y^34 + 560*y^32 - 5426*y^30 + 35554*y^28 - 166634*y^26 + 576201*y^24 - 1494747*y^22 + 2929464*y^20 - 4334718*y^18 + 4805781*y^16 - 3932287*y^14 + 2318239*y^12 - 949739*y^10 + 256105*y^8 - 41769*y^6 + 3570*y^4 - 120*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - 35*x^34 + 560*x^32 - 5426*x^30 + 35554*x^28 - 166634*x^26 + 576201*x^24 - 1494747*x^22 + 2929464*x^20 - 4334718*x^18 + 4805781*x^16 - 3932287*x^14 + 2318239*x^12 - 949739*x^10 + 256105*x^8 - 41769*x^6 + 3570*x^4 - 120*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 35*x^34 + 560*x^32 - 5426*x^30 + 35554*x^28 - 166634*x^26 + 576201*x^24 - 1494747*x^22 + 2929464*x^20 - 4334718*x^18 + 4805781*x^16 - 3932287*x^14 + 2318239*x^12 - 949739*x^10 + 256105*x^8 - 41769*x^6 + 3570*x^4 - 120*x^2 + 1)
 

\( x^{36} - 35 x^{34} + 560 x^{32} - 5426 x^{30} + 35554 x^{28} - 166634 x^{26} + 576201 x^{24} - 1494747 x^{22} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $36$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[36, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(799622233646074762983150698451178476894456963777140963130998784\) \(\medspace = 2^{36}\cdot 3^{18}\cdot 19^{34}\) 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:  \(55.89\)
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:  $2\cdot 3^{1/2}19^{17/18}\approx 55.88591389129187$
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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $36$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(228=2^{2}\cdot 3\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{228}(1,·)$, $\chi_{228}(131,·)$, $\chi_{228}(11,·)$, $\chi_{228}(23,·)$, $\chi_{228}(25,·)$, $\chi_{228}(29,·)$, $\chi_{228}(31,·)$, $\chi_{228}(35,·)$, $\chi_{228}(41,·)$, $\chi_{228}(173,·)$, $\chi_{228}(47,·)$, $\chi_{228}(49,·)$, $\chi_{228}(53,·)$, $\chi_{228}(185,·)$, $\chi_{228}(151,·)$, $\chi_{228}(61,·)$, $\chi_{228}(191,·)$, $\chi_{228}(65,·)$, $\chi_{228}(67,·)$, $\chi_{228}(73,·)$, $\chi_{228}(119,·)$, $\chi_{228}(79,·)$, $\chi_{228}(83,·)$, $\chi_{228}(85,·)$, $\chi_{228}(215,·)$, $\chi_{228}(89,·)$, $\chi_{228}(91,·)$, $\chi_{228}(221,·)$, $\chi_{228}(223,·)$, $\chi_{228}(103,·)$, $\chi_{228}(157,·)$, $\chi_{228}(113,·)$, $\chi_{228}(211,·)$, $\chi_{228}(169,·)$, $\chi_{228}(121,·)$, $\chi_{228}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$ Copy content Toggle raw display

