↑

Properties

Label 27.27.481...441.1
Degree $27$
Signature $(27, 0)$
Discriminant $4.810\times 10^{61}$
Root discriminant \(192.54\)
Ramified primes $3,7,13$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_9.C_3^2$ (as 27T28)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^27 - 297*x^25 - 333*x^24 + 36072*x^23 + 61137*x^22 - 2408229*x^21 - 4602204*x^20 + 100240344*x^19 + 188516023*x^18 - 2781715041*x^17 - 4662330939*x^16 + 53254728639*x^15 + 72347832963*x^14 - 710593217208*x^13 - 702999577791*x^12 + 6541339948164*x^11 + 4112713341072*x^10 - 40351928688652*x^9 - 13288812632739*x^8 + 159103682965773*x^7 + 20338350996222*x^6 - 374458043565018*x^5 - 13448413508607*x^4 + 467216524743780*x^3 + 21089835153009*x^2 - 231674121736407*x - 25404456460733)
 
Copy content gp:K = bnfinit(y^27 - 297*y^25 - 333*y^24 + 36072*y^23 + 61137*y^22 - 2408229*y^21 - 4602204*y^20 + 100240344*y^19 + 188516023*y^18 - 2781715041*y^17 - 4662330939*y^16 + 53254728639*y^15 + 72347832963*y^14 - 710593217208*y^13 - 702999577791*y^12 + 6541339948164*y^11 + 4112713341072*y^10 - 40351928688652*y^9 - 13288812632739*y^8 + 159103682965773*y^7 + 20338350996222*y^6 - 374458043565018*y^5 - 13448413508607*y^4 + 467216524743780*y^3 + 21089835153009*y^2 - 231674121736407*y - 25404456460733, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 297*x^25 - 333*x^24 + 36072*x^23 + 61137*x^22 - 2408229*x^21 - 4602204*x^20 + 100240344*x^19 + 188516023*x^18 - 2781715041*x^17 - 4662330939*x^16 + 53254728639*x^15 + 72347832963*x^14 - 710593217208*x^13 - 702999577791*x^12 + 6541339948164*x^11 + 4112713341072*x^10 - 40351928688652*x^9 - 13288812632739*x^8 + 159103682965773*x^7 + 20338350996222*x^6 - 374458043565018*x^5 - 13448413508607*x^4 + 467216524743780*x^3 + 21089835153009*x^2 - 231674121736407*x - 25404456460733);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^27 - 297*x^25 - 333*x^24 + 36072*x^23 + 61137*x^22 - 2408229*x^21 - 4602204*x^20 + 100240344*x^19 + 188516023*x^18 - 2781715041*x^17 - 4662330939*x^16 + 53254728639*x^15 + 72347832963*x^14 - 710593217208*x^13 - 702999577791*x^12 + 6541339948164*x^11 + 4112713341072*x^10 - 40351928688652*x^9 - 13288812632739*x^8 + 159103682965773*x^7 + 20338350996222*x^6 - 374458043565018*x^5 - 13448413508607*x^4 + 467216524743780*x^3 + 21089835153009*x^2 - 231674121736407*x - 25404456460733)
 

