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Properties

Label 44.0.681...976.1
Degree $44$
Signature $[0, 22]$
Discriminant $6.820\times 10^{84}$
Root discriminant \(84.73\)
Ramified primes $2,3,23$
Class number not computed
Class group not computed
Galois group $C_2\times C_{22}$ (as 44T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - 42*x^42 + 1004*x^40 - 16416*x^38 + 203056*x^36 - 1976896*x^34 + 15592256*x^32 - 100902784*x^30 + 540972800*x^28 - 2406430208*x^26 + 8885665792*x^24 - 27062865920*x^22 + 67552104448*x^20 - 136133230592*x^18 + 218558169088*x^16 - 271461023744*x^14 + 255644925952*x^12 - 171145035776*x^10 + 80371253248*x^8 - 21592276992*x^6 + 3817865216*x^4 - 138412032*x^2 + 4194304)
 
gp: K = bnfinit(y^44 - 42*y^42 + 1004*y^40 - 16416*y^38 + 203056*y^36 - 1976896*y^34 + 15592256*y^32 - 100902784*y^30 + 540972800*y^28 - 2406430208*y^26 + 8885665792*y^24 - 27062865920*y^22 + 67552104448*y^20 - 136133230592*y^18 + 218558169088*y^16 - 271461023744*y^14 + 255644925952*y^12 - 171145035776*y^10 + 80371253248*y^8 - 21592276992*y^6 + 3817865216*y^4 - 138412032*y^2 + 4194304, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - 42*x^42 + 1004*x^40 - 16416*x^38 + 203056*x^36 - 1976896*x^34 + 15592256*x^32 - 100902784*x^30 + 540972800*x^28 - 2406430208*x^26 + 8885665792*x^24 - 27062865920*x^22 + 67552104448*x^20 - 136133230592*x^18 + 218558169088*x^16 - 271461023744*x^14 + 255644925952*x^12 - 171145035776*x^10 + 80371253248*x^8 - 21592276992*x^6 + 3817865216*x^4 - 138412032*x^2 + 4194304);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - 42*x^42 + 1004*x^40 - 16416*x^38 + 203056*x^36 - 1976896*x^34 + 15592256*x^32 - 100902784*x^30 + 540972800*x^28 - 2406430208*x^26 + 8885665792*x^24 - 27062865920*x^22 + 67552104448*x^20 - 136133230592*x^18 + 218558169088*x^16 - 271461023744*x^14 + 255644925952*x^12 - 171145035776*x^10 + 80371253248*x^8 - 21592276992*x^6 + 3817865216*x^4 - 138412032*x^2 + 4194304)
 

\( x^{44} - 42 x^{42} + 1004 x^{40} - 16416 x^{38} + 203056 x^{36} - 1976896 x^{34} + 15592256 x^{32} + \cdots + 4194304 \) Copy content Toggle raw display

