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Properties

Label 14.14.838...952.1
Degree $14$
Signature $[14, 0]$
Discriminant $8.380\times 10^{25}$
Root discriminant \(71.07\)
Ramified primes $2,43$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{14}$ (as 14T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 49*x^12 + 130*x^11 + 690*x^10 - 2164*x^9 - 3225*x^8 + 13320*x^7 + 86*x^6 - 26074*x^5 + 20214*x^4 - 1996*x^3 - 1359*x^2 + 78*x + 7)
 
gp: K = bnfinit(y^14 - 2*y^13 - 49*y^12 + 130*y^11 + 690*y^10 - 2164*y^9 - 3225*y^8 + 13320*y^7 + 86*y^6 - 26074*y^5 + 20214*y^4 - 1996*y^3 - 1359*y^2 + 78*y + 7, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 2*x^13 - 49*x^12 + 130*x^11 + 690*x^10 - 2164*x^9 - 3225*x^8 + 13320*x^7 + 86*x^6 - 26074*x^5 + 20214*x^4 - 1996*x^3 - 1359*x^2 + 78*x + 7);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 2*x^13 - 49*x^12 + 130*x^11 + 690*x^10 - 2164*x^9 - 3225*x^8 + 13320*x^7 + 86*x^6 - 26074*x^5 + 20214*x^4 - 1996*x^3 - 1359*x^2 + 78*x + 7)
 

\( x^{14} - 2 x^{13} - 49 x^{12} + 130 x^{11} + 690 x^{10} - 2164 x^{9} - 3225 x^{8} + 13320 x^{7} + \cdots + 7 \) Copy content Toggle raw display

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

Invariants

Degree:  $14$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[14, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(83801419645740806624509952\) \(\medspace = 2^{21}\cdot 43^{12}\) 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:  \(71.07\)
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:  $2^{3/2}43^{6/7}\approx 71.06582671110044$
Ramified primes:   \(2\), \(43\) 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:  \(\Q(\sqrt{2}) \)
$\card{ \Gal(K/\Q) }$:  $14$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(344=2^{3}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{344}(1,·)$, $\chi_{344}(133,·)$, $\chi_{344}(97,·)$, $\chi_{344}(145,·)$, $\chi_{344}(41,·)$, $\chi_{344}(193,·)$, $\chi_{344}(269,·)$, $\chi_{344}(173,·)$, $\chi_{344}(305,·)$, $\chi_{344}(213,·)$, $\chi_{344}(121,·)$, $\chi_{344}(293,·)$, $\chi_{344}(317,·)$, $\chi_{344}(21,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}-\frac{1}{7}a^{5}+\frac{1}{7}a^{4}-\frac{1}{7}a^{3}+\frac{1}{7}a^{2}-\frac{1}{7}a$, $\frac{1}{7}a^{7}-\frac{1}{7}a$, $\frac{1}{7}a^{8}-\frac{1}{7}a^{2}$, $\frac{1}{7}a^{9}-\frac{1}{7}a^{3}$, $\frac{1}{49}a^{10}+\frac{3}{49}a^{9}-\frac{3}{49}a^{8}+\frac{2}{49}a^{7}+\frac{1}{49}a^{6}-\frac{15}{49}a^{5}-\frac{2}{7}a^{4}+\frac{17}{49}a^{3}+\frac{18}{49}a^{2}+\frac{11}{49}a-\frac{3}{7}$, $\frac{1}{49}a^{11}+\frac{2}{49}a^{9}-\frac{3}{49}a^{8}+\frac{2}{49}a^{7}+\frac{3}{49}a^{6}+\frac{10}{49}a^{5}-\frac{18}{49}a^{4}-\frac{19}{49}a^{3}-\frac{8}{49}a^{2}+\frac{16}{49}a+\frac{2}{7}$, $\frac{1}{343}a^{12}-\frac{2}{343}a^{11}+\frac{2}{343}a^{10}-\frac{2}{49}a^{9}+\frac{8}{343}a^{8}-\frac{22}{343}a^{7}+\frac{4}{343}a^{6}+\frac{158}{343}a^{5}+\frac{17}{343}a^{4}-\frac{110}{343}a^{3}-\frac{164}{343}a^{2}+\frac{3}{343}a+\frac{17}{49}$, $\frac{1}{26431843081}a^{13}+\frac{4344747}{26431843081}a^{12}-\frac{47361633}{26431843081}a^{11}+\frac{110836676}{26431843081}a^{10}+\frac{720011363}{26431843081}a^{9}+\frac{1762804521}{26431843081}a^{8}+\frac{852064088}{26431843081}a^{7}+\frac{261596610}{26431843081}a^{6}+\frac{5208466355}{26431843081}a^{5}-\frac{364560010}{26431843081}a^{4}+\frac{7921916222}{26431843081}a^{3}-\frac{7352728186}{26431843081}a^{2}-\frac{9948901142}{26431843081}a+\frac{249168117}{3775977583}$ 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:  $7$