\( x^{27} - 297 x^{25} - 333 x^{24} + 36072 x^{23} + 61137 x^{22} - 2408229 x^{21} - 4602204 x^{20} + \cdots - 25404456460733 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $27$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $(27, 0)$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(48101612257836942453739988389275689559050655489929124517167441\) \(\medspace = 3^{66}\cdot 7^{12}\cdot 13^{18}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(192.54\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $3^{22/9}7^{2/3}13^{2/3}\approx 296.70440583414893$
Ramified primes:   \(3\), \(7\), \(13\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_9$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}-\frac{3}{7}a^{10}+\frac{3}{7}a^{9}+\frac{1}{7}a^{8}-\frac{1}{7}a^{7}+\frac{2}{7}a^{6}+\frac{2}{7}a^{5}+\frac{1}{7}a^{4}+\frac{3}{7}a^{3}$, $\frac{1}{7}a^{13}-\frac{3}{7}a^{11}+\frac{3}{7}a^{10}+\frac{1}{7}a^{9}-\frac{1}{7}a^{8}+\frac{2}{7}a^{7}+\frac{2}{7}a^{6}+\frac{1}{7}a^{5}+\frac{3}{7}a^{4}$, $\frac{1}{7}a^{14}+\frac{3}{7}a^{11}-\frac{1}{7}a^{10}+\frac{1}{7}a^{9}-\frac{2}{7}a^{8}-\frac{1}{7}a^{7}+\frac{2}{7}a^{5}+\frac{3}{7}a^{4}+\frac{2}{7}a^{3}$, $\frac{1}{49}a^{15}-\frac{3}{49}a^{13}+\frac{3}{49}a^{12}+\frac{8}{49}a^{11}+\frac{6}{49}a^{10}+\frac{2}{49}a^{9}+\frac{16}{49}a^{8}+\frac{22}{49}a^{7}+\frac{24}{49}a^{6}-\frac{1}{7}a^{4}+\frac{2}{7}a^{3}$, $\frac{1}{49}a^{16}-\frac{3}{49}a^{14}+\frac{3}{49}a^{13}+\frac{1}{49}a^{12}+\frac{6}{49}a^{11}+\frac{23}{49}a^{10}-\frac{5}{49}a^{9}+\frac{15}{49}a^{8}-\frac{18}{49}a^{7}-\frac{2}{7}a^{6}-\frac{3}{7}a^{5}+\frac{1}{7}a^{4}-\frac{3}{7}a^{3}$, $\frac{1}{49}a^{17}+\frac{3}{49}a^{14}-\frac{1}{49}a^{13}+\frac{1}{49}a^{12}-\frac{23}{49}a^{11}-\frac{22}{49}a^{10}-\frac{2}{7}a^{9}+\frac{9}{49}a^{8}-\frac{18}{49}a^{7}-\frac{12}{49}a^{6}-\frac{2}{7}a^{5}+\frac{2}{7}a^{4}$, $\frac{1}{343}a^{18}-\frac{3}{343}a^{16}+\frac{3}{343}a^{15}+\frac{8}{343}a^{14}+\frac{6}{343}a^{13}+\frac{2}{343}a^{12}+\frac{114}{343}a^{11}-\frac{125}{343}a^{10}+\frac{171}{343}a^{9}+\frac{3}{7}a^{8}+\frac{6}{49}a^{7}-\frac{5}{49}a^{6}+\frac{2}{7}a^{5}+\frac{3}{7}a^{4}+\frac{1}{7}a^{3}$, $\frac{1}{343}a^{19}-\frac{3}{343}a^{17}+\frac{3}{343}a^{16}+\frac{1}{343}a^{15}+\frac{6}{343}a^{14}+\frac{23}{343}a^{13}-\frac{5}{343}a^{12}+\frac{162}{343}a^{11}+\frac{80}{343}a^{10}-\frac{23}{49}a^{9}-\frac{24}{49}a^{8}-\frac{13}{49}a^{7}+\frac{11}{49}a^{6}-\frac{1}{7}a^{5}-\frac{1}{7}a^{3}$, $\frac{1}{343}a^{20}+\frac{3}{343}a^{17}-\frac{1}{343}a^{16}+\frac{1}{343}a^{15}-\frac{23}{343}a^{14}-\frac{22}{343}a^{13}-\frac{2}{49}a^{12}+\frac{156}{343}a^{11}+\frac{80}{343}a^{10}+\frac{37}{343}a^{9}-\frac{9}{49}a^{8}+\frac{16}{49}a^{7}-\frac{1}{7}a^{6}-\frac{1}{7}a^{4}-\frac{1}{7}a^{3}$, $\frac{1}{2401}a^{21}-\frac{3}{2401}a^{19}+\frac{3}{2401}a^{18}+\frac{8}{2401}a^{17}+\frac{6