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

Invariants

Degree:  $44$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 22]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(681\!\cdots\!976\) \(\medspace = 2^{66}\cdot 3^{22}\cdot 23^{40}\) 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:  \(84.73\)
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:  $2^{3/2}3^{1/2}23^{10/11}\approx 84.73054838890124$
Ramified primes:   \(2\), \(3\), \(23\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $44$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(552=2^{3}\cdot 3\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{552}(1,·)$, $\chi_{552}(259,·)$, $\chi_{552}(257,·)$, $\chi_{552}(265,·)$, $\chi_{552}(395,·)$, $\chi_{552}(515,·)$, $\chi_{552}(25,·)$, $\chi_{552}(409,·)$, $\chi_{552}(545,·)$, $\chi_{552}(35,·)$, $\chi_{552}(163,·)$, $\chi_{552}(49,·)$, $\chi_{552}(41,·)$, $\chi_{552}(427,·)$, $\chi_{552}(307,·)$, $\chi_{552}(305,·)$, $\chi_{552}(449,·)$, $\chi_{552}(179,·)$, $\chi_{552}(371,·)$, $\chi_{552}(185,·)$, $\chi_{552}(59,·)$, $\chi_{552}(169,·)$, $\chi_{552}(193,·)$, $\chi_{552}(323,·)$, $\chi_{552}(289,·)$, $\chi_{552}(73,·)$, $\chi_{552}(331,·)$, $\chi_{552}(403,·)$, $\chi_{552}(209,·)$, $\chi_{552}(211,·)$, $\chi_{552}(139,·)$, $\chi_{552}(377,·)$, $\chi_{552}(473,·)$, $\chi_{552}(347,·)$, $\chi_{552}(353,·)$, $\chi_{552}(443,·)$, $\chi_{552}(131,·)$, $\chi_{552}(233,·)$, $\chi_{552}(491,·)$, $\chi_{552}(499,·)$, $\chi_{552}(187,·)$, $\chi_{552}(547,·)$, $\chi_{552}(361,·)$, $\chi_{552}(121,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2097152}$

