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Properties

Label 36.0.114...625.1
Degree $36$
Signature $[0, 18]$
Discriminant $1.146\times 10^{56}$
Root discriminant \(36.07\)
Ramified primes $5,19$
Class number $76$ (GRH)
Class group [76] (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 - x^35 + 2*x^34 - 3*x^33 + 5*x^32 - 8*x^31 + 13*x^30 - 21*x^29 + 34*x^28 - 55*x^27 + 89*x^26 - 144*x^25 + 233*x^24 - 377*x^23 + 610*x^22 - 987*x^21 + 1597*x^20 - 2584*x^19 + 4181*x^18 + 2584*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1)
 
gp: K = bnfinit(y^36 - y^35 + 2*y^34 - 3*y^33 + 5*y^32 - 8*y^31 + 13*y^30 - 21*y^29 + 34*y^28 - 55*y^27 + 89*y^26 - 144*y^25 + 233*y^24 - 377*y^23 + 610*y^22 - 987*y^21 + 1597*y^20 - 2584*y^19 + 4181*y^18 + 2584*y^17 + 1597*y^16 + 987*y^15 + 610*y^14 + 377*y^13 + 233*y^12 + 144*y^11 + 89*y^10 + 55*y^9 + 34*y^8 + 21*y^7 + 13*y^6 + 8*y^5 + 5*y^4 + 3*y^3 + 2*y^2 + y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - x^35 + 2*x^34 - 3*x^33 + 5*x^32 - 8*x^31 + 13*x^30 - 21*x^29 + 34*x^28 - 55*x^27 + 89*x^26 - 144*x^25 + 233*x^24 - 377*x^23 + 610*x^22 - 987*x^21 + 1597*x^20 - 2584*x^19 + 4181*x^18 + 2584*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - x^35 + 2*x^34 - 3*x^33 + 5*x^32 - 8*x^31 + 13*x^30 - 21*x^29 + 34*x^28 - 55*x^27 + 89*x^26 - 144*x^25 + 233*x^24 - 377*x^23 + 610*x^22 - 987*x^21 + 1597*x^20 - 2584*x^19 + 4181*x^18 + 2584*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1)
 

\( x^{36} - x^{35} + 2 x^{34} - 3 x^{33} + 5 x^{32} - 8 x^{31} + 13 x^{30} - 21 x^{29} + 34 x^{28} + \cdots + 1 \) 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:  $[0, 18]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(114573059505387793044837364496233492772337802886962890625\) \(\medspace = 5^{18}\cdot 19^{34}\) 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:  \(36.07\)
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:  $5^{1/2}19^{17/18}\approx 36.07420229809253$
Ramified primes:   \(5\), \(19\) 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:  \(95=5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{95}(1,·)$, $\chi_{95}(4,·)$, $\chi_{95}(6,·)$, $\chi_{95}(9,·)$, $\chi_{95}(11,·)$, $\chi_{95}(14,·)$, $\chi_{95}(16,·)$, $\chi_{95}(21,·)$, $\chi_{95}(24,·)$, $\chi_{95}(26,·)$, $\chi_{95}(29,·)$, $\chi_{95}(31,·)$, $\chi_{95}(34,·)$, $\chi_{95}(36,·)$, $\chi_{95}(39,·)$, $\chi_{95}(41,·)$, $\chi_{95}(44,·)$, $\chi_{95}(46,·)$, $\chi_{95}(49,·)$, $\chi_{95}(51,·)$, $\chi_{95}(54,·)$, $\chi_{95}(56,·)$, $\chi_{95}(59,·)$, $\chi_{95}(61,·)$, $\chi_{95}(64,·)$, $\chi_{95}(66,·)$, $\chi_{95}(69,·)$, $\chi_{95}(71,·)$, $\chi_{95}(74,·)$, $\chi_{95}(79,·)$, $\chi_{95}(81,·)$, $\chi_{95}(84,·)$, $\chi_{95}(86,·)$, $\chi_{95}(89,·)$, $\chi_{95}(91,·)$, $\chi_{95}(94,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{4181}a^{19}-\frac{1597}{4181}$, $\frac{1}{4181}a^{20}-\frac{1597}{4181}a$, $\frac{1}{4181}a^{21}-\frac{1597}{4181}a^{2}$, $\frac{1}{4181}a^{22}-\frac{1597}{4181}a^{3}$, $\frac{1}{4181}a^{23}-\frac{1597}{4181}a^{4}$, $\frac{1}{4181}a^{24}-\frac{1597}{4181}a^{5}$, $\frac{1}{4181}a^{25}-\frac{1597}{4181}a^{6}$, $\frac{1}{4181}a^{26}-\frac{1597}{4181}a^{7}$, $\frac{1}{4181}a^{27}-\frac{1597}{4181}a^{8}$, $\frac{1}{4181}a^{28}-\frac{1597}{4181}a^{9}$, $\frac{1}{4181}a^{29}-\frac{1597}{4181}a^{10}$, $\frac{1}{4181}a^{30}-\frac{1597}{4181}a^{11}$, $\frac{1}{4181}a^{31}-\frac{1597}{4181}a^{12}$, $\frac{1}{4181}a^{32}-\frac{1597}{4181}a^{13}$, $\frac{1}{4181}a^{33}-\frac{1597}{4181}a^{14}$, $\frac{1}{4181}a^{34}-\frac{1597}{4181}a^{15}$, $\frac{1}{4181}a^{35}-\frac{1597}{4181}a^{16}$ 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