}{2401}a^{16}+\frac{2}{2401}a^{15}+\frac{114}{2401}a^{14}-\frac{125}{2401}a^{13}+\frac{171}{2401}a^{12}+\frac{24}{49}a^{11}-\frac{43}{343}a^{10}-\frac{103}{343}a^{9}-\frac{5}{49}a^{8}+\frac{3}{49}a^{7}-\frac{6}{49}a^{6}-\frac{1}{7}a^{4}-\frac{1}{7}a^{3}$, $\frac{1}{2401}a^{22}-\frac{3}{2401}a^{20}+\frac{3}{2401}a^{19}+\frac{1}{2401}a^{18}+\frac{6}{2401}a^{17}+\frac{23}{2401}a^{16}-\frac{5}{2401}a^{15}+\frac{162}{2401}a^{14}+\frac{80}{2401}a^{13}-\frac{23}{343}a^{12}+\frac{25}{343}a^{11}-\frac{160}{343}a^{10}+\frac{11}{343}a^{9}+\frac{20}{49}a^{8}-\frac{1}{7}a^{7}-\frac{1}{49}a^{6}-\frac{1}{7}a^{5}+\frac{2}{7}a^{4}+\frac{2}{7}a^{3}$, $\frac{1}{2401}a^{23}+\frac{3}{2401}a^{20}-\frac{1}{2401}a^{19}+\frac{1}{2401}a^{18}-\frac{23}{2401}a^{17}-\frac{22}{2401}a^{16}-\frac{2}{343}a^{15}+\frac{156}{2401}a^{14}+\frac{80}{2401}a^{13}+\frac{37}{2401}a^{12}-\frac{107}{343}a^{11}+\frac{114}{343}a^{10}+\frac{20}{49}a^{9}-\frac{3}{7}a^{8}-\frac{8}{49}a^{7}-\frac{22}{49}a^{6}-\frac{3}{7}a^{5}-\frac{1}{7}a^{4}-\frac{1}{7}a^{3}$, $\frac{1}{3042067}a^{24}+\frac{26}{434581}a^{23}-\frac{563}{3042067}a^{22}+\frac{549}{3042067}a^{21}-\frac{853}{3042067}a^{20}+\frac{1455}{3042067}a^{19}+\frac{1801}{3042067}a^{18}-\frac{30602}{3042067}a^{17}-\frac{17653}{3042067}a^{16}-\frac{30678}{3042067}a^{15}+\frac{17215}{434581}a^{14}+\frac{11185}{434581}a^{13}-\frac{24814}{434581}a^{12}-\frac{19105}{62083}a^{11}-\frac{14909}{62083}a^{10}+\frac{23159}{62083}a^{9}+\frac{3762}{8869}a^{8}+\frac{412}{1267}a^{7}+\frac{1877}{8869}a^{6}-\frac{62}{181}a^{5}+\frac{552}{1267}a^{4}-\frac{93}{1267}a^{3}+\frac{64}{181}a^{2}+\frac{31}{181}a+\frac{74}{181}$, $\frac{1}{3042067}a^{25}+\frac{522}{3042067}a^{23}+\frac{388}{3042067}a^{22}+\frac{589}{3042067}a^{21}-\frac{407}{3042067}a^{20}+\frac{4328}{3042067}a^{19}+\frac{1444}{3042067}a^{18}+\frac{20189}{3042067}a^{17}+\frac{12134}{3042067}a^{16}+\frac{2334}{434581}a^{15}-\frac{19062}{434581}a^{14}+\frac{20}{434581}a^{13}+\frac{4143}{62083}a^{12}-\frac{8836}{62083}a^{11}-\frac{664}{8869}a^{10}-\frac{13477}{62083}a^{9}+\frac{1656}{8869}a^{8}-\frac{3179}{8869}a^{7}-\frac{2743}{8869}a^{6}-\frac{100}{1267}a^{5}-\frac{283}{1267}a^{4}-\frac{545}{1267}a^{3}-\frac{33}{181}a^{2}+\frac{43}{181}a-\frac{74}{181}$, $\frac{1}{22\cdots 37}a^{26}-\frac{58\cdots 65}{22\cdots 37}a^{25}+\frac{13\cdots 59}{22\cdots 37}a^{24}-\frac{15\cdots 30}{22\cdots 37}a^{23}+\frac{30\cdots 41}{22\cdots 37}a^{22}-\frac{84\cdots 22}{22\cdots 37}a^{21}+\frac{16\cdots 50}{22\cdots 37}a^{20}+\frac{23\cdots 98}{22\cdots 37}a^{19}-\frac{18\cdots 91}{22\cdots 37}a^{18}+\frac{13\cdots 