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

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{2048}a^{22}$, $\frac{1}{2048}a^{23}$, $\frac{1}{4096}a^{24}$, $\frac{1}{4096}a^{25}$, $\frac{1}{8192}a^{26}$, $\frac{1}{8192}a^{27}$, $\frac{1}{16384}a^{28}$, $\frac{1}{16384}a^{29}$, $\frac{1}{32768}a^{30}$, $\frac{1}{32768}a^{31}$, $\frac{1}{65536}a^{32}$, $\frac{1}{65536}a^{33}$, $\frac{1}{131072}a^{34}$, $\frac{1}{131072}a^{35}$, $\frac{1}{262144}a^{36}$, $\frac{1}{262144}a^{37}$, $\frac{1}{524288}a^{38}$, $\frac{1}{524288}a^{39}$, $\frac{1}{40373321728}a^{40}-\frac{7487}{10093330432}a^{38}-\frac{7545}{10093330432}a^{36}-\frac{11609}{5046665216}a^{34}+\frac{12115}{2523332608}a^{32}+\frac{247}{78854144}a^{30}+\frac{473}{39427072}a^{28}-\frac{2463}{78854144}a^{26}+\frac{3271}{157708288}a^{24}-\frac{1295}{39427072}a^{22}-\frac{2219}{19713536}a^{20}+\frac{12705}{19713536}a^{18}-\frac{13257}{9856768}a^{16}+\frac{10861}{4928384}a^{14}-\frac{1133}{154012}a^{12}-\frac{5037}{1232096}a^{10}+\frac{8413}{308024}a^{8}-\frac{11491}{308024}a^{6}-\frac{15191}{154012}a^{4}-\frac{10591}{77006}a^{2}+\frac{10893}{38503}$, $\frac{1}{40373321728}a^{41}-\frac{7487}{10093330432}a^{39}-\frac{7545}{10093330432}a^{37}-\frac{11609}{5046665216}a^{35}+\frac{12115}{2523332608}a^{33}+\frac{247}{78854144}a^{31}+\frac{473}{39427072}a^{29}-\frac{2463}{78854144}a^{27}+\frac{3271}{157708288}a^{25}-\frac{1295}{39427072}a^{23}-\frac{2219}{19713536}a^{21}+\frac{12705}{19713536}a^{19}-\frac{13257}{9856768}a^{17}+\frac{10861}{4928384}a^{15}-\frac{1133}{154012}a^{13}-\frac{5037}{1232096}a^{11}+\frac{8413}{308024}a^{9}-\frac{11491}{308024}a^{7}-\frac{15191}{154012}a^{5}-\frac{10591}{77006}a^{3}+\frac{10893}{38503}a$, $\frac{1}{82\!\cdots\!56}a^{42}-\frac{19\!\cdots\!75}{41\!\cdots\!28}a^{40}-\frac{78\!\cdots\!25}{10\!\cdots\!32}a^{38}-\frac{30\!\cdots\!43}{51\!\cdots\!16}a^{36}+\frac{48\!\cdots\!49}{25\!\cdots\!08}a^{34}-\frac{42\!\cdots\!51}{25\!\cdots\!08}a^{32}+\frac{65\!\cdots\!33}{12\!\cdots\!04}a^{30}-\frac{14\!\cdots\!11}{64\!\cdots\!52}a^{28}+\frac{63\!\cdots\!03}{16\!\cdots\!88}a^{26}+\frac{22\!\cdots\!57}{80\!\cdots\!44}a^{24}+\frac{10\!\cdots\!29}{80\!\cdots\!44}a^{22}-\frac{12\!\cdots\!55}{40\!\cdots\!72}a^{20}+\frac{31\!\cdots\!59}{50\!\cdots\!84}a^{18}-\frac{13\!\cdots\!33}{10\!\cdots\!68}a^{16}-\frac{35\!\cdots\!47}{31\!\cdots\!24}a^{14}-\frac{69\!\cdots\!19}{25\!\cdots\!92}a^{12}-\frac{42\!\cdots\!87}{39\!\cdots\!53}a^{10}+\frac{16\!\cdots\!65}{63\!\cdots\!48}a^{8}-\frac{85\!\cdots\!17}{15\!\cdots\!12}a^{6}-\frac{14\!\cdots\!05}{15\!\cdots\!12}a^{4}-\frac{85\!\cdots\!19}{39\!\cdots\!53}a^{2}+\frac{18\!\cdots\!21}{39\!\cdots\!53}$, $\frac{1}{82\!\cdots\!56}a^{43}-\frac{19\!\cdots\!75}{41\!\cdots\!28}a^{41}-\frac{78\!\cdots\!25}{10\!\cdots\!32}a^{39}-\frac{30\!\cdots\!43}{51\!\cdots\!16}a^{37}+\frac{48\!\cdots\!49}{25\!\cdots\!08}a^{35}-\frac{42\!\cdots\!51}{25\!\cdots\!08}a^{33}+\frac{65\!\cdots\!33}{12\!\cdots\!04}a^{31}-\frac{14\!\cdots\!11}{64\!\cdots\!52}a^{29}+\frac{63\!\cdots\!03}{16\!\cdots\!88}a^{27}+\frac{22\!\cdots\!57}{80\!\cdots\!44}a^{25}+\frac{10\!\cdots\!29}{80\!\cdots\!44}a^{23}-\frac{12\!\cdots\!55}{40\!\cdots\!72}a^{21}+\frac{31\!\cdots\!59}{50\!\cdots\!84}a^{19}-\frac{13\!\cdots\!33}{10\!\cdots\!68}a^{17}-\frac{35\!\cdots\!47}{31\!\cdots\!24}a^{15}-\frac{69\!\cdots\!19}{25\!\cdots\!92}a^{13}-\frac{42\!\cdots\!87}{39\!\cdots\!53}a^{11}+\frac{16\!\cdots\!65}{63\!\cdots\!48}a^{9}-\frac{85\!\cdots\!17}{15\!\cdots\!12}a^{7}-\frac{14\!\cdots\!05}{15\!\cdots\!12}a^{5}-\frac{85\!\cdots\!19}{39\!\cdots\!53}a^{3}+\frac{18\!\cdots\!21}{39\!\cdots\!53}a$ 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