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

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$ (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:  $35$
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:   $a^{21}-21a^{19}+188a^{17}-935a^{15}+2821a^{13}-5291a^{11}+6072a^{9}-4026a^{7}+1365a^{5}-181a^{3}+4a$, $a^{17}-17a^{15}+119a^{13}-442a^{11}+935a^{9}-1122a^{7}+714a^{5}-204a^{3}+17a$, $a^{18}-18a^{16}+135a^{14}-546a^{12}+1287a^{10}-1782a^{8}+1386a^{6}-540a^{4}+81a^{2}-2$, $a^{25}-25a^{23}+275a^{21}-1750a^{19}+7125a^{17}-19380a^{15}+35700a^{13}-44200a^{11}+35750a^{9}-17875a^{7}+5005a^{5}-650a^{3}+25a$, $a^{6}-6a^{4}+9a^{2}-2$, $a^{30}-30a^{28}+405a^{26}-3251a^{24}+17273a^{22}-63986a^{20}+169556a^{18}-324461a^{16}+446982a^{14}-436540a^{12}+293227a^{10}-128547a^{8}+33558a^{6}-4376a^{4}+193a^{2}-3$, $a^{23}-23a^{21}+230a^{19}-1311a^{17}+4692a^{15}-10948a^{13}+16744a^{11}-16445a^{9}+9867a^{7}-3289a^{5}+506a^{3}-23a$, $a^{31}-31a^{29}+434a^{27}-3627a^{25}+20150a^{23}-78430a^{21}+219604a^{19}-447051a^{17}+660858a^{15}-700910a^{13}+520676a^{11}-260338a^{9}+82212a^{7}-14756a^{5}+1240a^{3}-31a$, $a^{22}-22a^{20}+209a^{18}-1123a^{16}+3756a^{14}-8112a^{12}+11363a^{10}-10098a^{8}+5391a^{6}-1546a^{4}+185a^{2}-4$, $a^{30}-30a^{28}+405a^{26}-3251a^{24}+17274a^{22}-64008a^{20}+169765a^{18}-325584a^{16}+450738a^{14}-444652a^{12}+304590a^{10}-138645a^{8}+38948a^{6}-5916a^{4}+369a^{2}-4$, $2a^{35}-69a^{33}+1086a^{31}-10326a^{29}+66207a^{27}-302586a^{25}+1016073a^{23}-2546721a^{21}+4792257a^{19}-6755054a^{17}+7062633a^{15}-5378205a^{13}+2898483a^{11}-1058505a^{9}+245055a^{7}-32283a^{5}+1989a^{3}-36a$, $a^{28}-28a^{26}+350a^{24}-2576a^{22}+12397a^{20}-40964a^{18}+94962a^{16}-155040a^{14}+176358a^{12}-136137a^{10}+68078a^{8}-20419a^{6}+3235a^{4}-221a^{2}+4$, $a^{18}-18a^{16}+135a^{14}-546a^{12}+1287a^{10}-1782a^{8}+1387a^{6}-546a^{4}+90a^{2}-4$, $a^{29}-29a^{27}+377a^{25}-2900a^{23}+14674a^{21}-51359a^{19}+127281a^{17}-224808a^{15}+281010a^{13}-243542a^{11}+140997a^{9}-51263a^{7}+10529a^{5}-985a^{3}+20a$, $a^{5}-5a^{3}+5a$, $a^{35}-35a^{33}+560a^{31}-5425a^{29}+35525a^{27}-166257a^{25}+573300a^{23}-1480050a^{21}+2877875a^{19}-4206125a^{17}+4576264a^{15}-3640210a^{13}+2057510a^{11}-791350a^{9}+193800a^{7}-27132a^{5}+1784a^{3}-32a$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107405a^{11}-72919a^{9}+30844a^{7}-7294a^{5}+764a^{3}-16a$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a-1$, $a^{31}-31a^{29}+434a^{27}-3627a^{25}+20150a^{23}-78430a^{21}+219604a^{19}-447051a^{17}+660858a^{15}-700910a^{13}+520676a^{11}-260338a^{9}+82211a^{7}-a^{6}-14749a^{5}+6a^{4}+1226a^{3}-9a^{2}-24a+2$, $a^{25}-a^{24}-25a^{23}+24a^{22}+275a^{21}-252a^{20}-1750a^{19}+1520a^{18}+7125a^{17}-5814a^{16}-19380a^{15}+14688a^{14}+35699a^{13}-24752a^{12}-44187a^{11}+27456a^{10}+35685a^{9}-19305a^{8}-17719a^{7}+8008a^{6}+4823a^{5}-1716a^{4}-559a^{3}+144a^{2}+12a-2$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-a^{3}-36a^{2}+3a+2$, $a^{31}-31a^{29}+434a^{27}-3627a^{25}+20150a^{23}-78430a^{21}+219604a^{19}-a^{18}-447051a^{17}+18a^{16}+660858a^{15}-135a^{14}-700910a^{13}+546