34}{22\cdots 37}a^{17}+\frac{24\cdots 87}{22\cdots 37}a^{16}+\frac{22\cdots 74}{24\cdots 01}a^{15}+\frac{17\cdots 51}{32\cdots 91}a^{14}+\frac{16\cdots 33}{32\cdots 91}a^{13}+\frac{18\cdots 42}{32\cdots 91}a^{12}-\frac{21\cdots 52}{46\cdots 13}a^{11}+\frac{17\cdots 66}{46\cdots 13}a^{10}-\frac{18\cdots 32}{46\cdots 13}a^{9}+\frac{19\cdots 42}{66\cdots 59}a^{8}-\frac{11\cdots 11}{66\cdots 59}a^{7}+\frac{23\cdots 01}{66\cdots 59}a^{6}-\frac{34\cdots 95}{94\cdots 37}a^{5}-\frac{30\cdots 12}{15\cdots 09}a^{4}+\frac{24\cdots 07}{94\cdots 37}a^{3}-\frac{16\cdots 95}{13\cdots 91}a^{2}-\frac{80\cdots 83}{13\cdots 91}a-\frac{69\cdots 52}{13\cdots 91}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $26$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{27\cdots 96}{61\cdots 41}a^{26}-\frac{98\cdots 50}{61\cdots 41}a^{25}+\frac{76\cdots 29}{61\cdots 41}a^{24}+\frac{52\cdots 02}{88\cdots 63}a^{23}-\frac{83\cdots 37}{61\cdots 41}a^{22}-\frac{47\cdots 95}{61\cdots 41}a^{21}+\frac{47\cdots 60}{61\cdots 41}a^{20}+\frac{30\cdots 43}{61\cdots 41}a^{19}-\frac{16\cdots 18}{61\cdots 41}a^{18}-\frac{15\cdots 42}{88\cdots 63}a^{17}+\frac{34\cdots 48}{61\cdots 41}a^{16}+\frac{27\cdots 65}{65\cdots 93}a^{15}-\frac{71\cdots 19}{88\cdots 63}a^{14}-\frac{54\cdots 11}{88\cdots 63}a^{13}+\frac{72\cdots 06}{88\cdots 63}a^{12}+\frac{77\cdots 20}{12\cdots 09}a^{11}-\frac{73\cdots 11}{12\cdots 09}a^{10}-\frac{50\cdots 29}{12\cdots 09}a^{9}+\frac{71\cdots 45}{25\cdots 41}a^{8}+\frac{29\cdots 49}{17\cdots 87}a^{7}-\frac{14\cdots 61}{17\cdots 87}a^{6}-\frac{14\cdots 63}{36\cdots 63}a^{5}+\frac{31\cdots 14}{25\cdots 41}a^{4}+\frac{13\cdots 39}{25\cdots 41}a^{3}-\frac{17\cdots 08}{36\cdots 63}a^{2}-\frac{10\cdots 77}{36\cdots 63}a-\frac{10\cdots 18}{36\cdots 63}$, $\frac{46\cdots 72}{61\cdots 41}a^{26}-\frac{16\cdots 76}{61\cdots 41}a^{25}+\frac{27\cdots 55}{12\cdots 09}a^{24}+\frac{62\cdots 21}{61\cdots 41}a^{23}-\frac{14\cdots 46}{61\cdots 41}a^{22}-\frac{79\cdots 17}{61\cdots 41}a^{21}+\frac{84\cdots 79}{61\cdots 41}a^{20}+\frac{50\cdots 32}{61\cdots 41}a^{19}-\frac{28\cdots 15}{61\cdots 41}a^{18}-\frac{18\cdots 51}{61\cdots 41}a^{17}+\frac{63\cdots 42}{61\cdots 41}a^{16}+\frac{46\cdots 93}{65\cdots 93}a^{15}-\frac{13\cdots 25}{88\cdots 63}a^{14}-\frac{95\cdots 80}{88\cdots 63}a^{13}+\frac{13\cdots 42}{88\cdots 63}a^{12}+\frac{13\cdots 61}{12\cdots 09}a^{11}-\frac{14\cdots 88}{12\cdots 09}a^{10}-\frac{89\cdots 62}{12\cdots 09}a^{9}+\frac{20\cdots 17}{36\cdots 63}a^{8}+\frac{53\cdots 96}{17\cdots 87}a^{7}-\frac{30\cdots 63}{17\cdots 87}a^{6}-\frac{27\cdots 94}{36\cdots 