$C_{76}$, which has order $76$ (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:   \( -\frac{987}{4181} a^{35} - \frac{9227465}{4181} a^{16} \)  (order $38$) 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:   $\frac{2}{4181}a^{23}+\frac{17711}{4181}a^{4}$, $\frac{3}{4181}a^{23}+\frac{28657}{4181}a^{4}+1$, $\frac{377}{4181}a^{33}+\frac{89}{4181}a^{30}+\frac{2}{4181}a^{22}+\frac{3524578}{4181}a^{14}+\frac{832040}{4181}a^{11}+\frac{17711}{4181}a^{3}$, $\frac{377}{4181}a^{33}+\frac{3524578}{4181}a^{14}+1$, $\frac{3}{4181}a^{23}+\frac{1}{4181}a^{21}+\frac{28657}{4181}a^{4}+\frac{10946}{4181}a^{2}$, $\frac{34}{4181}a^{28}+\frac{317811}{4181}a^{9}+1$, $\frac{377}{4181}a^{33}+\frac{13}{4181}a^{26}+\frac{3524578}{4181}a^{14}+\frac{121393}{4181}a^{7}$, $\frac{89}{4181}a^{30}+\frac{2}{4181}a^{22}+\frac{832040}{4181}a^{11}+\frac{17711}{4181}a^{3}$, $\frac{987}{4181}a^{35}+\frac{21}{4181}a^{27}+\frac{5}{4181}a^{24}+\frac{9227465}{4181}a^{16}+\frac{196418}{4181}a^{8}+\frac{46368}{4181}a^{5}+1$, $\frac{610}{4181}a^{34}+\frac{233}{4181}a^{32}-\frac{89}{4181}a^{31}+\frac{89}{4181}a^{30}-\frac{34}{4181}a^{29}+\frac{34}{4181}a^{28}-\frac{13}{4181}a^{27}+\frac{13}{4181}a^{26}-\frac{5}{4181}a^{25}+\frac{5}{4181}a^{24}+\frac{2}{4181}a^{22}+\frac{5702887}{4181}a^{15}+\frac{2178309}{4181}a^{13}-\frac{832040}{4181}a^{12}+\frac{832040}{4181}a^{11}-\frac{317811}{4181}a^{10}+\frac{317811}{4181}a^{9}-\frac{121393}{4181}a^{8}+\frac{121393}{4181}a^{7}-\frac{46368}{4181}a^{6}+\frac{46368}{4181}a^{5}+\frac{17711}{4181}a^{3}$, $\frac{1597}{4181}a^{35}-\frac{3194}{4181}a^{34}+\frac{4791}{4181}a^{33}-\frac{8129}{4181}a^{32}+\frac{12776}{4181}a^{31}-\frac{20761}{4181}a^{30}+\frac{33537}{4181}a^{29}-\frac{54264}{4181}a^{28}+\frac{87835}{4181}a^{27}-\frac{142133}{4181}a^{26}+\frac{229968}{4181}a^{25}-\frac{372101}{4181}a^{24}+\frac{602069}{4181}a^{23}-\frac{974170}{4181}a^{22}+\frac{1576239}{4181}a^{21}-\frac{2550409}{4181}a^{20}+\frac{4126648}{4181}a^{19}-1597a^{18}+2584a^{17}-\frac{2550409}{4181}a^{16}-\frac{1576239}{4181}a^{15}-\frac{974170}{4181}a^{14}-\frac{1948338}{4181}a^{13}-\frac{372101}{4181}a^{12}-\frac{229968}{4181}a^{11}-\frac{142133}{4181}a^{10}+\frac{229976}{4181}a^{9}-\frac{54298}{4181}a^{8}-\frac{33537}{4181}a^{7}-\frac{20761}{4181}a^{6}-\frac{12776}{4181}a^{5}-\frac{7985}{4181}a^{4}-\frac{4791}{4181}a^{3}-\frac{3194}{4181}a^{2}-\frac{1597}{4181}a-\frac{1597}{4181}$, $\frac{233}{4181}a^{32}+\frac{144}{4181}a^{30}+\frac{34}{4181}a^{28}+\frac{2178309}{4181}a^{13}+\frac{1346269}{4181}a^{11}+\frac{317811}{4181}a^{9}$, $\frac{1597}{4181}a^{35}-\frac{3194}{4181}a^{34}+\frac{5168}{4181}a^{33}-2a^{32}+\frac{12776}{4181}a^{31}-\frac{20761}{4181}a^{30}+8a^{29}-\frac{54264}{4181}a^{28}+\