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:  $13$
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:   $\frac{213760802}{26431843081}a^{13}-\frac{377163042}{26431843081}a^{12}-\frac{10508174836}{26431843081}a^{11}+\frac{25234705479}{26431843081}a^{10}+\frac{150729235828}{26431843081}a^{9}-\frac{421532129831}{26431843081}a^{8}-\frac{748800010606}{26431843081}a^{7}+\frac{2578266675298}{26431843081}a^{6}+\frac{408518351286}{26431843081}a^{5}-\frac{4912893246074}{26431843081}a^{4}+\frac{3451320188878}{26431843081}a^{3}-\frac{717479568089}{26431843081}a^{2}+\frac{14193417209}{26431843081}a+\frac{1995251138}{3775977583}$, $\frac{273901106}{26431843081}a^{13}-\frac{692214408}{26431843081}a^{12}-\frac{13254956314}{26431843081}a^{11}+\frac{42779938947}{26431843081}a^{10}+\frac{176266910452}{26431843081}a^{9}-\frac{701182639844}{26431843081}a^{8}-\frac{662586876424}{26431843081}a^{7}+\frac{4274778165223}{26431843081}a^{6}-\frac{1364526806472}{26431843081}a^{5}-\frac{8204210497058}{26431843081}a^{4}+\frac{8361698193942}{26431843081}a^{3}-\frac{1216111332398}{26431843081}a^{2}-\frac{572613680206}{26431843081}a+\frac{6434598449}{3775977583}$, $\frac{925165984}{26431843081}a^{13}-\frac{1695190356}{26431843081}a^{12}-\frac{45579377722}{26431843081}a^{11}+\frac{112482896826}{26431843081}a^{10}+\frac{655202429834}{26431843081}a^{9}-\frac{1884116561892}{26431843081}a^{8}-\frac{3268723119864}{26431843081}a^{7}+\frac{11646358221668}{26431843081}a^{6}+\frac{1866289677120}{26431843081}a^{5}-\frac{23030697925980}{26431843081}a^{4}+\frac{14978703054626}{26431843081}a^{3}-\frac{865206096915}{26431843081}a^{2}-\frac{663839917326}{26431843081}a+\frac{9517416540}{3775977583}$, $\frac{198463628}{26431843081}a^{13}-\frac{436536598}{26431843081}a^{12}-\frac{9727315244}{26431843081}a^{11}+\frac{27772460486}{26431843081}a^{10}+\frac{135825733392}{26431843081}a^{9}-\frac{461104726959}{26431843081}a^{8}-\frac{616534341984}{26431843081}a^{7}+\frac{2854777513222}{26431843081}a^{6}-\frac{136079418000}{26431843081}a^{5}-\frac{5732815054166}{26431843081}a^{4}+\frac{4328844345732}{26431843081}a^{3}-\frac{33033749036}{26431843081}a^{2}-\frac{396270477444}{26431843081}a-\frac{4574737232}{3775977583}$, $\frac{230199070}{26431843081}a^{13}-\frac{565744281}{26431843081}a^{12}-\frac{11107168490}{26431843081}a^{11}+\frac{35110587957}{26431843081}a^{10}+\frac{147001301464}{26431843081}a^{9}-\frac{573239955793}{26431843081}a^{8}-\frac{542032646264}{26431843081}a^{7}+\frac{3450845057435}{26431843081}a^{6}-\frac{1232343233076}{26431843081}a^{5}-\frac{6307182950266}{26431843081}a^{4}+\frac{7204568358626}{26431843081}a^{3}-\frac{1949314043422}{26431843081}a^{2}-\frac{402725502362}{26431843081}a+\frac{21732339026}{3775977583}$, $\frac{213760802}{26431843081}a^{13}-\frac{377163042}{26431843081}a^{12}-\frac{10508174836}{26431843081}a^{11}+\frac{25234705479}{26431843081}a^{10}+\frac{150729235828}{26431843081}a^{9}-\frac{421532129831}{26431843081}a^{8}-\frac{748800010606}{26431843081}a^{7}+\frac{2578266675298}{26431843081}a^{6}+\frac{408518351286}{26431843081}a^{5}-\frac{4912893246074}{26431843081}a^{4}+\frac{3451320188878}{26431843081}a^{3}-\frac{717479568089}{26431843081}a^{2}+\frac{40625260290}{26431843081}a-\frac{1780726445}{3775977583}$, $\frac{101612802}{26431843081}a^{13}-\frac{373403411}{26431843081}a^{12}-\frac{4699493906}{26431843081}a^{11}+\frac{21626369971}{26431843081}a^{10}+\frac{51010859274}{26431843081}a^{9}-\frac{342929512841}{26431843081}a^{8}-\frac{4894718230}{26431843081}a^{7}+\frac{1996118583754}{26431843081}a^{6}-\frac{1993734277986}{26431843081}a^{5}-\frac{3239140633573}{26431843081}a^{4}+\frac{6014986732088}{26431843081}a^{3}-\frac{2462329644932}{26431