63}a^{5}+\frac{65\cdots 07}{25\cdots 41}a^{4}+\frac{25\cdots 14}{25\cdots 41}a^{3}-\frac{38\cdots 35}{36\cdots 63}a^{2}-\frac{19\cdots 56}{36\cdots 63}a-\frac{20\cdots 76}{36\cdots 63}$, $\frac{10\cdots 20}{48\cdots 49}a^{26}+\frac{69\cdots 82}{69\cdots 07}a^{25}-\frac{29\cdots 68}{48\cdots 49}a^{24}-\frac{17\cdots 26}{48\cdots 49}a^{23}+\frac{43\cdots 03}{69\cdots 07}a^{22}+\frac{30\cdots 00}{69\cdots 07}a^{21}-\frac{23\cdots 35}{69\cdots 07}a^{20}-\frac{12\cdots 82}{48\cdots 49}a^{19}+\frac{49\cdots 77}{48\cdots 49}a^{18}+\frac{45\cdots 47}{48\cdots 49}a^{17}-\frac{93\cdots 09}{48\cdots 49}a^{16}-\frac{10\cdots 55}{52\cdots 77}a^{15}+\frac{16\cdots 83}{69\cdots 07}a^{14}+\frac{20\cdots 26}{69\cdots 07}a^{13}-\frac{13\cdots 00}{69\cdots 07}a^{12}-\frac{27\cdots 98}{99\cdots 01}a^{11}+\frac{17\cdots 85}{14\cdots 43}a^{10}+\frac{17\cdots 76}{99\cdots 01}a^{9}-\frac{80\cdots 82}{14\cdots 43}a^{8}-\frac{97\cdots 42}{14\cdots 43}a^{7}+\frac{23\cdots 21}{14\cdots 43}a^{6}+\frac{32\cdots 38}{20\cdots 49}a^{5}-\frac{50\cdots 33}{20\cdots 49}a^{4}-\frac{40\cdots 14}{20\cdots 49}a^{3}+\frac{16\cdots 91}{29\cdots 07}a^{2}+\frac{29\cdots 29}{29\cdots 07}a+\frac{31\cdots 15}{29\cdots 07}$, $\frac{37\cdots 05}{48\cdots 49}a^{26}+\frac{18\cdots 12}{69\cdots 07}a^{25}-\frac{10\cdots 82}{48\cdots 49}a^{24}-\frac{50\cdots 47}{48\cdots 49}a^{23}+\frac{11\cdots 97}{48\cdots 49}a^{22}+\frac{64\cdots 33}{48\cdots 49}a^{21}-\frac{68\cdots 62}{48\cdots 49}a^{20}-\frac{41\cdots 23}{48\cdots 49}a^{19}+\frac{23\cdots 06}{48\cdots 49}a^{18}+\frac{15\cdots 91}{48\cdots 49}a^{17}-\frac{51\cdots 22}{48\cdots 49}a^{16}-\frac{38\cdots 30}{52\cdots 77}a^{15}+\frac{10\cdots 92}{69\cdots 07}a^{14}+\frac{77\cdots 03}{69\cdots 07}a^{13}-\frac{11\cdots 67}{69\cdots 07}a^{12}-\frac{11\cdots 24}{99\cdots 01}a^{11}+\frac{11\cdots 36}{99\cdots 01}a^{10}+\frac{72\cdots 21}{99\cdots 01}a^{9}-\frac{11\cdots 33}{20\cdots 49}a^{8}-\frac{43\cdots 04}{14\cdots 43}a^{7}+\frac{24\cdots 02}{14\cdots 43}a^{6}+\frac{15\cdots 58}{20\cdots 49}a^{5}-\frac{52\cdots 08}{20\cdots 49}a^{4}-\frac{20\cdots 45}{20\cdots 49}a^{3}+\frac{30\cdots 45}{29\cdots 07}a^{2}+\frac{15\cdots 01}{29\cdots 07}a+\frac{16\cdots 74}{29\cdots 07}$, $\frac{93\cdots 52}{38\cdots 09}a^{26}-\frac{47\cdots 50}{38\cdots 09}a^{25}+\frac{25\cdots 54}{38\cdots 09}a^{24}+\frac{16\cdots 30}{38\cdots 09}a^{23}-\frac{25\cdots 48}{38\cdots 09}a^{22}-\frac{19\cdots 85}{38\cdots 09}a^{21}+\frac{12\cdots 39}{38\cdots 09}a^{20}+\frac{11\cdots 57}{38\cdots 09}a^{19}-\frac{35\cdots 97}{38\cdots 09}a^{18}-\frac{39\cdots 81}{38\cdots 09}a^{17}+\frac{56\cdots 