not computed

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:  $21$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{414556177818937525350558483019015320605052303}{41471323485685225376340127941048825823595706924924928} a^{42} + \frac{17385663921699872161341757065009211580446431211}{41471323485685225376340127941048825823595706924924928} a^{40} - \frac{207571435231570322670936516059315369508076226499}{20735661742842612688170063970524412911797853462462464} a^{38} + \frac{423742173509196008995238865786615125793968494271}{2591957717855326586021257996315551613974731682807808} a^{36} - \frac{10470482522236172424103405482756392899998492353119}{5183915435710653172042515992631103227949463365615616} a^{34} + \frac{6362807107977430776479524171323280007627947299131}{323994714731915823252657249539443951746841460350976} a^{32} - \frac{100227662715035713281310453796142886276182769225661}{647989429463831646505314499078887903493682920701952} a^{30} + \frac{161889454796178142746571197659340991714810819002625}{161997357365957911626328624769721975873420730175488} a^{28} - \frac{1732686555624062704122396534404865341348735205808011}{323994714731915823252657249539443951746841460350976} a^{26} + \frac{1922700536593001496491465570338732439893212888516475}{80998678682978955813164312384860987936710365087744} a^{24} - \frac{7081243878823882957966445691871830577051513278050789}{80998678682978955813164312384860987936710365087744} a^{22} + \frac{10749514882813051417553785399715272103359330343705585}{40499339341489477906582156192430493968355182543872} a^{20} - \frac{13364002198568950540869434595421413010164419072808287}{20249669670744738953291078096215246984177591271936} a^{18} + \frac{13398445484858974345060546099950451427258482198520615}{10124834835372369476645539048107623492088795635968} a^{16} - \frac{5343417171040549871852619428971842726985846840924477}{2531208708843092369161384762026905873022198908992} a^{14} + \frac{6578200791580392859217287085261064583363924327001495}{2531208708843092369161384762026905873022198908992} a^{12} - \frac{3061668823017677054286700470270645279848404039106171}{1265604354421546184580692381013452936511099454496} a^{10} + \frac{62876845523727485055108361330912641191341407209893}{39550136075673318268146636906670404265971857953} a^{8} - \frac{28883149982394774867804473309861945316069853721436}{39550136075673318268146636906670404265971857953} a^{6} + \frac{106108068317830159407054267426559430134303497265}{571121098565679686182622915619789231277571956} a^{4} - \frac{2664524270969116664641531806015314757346935267351}{79100272151346636536293273813340808531943715906} a^{2} + \frac{48262296360680219822595451880150324418269390595}{39550136075673318268146636906670404265971857953} \)  (order $6$) 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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^44 - 42*x^42 + 1004*x^40 - 16416*x^38 + 203056*x^36 - 1976896*x^34 + 15592256*x^32 - 100902784*x^30 + 540972800*x^28 - 2406430208*x^26 + 8885665792*x^24 - 27062865920*x^22 + 67552104448*x^20 - 136133230592*x^18 + 218558169088*x^16 - 271461023744*x^14 + 255644925952*x^12 - 171145035776*x^10 + 80371253248*x^8 - 21592276992*x^6 + 3817865216*x^4 - 138412032*x^2 + 4194304)
 
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^44 - 42*x^42 + 1004*x^40 - 16416*x^38 + 203056*x^36 - 1976896*x^34 + 15592256*x^32 - 100902784*x^30 + 540972800*x^28 - 2406430208*x^26 + 8885665792*x^24 - 27062865920*x^22 + 67552104448*x^20 - 136133230592*x^18 + 218558169088*x^16 - 271461023744*x^14 + 255644925952*x^12 - 171145035776*x^10 + 80371253248*x^8 - 21592276992*x^6 + 3817865216*x^4 - 138412032*x^2 + 4194304, 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^44 - 42*x^42 + 1004*x^40 - 16416*x^38 + 203056*x^36 - 1976896*x^34 + 15592256*x^32 - 100902784*x^30 + 540972800*x^28 - 2406430208*x^26 + 8885665792*x^24 - 27062865920*x^22 + 67552104448*x^20 - 136133230592*x^18 + 218558169088*x^16 - 271461023744*x^14 + 255644925952*x^12 - 171145035776*x^10 + 80371253248*x^8 - 21592276992*x^6 + 3817865216*x^4 - 138412032*x^2 + 4194304);
 