a^{12}+520676a^{11}-1287a^{10}-260338a^{9}+1782a^{8}+82211a^{7}-1386a^{6}-14749a^{5}+540a^{4}+1226a^{3}-81a^{2}-24a+2$, $a^{13}-13a^{11}+65a^{9}-156a^{7}+182a^{5}-91a^{3}+13a+1$, $a^{29}-a^{28}-29a^{27}+28a^{26}+377a^{25}-350a^{24}-2900a^{23}+2576a^{22}+14674a^{21}-12397a^{20}-51359a^{19}+40964a^{18}+127281a^{17}-94962a^{16}-224808a^{15}+155040a^{14}+281010a^{13}-176358a^{12}-243542a^{11}+136136a^{10}+140997a^{9}-68068a^{8}-51263a^{7}+20384a^{6}+10529a^{5}-3185a^{4}-985a^{3}+196a^{2}+20a-2$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a-1$, $a^{16}-16a^{14}+104a^{12}-352a^{10}+660a^{8}+a^{7}-672a^{6}-7a^{5}+336a^{4}+14a^{3}-64a^{2}-7a+2$, $a^{29}-29a^{27}+377a^{25}-2900a^{23}+14674a^{21}-51359a^{19}+127281a^{17}-224808a^{15}+281010a^{13}-243542a^{11}+140998a^{9}-51272a^{7}+10556a^{5}-1015a^{3}+29a-1$, $a^{21}-21a^{19}+189a^{17}-952a^{15}+2940a^{13}-5733a^{11}+7007a^{9}-5148a^{7}+2079a^{5}-385a^{3}+21a+1$, $2a^{35}-69a^{33}+1086a^{31}-10326a^{29}+66207a^{27}-302586a^{25}+1016073a^{23}-2546721a^{21}+4792257a^{19}-6755054a^{17}+7062633a^{15}-5378205a^{13}+2898483a^{11}-1058505a^{9}+245055a^{7}-32283a^{5}+1989a^{3}-35a+1$, $a^{22}+a^{21}-22a^{20}-21a^{19}+209a^{18}+189a^{17}-1123a^{16}-952a^{15}+3756a^{14}+2940a^{13}-8112a^{12}-5733a^{11}+11363a^{10}+7007a^{9}-10098a^{8}-5148a^{7}+5391a^{6}+2079a^{5}-1546a^{4}-385a^{3}+185a^{2}+21a-4$, $a^{33}-33a^{31}+495a^{29}-4466a^{27}+27027a^{25}-115830a^{23}+361790a^{21}-834900a^{19}+1427679a^{17}-1797818a^{15}+1641486a^{13}-1058148a^{11}+461890a^{9}-127908a^{7}+20195a^{5}-1491a^{3}+28a-1$, $2a^{35}-69a^{33}+1086a^{31}-10326a^{29}+66207a^{27}-302586a^{25}+1016073a^{23}-2546721a^{21}+4792257a^{19}-6755054a^{17}+7062633a^{15}-5378205a^{13}+2898483a^{11}-1058505a^{9}+245055a^{7}-32283a^{5}+1989a^{3}-36a+1$, $2a^{35}+a^{34}-70a^{33}-34a^{32}+1120a^{31}+527a^{30}-10851a^{29}-4931a^{28}+71079a^{27}+31087a^{26}-332891a^{25}-139580a^{24}+1149501a^{23}+460047a^{22}-2974797a^{21}-1130679a^{20}+5807339a^{19}+2084148a^{18}-8540843a^{17}-2874531a^{16}+9382045a^{15}+2937243a^{14}-7572497a^{13}-2183208a^{12}+4375749a^{11}+1146939a^{10}-1741089a^{9}-407847a^{8}+449904a^{7}+92004a^{6}-68894a^{5}-11934a^{4}+5340a^{3}+765a^{2}-144a-18$, $a^{35}-35a^{33}+560a^{31}-5425a^{29}+35525a^{27}-166257a^{25}+573300a^{23}-1480050a^{21}+2877875a^{19}-4206125a^{17}+4576264a^{15}-3640210a^{13}+2057510a^{11}-791350a^{9}+193800a^{7}-27132a^{5}+1784a^{3}-32a-1$, $a^{21}-21a^{19}+188a^{17}-935a^{15}+2821a^{13}-5291a^{11}+6072a^{9}-4026a^{7}+1365a^{5}-181a^{3}+4a+1$ 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:  \( 104443369202940970000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{36}\cdot(2\pi)^{0}\cdot 104443369202940970000 \cdot 1}{2\cdot\sqrt{799622233646074762983150698451178476894456963777140963130998784}}\cr\approx \mathstrut & 0.126907792767333 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^36 - 35*x^34 + 560*x^32 - 5426*x^30 + 35554*x^28 - 166634*x^26 + 576201*x^24 - 1494747*x^22 + 2929464*x^20 - 4334718*x^18 + 4805781*x^16 - 3932287*x^14 + 2318239*x^12 - 949739*x^10 + 256105*x^8 - 41769*x^6 + 3570*x^4 - 120*x^2 + 1)
 