92}{38\cdots 09}a^{16}+\frac{91\cdots 72}{40\cdots 57}a^{15}-\frac{72\cdots 88}{54\cdots 87}a^{14}-\frac{24\cdots 81}{77\cdots 41}a^{13}+\frac{31\cdots 24}{54\cdots 87}a^{12}+\frac{22\cdots 68}{77\cdots 41}a^{11}-\frac{44\cdots 33}{77\cdots 41}a^{10}-\frac{18\cdots 18}{11\cdots 63}a^{9}-\frac{45\cdots 09}{11\cdots 63}a^{8}+\frac{71\cdots 05}{11\cdots 63}a^{7}+\frac{16\cdots 36}{11\cdots 63}a^{6}-\frac{32\cdots 57}{22\cdots 87}a^{5}-\frac{59\cdots 43}{15\cdots 09}a^{4}+\frac{39\cdots 24}{22\cdots 87}a^{3}+\frac{22\cdots 80}{22\cdots 87}a^{2}-\frac{18\cdots 20}{22\cdots 87}a-\frac{20\cdots 73}{22\cdots 87}$, $\frac{86\cdots 68}{38\cdots 09}a^{26}+\frac{24\cdots 34}{38\cdots 09}a^{25}-\frac{10\cdots 21}{38\cdots 09}a^{24}-\frac{68\cdots 10}{38\cdots 09}a^{23}-\frac{19\cdots 04}{38\cdots 09}a^{22}+\frac{37\cdots 30}{21\cdots 89}a^{21}+\frac{37\cdots 56}{38\cdots 09}a^{20}-\frac{30\cdots 12}{38\cdots 09}a^{19}-\frac{24\cdots 28}{38\cdots 09}a^{18}+\frac{67\cdots 02}{38\cdots 09}a^{17}+\frac{84\cdots 91}{38\cdots 09}a^{16}-\frac{55\cdots 42}{40\cdots 57}a^{15}-\frac{24\cdots 40}{54\cdots 87}a^{14}-\frac{15\cdots 58}{77\cdots 41}a^{13}+\frac{30\cdots 70}{54\cdots 87}a^{12}+\frac{44\cdots 94}{77\cdots 41}a^{11}-\frac{35\cdots 11}{77\cdots 41}a^{10}-\frac{46\cdots 79}{77\cdots 41}a^{9}+\frac{36\cdots 26}{15\cdots 09}a^{8}+\frac{36\cdots 74}{11\cdots 63}a^{7}-\frac{74\cdots 64}{11\cdots 63}a^{6}-\frac{16\cdots 52}{15\cdots 09}a^{5}+\frac{22\cdots 68}{22\cdots 87}a^{4}+\frac{25\cdots 69}{15\cdots 09}a^{3}-\frac{11\cdots 41}{22\cdots 87}a^{2}-\frac{21\cdots 64}{22\cdots 87}a-\frac{22\cdots 07}{22\cdots 87}$, $\frac{14\cdots 95}{38\cdots 09}a^{26}-\frac{49\cdots 14}{38\cdots 09}a^{25}+\frac{42\cdots 48}{38\cdots 09}a^{24}+\frac{19\cdots 24}{38\cdots 09}a^{23}-\frac{47\cdots 03}{38\cdots 09}a^{22}-\frac{24\cdots 50}{38\cdots 09}a^{21}+\frac{28\cdots 75}{38\cdots 09}a^{20}+\frac{16\cdots 37}{38\cdots 09}a^{19}-\frac{98\cdots 31}{38\cdots 09}a^{18}-\frac{60\cdots 20}{38\cdots 09}a^{17}+\frac{22\cdots 85}{38\cdots 09}a^{16}+\frac{15\cdots 95}{40\cdots 57}a^{15}-\frac{48\cdots 99}{54\cdots 87}a^{14}-\frac{31\cdots 35}{54\cdots 87}a^{13}+\frac{50\cdots 81}{54\cdots 87}a^{12}+\frac{45\cdots 35}{77\cdots 41}a^{11}-\frac{52\cdots 37}{77\cdots 41}a^{10}-\frac{30\cdots 88}{77\cdots 41}a^{9}+\frac{36\cdots 52}{11\cdots 63}a^{8}+\frac{18\cdots 58}{11\cdots 63}a^{7}-\frac{10\cdots 91}{11\cdots 63}a^{6}-\frac{65\cdots 32}{15\cdots 09}a^{5}+\frac{23\cdots 97}{15\cdots 09}a^{4}+\frac{88\cdots 58}{15\cdots 09}a^{3}-\frac{13\cdots 73}{22\cdots 87}a^{2}-\frac{67\cdots 