843081}a^{2}-\frac{8573643280}{26431843081}a+\frac{674751799}{3775977583}$, $\frac{796488508}{26431843081}a^{13}-\frac{1686191482}{26431843081}a^{12}-\frac{38899053125}{26431843081}a^{11}+\frac{108117023633}{26431843081}a^{10}+\frac{540258976023}{26431843081}a^{9}-\frac{1790336457622}{26431843081}a^{8}-\frac{2409750794545}{26431843081}a^{7}+\frac{10962706977783}{26431843081}a^{6}-\frac{918626157228}{26431843081}a^{5}-\frac{21164502170635}{26431843081}a^{4}+\frac{18067958234842}{26431843081}a^{3}-\frac{2529829212179}{26431843081}a^{2}-\frac{1120801022081}{26431843081}a+\frac{10364389309}{3775977583}$, $\frac{2716681643}{26431843081}a^{13}-\frac{5164654288}{26431843081}a^{12}-\frac{133744583079}{26431843081}a^{11}+\frac{339940195115}{26431843081}a^{10}+\frac{1913690342267}{26431843081}a^{9}-\frac{5693821628445}{26431843081}a^{8}-\frac{9406246475167}{26431843081}a^{7}+\frac{35346919605852}{26431843081}a^{6}+\frac{4180992405139}{26431843081}a^{5}-\frac{71056442918116}{26431843081}a^{4}+\frac{47221198141136}{26431843081}a^{3}+\frac{650140434962}{26431843081}a^{2}-\frac{4189696615642}{26431843081}a-\frac{29477972642}{3775977583}$, $\frac{597182086}{26431843081}a^{13}-\frac{1819317832}{26431843081}a^{12}-\frac{28545341337}{26431843081}a^{11}+\frac{108893866224}{26431843081}a^{10}+\frac{357421170166}{26431843081}a^{9}-\frac{1771521047302}{26431843081}a^{8}-\frac{978384878439}{26431843081}a^{7}+\frac{10800774937541}{26431843081}a^{6}-\frac{5912342677097}{26431843081}a^{5}-\frac{20852830051912}{26431843081}a^{4}+\frac{24283806384248}{26431843081}a^{3}-\frac{3133377928985}{26431843081}a^{2}-\frac{2072460606453}{26431843081}a-\frac{5504553116}{3775977583}$, $\frac{2252093259}{26431843081}a^{13}-\frac{4824678681}{26431843081}a^{12}-\frac{109787357349}{26431843081}a^{11}+\frac{308599921571}{26431843081}a^{10}+\frac{1516174754727}{26431843081}a^{9}-\frac{5102952458927}{26431843081}a^{8}-\frac{6630483047493}{26431843081}a^{7}+\frac{31172130698316}{26431843081}a^{6}-\frac{3709935437044}{26431843081}a^{5}-\frac{59630887785772}{26431843081}a^{4}+\frac{53242682351609}{26431843081}a^{3}-\frac{9183348565402}{26431843081}a^{2}-\frac{2876041450360}{26431843081}a+\frac{68396978278}{3775977583}$, $\frac{618313987}{26431843081}a^{13}-\frac{948984459}{26431843081}a^{12}-\frac{30689838916}{26431843081}a^{11}+\frac{65982515294}{26431843081}a^{10}+\frac{455009417986}{26431843081}a^{9}-\frac{1119386459547}{26431843081}a^{8}-\frac{2484430801800}{26431843081}a^{7}+\frac{6981120898718}{26431843081}a^{6}+\frac{3154227384893}{26431843081}a^{5}-\frac{14141009269105}{26431843081}a^{4}+\frac{6117494680841}{26431843081}a^{3}+\frac{737501536475}{26431843081}a^{2}-\frac{245298330766}{26431843081}a+\frac{420890779}{3775977583}$, $\frac{573671241}{26431843081}a^{13}-\frac{1051686857}{26431843081}a^{12}-\frac{28202138081}{26431843081}a^{11}+\frac{69924130336}{26431843081}a^{10}+\frac{403650710823}{26431843081}a^{9}-\frac{1173395572252}{26431843081}a^{8}-\frac{1990392189748}{26431843081}a^{7}+\frac{7281041908461}{26431843081}a^{6}+\frac{949808191573}{26431843081}a^{5}-\frac{14579365036889}{26431843081}a^{4}+\frac{9755634471459}{26431843081}a^{3}-\frac{49896535637}{26431843081}a^{2}-\frac{776407295637}{26431843081}a-\frac{5756289477}{3775977583}$ 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:  \( 1198577302.5055304 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{0}\cdot 1198577302.5055304 \cdot 1}{2\cdot\sqrt{83801419645740806624509952}}\cr\approx \mathstrut & 1.07258150725833 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 49*x^12 + 130*x^11 + 690*x^10 - 2164*x^9 - 3225*x^8 + 13320*x^7 + 86*x^6 - 26074*x^5 + 20214*x^4 - 1996*x^3 - 1359*x^2 + 78*x + 7)
 