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^44 - 42*x^42 + 1004*x^40 - 16416*x^38 + 203056*x^36 - 1976896*x^34 + 15592256*x^32 - 100902784*x^30 + 540972800*x^28 - 2406430208*x^26 + 8885665792*x^24 - 27062865920*x^22 + 67552104448*x^20 - 136133230592*x^18 + 218558169088*x^16 - 271461023744*x^14 + 255644925952*x^12 - 171145035776*x^10 + 80371253248*x^8 - 21592276992*x^6 + 3817865216*x^4 - 138412032*x^2 + 4194304);
 
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

$C_2\times C_{22}$ (as 44T2):

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 44
The 44 conjugacy class representatives for $C_2\times C_{22}$
Character table for $C_2\times C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\zeta_{23})^+\), 22.0.14741666340843480753092741810452692992.1, 22.22.2611441967281400084968119933496263205453824.1, 22.0.304011857053427966889939263171547.1

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 $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/padicField/19.11.0.1}{11} }^{4}$ R $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/padicField/43.11.0.1}{11} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{22}$ $22^{2}$ $22^{2}$

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 Deg $44$$2$$22$$66$
\(3\) Copy content Toggle raw display 3.22.11.2$x^{22} + 33 x^{20} + 495 x^{18} + 4455 x^{16} + 26730 x^{14} + 4 x^{13} + 112266 x^{12} - 406 x^{11} + 336798 x^{10} + 1650 x^{9} + 721710 x^{8} + 20196 x^{7} + 1082565 x^{6} - 35640 x^{5} + 1082569 x^{4} - 37422 x^{3} + 649567 x^{2} + 20898 x + 177172$$2$$11$$11$22T1$[\ ]_{2}^{11}$
3.22.11.2$x^{22} + 33 x^{20} + 495 x^{18} + 4455 x^{16} + 26730 x^{14} + 4 x^{13} + 112266 x^{12} - 406 x^{11} + 336798 x^{10} + 1650 x^{9} + 721710 x^{8} + 20196 x^{7} + 1082565 x^{6} - 35640 x^{5} + 1082569 x^{4} - 37422 x^{3} + 649567 x^{2} + 20898 x + 177172$$2$$11$$11$22T1$[\ ]_{2}^{11}$
\(23\) Copy content Toggle raw display 23.22.20.1$x^{22} + 231 x^{21} + 24310 x^{20} + 1539615 x^{19} + 65271580 x^{18} + 1948237137 x^{17} + 41892053577 x^{16} + 652037968890 x^{15} + 7265319209115 x^{14} + 56290884204555 x^{13} + 287278948122936 x^{12} + 881069580352567 x^{11} + 1436394740619993 x^{10} + 1407272105659090 x^{9} + 908164935089445 x^{8} + 407525140102740 x^{7} + 130953632793951 x^{6} + 31291611122541 x^{5} + 17709125163290 x^{4} + 131485551467820 x^{3} + 905728303498040 x^{2} + 3760233666129578 x + 7096289029978197$$11$$2$$20$22T1$[\ ]_{11}^{2}$
23.22.20.1$x^{22} + 231 x^{21} + 24310 x^{20} + 1539615 x^{19} + 65271580 x^{18} + 1948237137 x^{17} + 41892053577 x^{16} + 652037968890 x^{15} + 7265319209115 x^{14} + 56290884204555 x^{13} + 287278948122936 x^{12} + 881069580352567 x^{11} + 1436394740619993 x^{10} + 1407272105659090 x^{9} + 908164935089445 x^{8} + 407525140102740 x^{7} + 130953632793951 x^{6} + 31291611122541 x^{5} + 17709125163290 x^{4} + 131485551467820 x^{3} + 905728303498040 x^{2} + 3760233666129578 x + 7096289029978197$$11$$2$$20$22T1$[\ ]_{11}^{2}$