45}{22\cdots 87}a-\frac{71\cdots 01}{22\cdots 87}$, $\frac{67\cdots 33}{38\cdots 09}a^{26}+\frac{24\cdots 02}{38\cdots 09}a^{25}-\frac{19\cdots 10}{38\cdots 09}a^{24}-\frac{13\cdots 33}{54\cdots 87}a^{23}+\frac{30\cdots 23}{54\cdots 87}a^{22}+\frac{11\cdots 12}{38\cdots 09}a^{21}-\frac{17\cdots 37}{54\cdots 87}a^{20}-\frac{74\cdots 68}{38\cdots 09}a^{19}+\frac{41\cdots 85}{38\cdots 09}a^{18}+\frac{27\cdots 33}{38\cdots 09}a^{17}-\frac{90\cdots 17}{38\cdots 09}a^{16}-\frac{67\cdots 22}{40\cdots 57}a^{15}+\frac{19\cdots 52}{54\cdots 87}a^{14}+\frac{13\cdots 52}{54\cdots 87}a^{13}-\frac{19\cdots 43}{54\cdots 87}a^{12}-\frac{19\cdots 70}{77\cdots 41}a^{11}+\frac{20\cdots 28}{77\cdots 41}a^{10}+\frac{12\cdots 30}{77\cdots 41}a^{9}-\frac{14\cdots 82}{11\cdots 63}a^{8}-\frac{76\cdots 75}{11\cdots 63}a^{7}+\frac{43\cdots 48}{11\cdots 63}a^{6}+\frac{27\cdots 75}{15\cdots 09}a^{5}-\frac{13\cdots 53}{22\cdots 87}a^{4}-\frac{36\cdots 57}{15\cdots 09}a^{3}+\frac{55\cdots 30}{22\cdots 87}a^{2}+\frac{27\cdots 69}{22\cdots 87}a+\frac{29\cdots 08}{22\cdots 87}$, $\frac{73\cdots 91}{32\cdots 91}a^{26}-\frac{17\cdots 35}{22\cdots 37}a^{25}+\frac{14\cdots 47}{22\cdots 37}a^{24}+\frac{68\cdots 78}{22\cdots 37}a^{23}-\frac{23\cdots 76}{32\cdots 91}a^{22}-\frac{88\cdots 68}{22\cdots 37}a^{21}+\frac{13\cdots 40}{32\cdots 91}a^{20}+\frac{56\cdots 87}{22\cdots 37}a^{19}-\frac{32\cdots 89}{22\cdots 37}a^{18}-\frac{21\cdots 62}{22\cdots 37}a^{17}+\frac{70\cdots 79}{22\cdots 37}a^{16}+\frac{52\cdots 71}{24\cdots 01}a^{15}-\frac{14\cdots 48}{32\cdots 91}a^{14}-\frac{10\cdots 38}{32\cdots 91}a^{13}+\frac{15\cdots 72}{32\cdots 91}a^{12}+\frac{15\cdots 29}{46\cdots 13}a^{11}-\frac{15\cdots 37}{46\cdots 13}a^{10}-\frac{99\cdots 99}{46\cdots 13}a^{9}+\frac{10\cdots 13}{66\cdots 59}a^{8}+\frac{58\cdots 89}{66\cdots 59}a^{7}-\frac{29\cdots 56}{66\cdots 59}a^{6}-\frac{20\cdots 65}{94\cdots 37}a^{5}+\frac{10\cdots 93}{15\cdots 09}a^{4}+\frac{38\cdots 83}{13\cdots 91}a^{3}-\frac{32\cdots 78}{13\cdots 91}a^{2}-\frac{19\cdots 04}{13\cdots 91}a-\frac{21\cdots 85}{13\cdots 91}$, $\frac{17\cdots 49}{22\cdots 37}a^{26}+\frac{61\cdots 73}{22\cdots 37}a^{25}-\frac{27\cdots 94}{12\cdots 77}a^{24}-\frac{33\cdots 28}{32\cdots 91}a^{23}+\frac{54\cdots 77}{22\cdots 37}a^{22}+\frac{29\cdots 61}{22\cdots 37}a^{21}-\frac{31\cdots 56}{22\cdots 37}a^{20}-\frac{19\cdots 73}{22\cdots 37}a^{19}+\frac{10\cdots 40}{22\cdots 37}a^{18}+\frac{14\cdots 45}{46\cdots 13}a^{17}-\frac{23\cdots 45}{22\cdots 37}a^{16}-\frac{17\cdots 47}{24\cdots 01}a^{15}+\frac{49\cdots 13}{32\cdots 91}a^{14}+\frac{35\cdots 