frac{87835}{4181}a^{27}-\frac{142133}{4181}a^{26}+\frac{229976}{4181}a^{25}-\frac{372101}{4181}a^{24}+\frac{602069}{4181}a^{23}-\frac{974170}{4181}a^{22}+\frac{1576238}{4181}a^{21}-\frac{2550409}{4181}a^{20}+\frac{4126648}{4181}a^{19}-1597a^{18}+2584a^{17}-\frac{2550409}{4181}a^{16}-\frac{1576239}{4181}a^{15}+\frac{2550408}{4181}a^{14}-987a^{13}-\frac{372101}{4181}a^{12}-\frac{229968}{4181}a^{11}-233a^{10}+\frac{229976}{4181}a^{9}-\frac{54298}{4181}a^{8}-\frac{33537}{4181}a^{7}+\frac{54264}{4181}a^{6}-\frac{12776}{4181}a^{5}-\frac{7985}{4181}a^{4}-\frac{4791}{4181}a^{3}-\frac{9959}{4181}a^{2}-\frac{1597}{4181}a-\frac{1597}{4181}$, $\frac{233}{4181}a^{32}+\frac{8}{4181}a^{25}-\frac{1}{4181}a^{19}+\frac{2178309}{4181}a^{13}+\frac{75025}{4181}a^{6}-\frac{2584}{4181}$, $\frac{34}{4181}a^{28}+\frac{13}{4181}a^{25}+\frac{2}{4181}a^{22}+\frac{317811}{4181}a^{9}+\frac{121393}{4181}a^{6}+\frac{17711}{4181}a^{3}$, $\frac{5}{4181}a^{24}-\frac{2}{4181}a^{23}+\frac{2}{4181}a^{22}+\frac{46368}{4181}a^{5}-\frac{17711}{4181}a^{4}+\frac{17711}{4181}a^{3}$, $\frac{987}{4181}a^{34}+\frac{55}{4181}a^{29}+\frac{1}{4181}a^{20}+\frac{9227465}{4181}a^{15}+\frac{514229}{4181}a^{10}+\frac{6765}{4181}a$ 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:  \( 3431432369157.3267 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 3431432369157.3267 \cdot 76}{38\cdot\sqrt{114573059505387793044837364496233492772337802886962890625}}\cr\approx \mathstrut & 0.149348826529527 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 + 2*x^34 - 3*x^33 + 5*x^32 - 8*x^31 + 13*x^30 - 21*x^29 + 34*x^28 - 55*x^27 + 89*x^26 - 144*x^25 + 233*x^24 - 377*x^23 + 610*x^22 - 987*x^21 + 1597*x^20 - 2584*x^19 + 4181*x^18 + 2584*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 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 - x^35 + 2*x^34 - 3*x^33 + 5*x^32 - 8*x^31 + 13*x^30 - 21*x^29 + 34*x^28 - 55*x^27 + 89*x^26 - 144*x^25 + 233*x^24 - 377*x^23 + 610*x^22 - 987*x^21 + 1597*x^20 - 2584*x^19 + 4181*x^18 + 2584*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 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 - x^35 + 2*x^34 - 3*x^33 + 5*x^32 - 8*x^31 + 13*x^30 - 21*x^29 + 34*x^28 - 55*x^27 + 89*x^26 - 144*x^25 + 233*x^24 - 377*x^23 + 610*x^22 - 987*x^21 + 1597*x^20 - 2584*x^19 + 4181*x^18 + 2584*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 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 - x^35 + 2*x^34 - 3*x^33 + 5*x^32 - 8*x^31 + 13*x^30 - 21*x^29 + 34*x^28 - 55*x^27 + 89*x^26 - 144*x^25 + 233*x^24 - 377*x^23 + 610*x^22 - 987*x^21 + 1597*x^20 - 2584*x^19 + 4181*x^18 + 2584*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 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