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^14 - 2*x^13 - 49*x^12 + 130*x^11 + 690*x^10 - 2164*x^9 - 3225*x^8 + 13320*x^7 + 86*x^6 - 26074*x^5 + 20214*x^4 - 1996*x^3 - 1359*x^2 + 78*x + 7, 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^14 - 2*x^13 - 49*x^12 + 130*x^11 + 690*x^10 - 2164*x^9 - 3225*x^8 + 13320*x^7 + 86*x^6 - 26074*x^5 + 20214*x^4 - 1996*x^3 - 1359*x^2 + 78*x + 7);
 
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^14 - 2*x^13 - 49*x^12 + 130*x^11 + 690*x^10 - 2164*x^9 - 3225*x^8 + 13320*x^7 + 86*x^6 - 26074*x^5 + 20214*x^4 - 1996*x^3 - 1359*x^2 + 78*x + 7);
 
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

$C_{14}$ (as 14T1):

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 14
The 14 conjugacy class representatives for $C_{14}$
Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{2}) \), 7.7.6321363049.1

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 ${\href{/padicField/3.14.0.1}{14} }$ ${\href{/padicField/5.14.0.1}{14} }$ ${\href{/padicField/7.1.0.1}{1} }^{14}$ ${\href{/padicField/11.14.0.1}{14} }$ ${\href{/padicField/13.14.0.1}{14} }$ ${\href{/padicField/17.7.0.1}{7} }^{2}$ ${\href{/padicField/19.14.0.1}{14} }$ ${\href{/padicField/23.7.0.1}{7} }^{2}$ ${\href{/padicField/29.14.0.1}{14} }$ ${\href{/padicField/31.7.0.1}{7} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{7}$ ${\href{/padicField/41.7.0.1}{7} }^{2}$ R ${\href{/padicField/47.7.0.1}{7} }^{2}$ ${\href{/padicField/53.14.0.1}{14} }$ ${\href{/padicField/59.14.0.1}{14} }$

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 2.14.21.34$x^{14} + 4 x^{13} + 14 x^{12} + 656 x^{11} + 1236 x^{10} - 43472 x^{9} - 244456 x^{8} + 434816 x^{7} + 8570160 x^{6} + 35893184 x^{5} + 77018272 x^{4} + 105671936 x^{3} + 121134528 x^{2} + 74194176 x - 52979584$$2$$7$$21$$C_{14}$$[3]^{7}$
\(43\) Copy content Toggle raw display 43.14.12.1$x^{14} + 294 x^{13} + 37065 x^{12} + 2598372 x^{11} + 109465209 x^{10} + 2775672522 x^{9} + 39406741353 x^{8} + 247146613646 x^{7} + 118220236701 x^{6} + 24982640172 x^{5} + 3066494193 x^{4} + 4861739862 x^{3} + 117021374001 x^{2} + 1635382592172 x + 9795578218654$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$