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^36 - 35*x^34 + 560*x^32 - 5426*x^30 + 35554*x^28 - 166634*x^26 + 576201*x^24 - 1494747*x^22 + 2929464*x^20 - 4334718*x^18 + 4805781*x^16 - 3932287*x^14 + 2318239*x^12 - 949739*x^10 + 256105*x^8 - 41769*x^6 + 3570*x^4 - 120*x^2 + 1, 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^36 - 35*x^34 + 560*x^32 - 5426*x^30 + 35554*x^28 - 166634*x^26 + 576201*x^24 - 1494747*x^22 + 2929464*x^20 - 4334718*x^18 + 4805781*x^16 - 3932287*x^14 + 2318239*x^12 - 949739*x^10 + 256105*x^8 - 41769*x^6 + 3570*x^4 - 120*x^2 + 1);
 
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^36 - 35*x^34 + 560*x^32 - 5426*x^30 + 35554*x^28 - 166634*x^26 + 576201*x^24 - 1494747*x^22 + 2929464*x^20 - 4334718*x^18 + 4805781*x^16 - 3932287*x^14 + 2318239*x^12 - 949739*x^10 + 256105*x^8 - 41769*x^6 + 3570*x^4 - 120*x^2 + 1);
 
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

$C_2\times C_{18}$ (as 36T2):

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)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_2\times C_{18}$
Character table for $C_2\times C_{18}$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{19}) \), \(\Q(\sqrt{57}) \), 3.3.361.1, \(\Q(\sqrt{3}, \sqrt{19})\), 6.6.225194688.1, 6.6.158470336.1, 6.6.66854673.1, \(\Q(\zeta_{19})^+\), 12.12.18307265748733661184.1, 18.18.1488294338429317924379721203712.1, \(\Q(\zeta_{76})^+\), \(\Q(\zeta_{57})^+\)

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 $18^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{6}$ ${\href{/padicField/11.6.0.1}{6} }^{6}$ $18^{2}$ $18^{2}$ R $18^{2}$ $18^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{6}$ ${\href{/padicField/37.2.0.1}{2} }^{18}$ $18^{2}$ $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/padicField/59.9.0.1}{9} }^{4}$

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
\(2\) 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}$$[2]^{9}$
2.18.18.118$x^{18} + 18 x^{17} + 198 x^{16} + 1536 x^{15} + 9312 x^{14} + 45696 x^{13} + 187776 x^{12} + 655872 x^{11} + 2010400 x^{10} + 5500224 x^{9} + 14116288 x^{8} + 34058240 x^{7} + 79898624 x^{6} + 169960448 x^{5} + 335809536 x^{4} + 542121984 x^{3} + 798549248 x^{2} + 783239680 x + 807955968$$2$$9$$18$$C_{18}$$[2]^{9}$
\(3\) Copy content Toggle raw display Deg $18$$2$$9$$9$
Deg $18$$2$$9$$9$
\(19\) Copy content Toggle raw display Deg $36$$18$$2$$34$