\(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-95}) \), \(\Q(\sqrt{5}) \), 3.3.361.1, \(\Q(\sqrt{5}, \sqrt{-19})\), 6.0.2476099.1, 6.0.309512375.1, 6.6.16290125.1, \(\Q(\zeta_{19})^+\), 12.0.95797910278140625.1, \(\Q(\zeta_{19})\), 18.0.10703880581610941769412109375.1, 18.18.563362135874260093126953125.1

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 $18^{2}$ $18^{2}$ R ${\href{/padicField/7.6.0.1}{6} }^{6}$ ${\href{/padicField/11.3.0.1}{3} }^{12}$ $18^{2}$ $18^{2}$ R $18^{2}$ $18^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{6}$ ${\href{/padicField/37.2.0.1}{2} }^{18}$ $18^{2}$ $18^{2}$ $18^{2}$ $18^{2}$ $18^{2}$

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
\(5\) Copy content Toggle raw display 5.18.9.1$x^{18} + 180 x^{17} + 14445 x^{16} + 679200 x^{15} + 20664900 x^{14} + 423486000 x^{13} + 5887570504 x^{12} + 54397260480 x^{11} + 316143109712 x^{10} + 1034969211206 x^{9} + 1580754753720 x^{8} + 1360718216520 x^{7} + 746510415004 x^{6} + 357128191140 x^{5} + 552895560364 x^{4} + 1314509471572 x^{3} + 1121303668936 x^{2} + 1315877100296 x + 1500010785049$$2$$9$$9$$C_{18}$$[\ ]_{2}^{9}$
5.18.9.1$x^{18} + 180 x^{17} + 14445 x^{16} + 679200 x^{15} + 20664900 x^{14} + 423486000 x^{13} + 5887570504 x^{12} + 54397260480 x^{11} + 316143109712 x^{10} + 1034969211206 x^{9} + 1580754753720 x^{8} + 1360718216520 x^{7} + 746510415004 x^{6} + 357128191140 x^{5} + 552895560364 x^{4} + 1314509471572 x^{3} + 1121303668936 x^{2} + 1315877100296 x + 1500010785049$$2$$9$$9$$C_{18}$$[\ ]_{2}^{9}$
\(19\) Copy content Toggle raw display 19.18.17.14$x^{18} + 19$$18$$1$$17$$C_{18}$$[\ ]_{18}$
19.18.17.14$x^{18} + 19$$18$$1$$17$$C_{18}$$[\ ]_{18}$