43}{32\cdots 91}a^{13}-\frac{50\cdots 99}{32\cdots 91}a^{12}-\frac{72\cdots 74}{66\cdots 59}a^{11}+\frac{74\cdots 10}{66\cdots 59}a^{10}+\frac{33\cdots 78}{46\cdots 13}a^{9}-\frac{36\cdots 04}{66\cdots 59}a^{8}-\frac{19\cdots 95}{66\cdots 59}a^{7}+\frac{10\cdots 24}{66\cdots 59}a^{6}+\frac{69\cdots 32}{94\cdots 37}a^{5}-\frac{39\cdots 46}{15\cdots 09}a^{4}-\frac{13\cdots 87}{13\cdots 91}a^{3}+\frac{13\cdots 92}{13\cdots 91}a^{2}+\frac{69\cdots 07}{13\cdots 91}a+\frac{74\cdots 79}{13\cdots 91}$, $\frac{23\cdots 01}{22\cdots 37}a^{26}+\frac{87\cdots 74}{22\cdots 37}a^{25}-\frac{65\cdots 99}{22\cdots 37}a^{24}-\frac{32\cdots 15}{22\cdots 37}a^{23}+\frac{71\cdots 55}{22\cdots 37}a^{22}+\frac{41\cdots 21}{22\cdots 37}a^{21}-\frac{40\cdots 11}{22\cdots 37}a^{20}-\frac{26\cdots 70}{22\cdots 37}a^{19}+\frac{13\cdots 89}{22\cdots 37}a^{18}+\frac{96\cdots 41}{22\cdots 37}a^{17}-\frac{28\cdots 94}{22\cdots 37}a^{16}-\frac{13\cdots 64}{13\cdots 21}a^{15}+\frac{57\cdots 37}{32\cdots 91}a^{14}+\frac{67\cdots 88}{46\cdots 13}a^{13}-\frac{56\cdots 64}{32\cdots 91}a^{12}-\frac{67\cdots 88}{46\cdots 13}a^{11}+\frac{54\cdots 03}{46\cdots 13}a^{10}+\frac{43\cdots 30}{46\cdots 13}a^{9}-\frac{35\cdots 49}{66\cdots 59}a^{8}-\frac{25\cdots 21}{66\cdots 59}a^{7}+\frac{10\cdots 49}{66\cdots 59}a^{6}+\frac{88\cdots 09}{94\cdots 37}a^{5}-\frac{32\cdots 28}{15\cdots 09}a^{4}-\frac{11\cdots 27}{94\cdots 37}a^{3}+\frac{53\cdots 32}{13\cdots 91}a^{2}+\frac{84\cdots 80}{13\cdots 91}a+\frac{11\cdots 41}{13\cdots 91}$, $\frac{30\cdots 09}{20\cdots 93}a^{26}-\frac{96\cdots 60}{20\cdots 93}a^{25}+\frac{87\cdots 89}{20\cdots 93}a^{24}+\frac{37\cdots 33}{20\cdots 93}a^{23}-\frac{97\cdots 53}{20\cdots 93}a^{22}-\frac{49\cdots 84}{20\cdots 93}a^{21}+\frac{57\cdots 55}{20\cdots 93}a^{20}+\frac{32\cdots 09}{20\cdots 93}a^{19}-\frac{20\cdots 71}{20\cdots 93}a^{18}-\frac{12\cdots 33}{20\cdots 93}a^{17}+\frac{65\cdots 42}{29\cdots 99}a^{16}+\frac{30\cdots 05}{22\cdots 89}a^{15}-\frac{10\cdots 22}{29\cdots 99}a^{14}-\frac{63\cdots 71}{29\cdots 99}a^{13}+\frac{10\cdots 15}{29\cdots 99}a^{12}+\frac{92\cdots 51}{42\cdots 57}a^{11}-\frac{11\cdots 56}{42\cdots 57}a^{10}-\frac{61\cdots 75}{42\cdots 57}a^{9}+\frac{79\cdots 90}{60\cdots 51}a^{8}+\frac{37\cdots 98}{60\cdots 51}a^{7}-\frac{23\cdots 78}{60\cdots 51}a^{6}-\frac{13\cdots 07}{86\cdots 93}a^{5}+\frac{86\cdots 88}{14\cdots 01}a^{4}+\frac{18\cdots 40}{86\cdots 93}a^{3}-\frac{31\cdots 54}{12\cdots 99}a^{2}-\frac{13\cdots 32}{12\cdots 99}a-\frac{14\cdots 54}{12\cdots 99}$, $\frac{19\cdots 36}{22\cdots 37}a^{26}+\frac{67\cdots 82}{22\cdots 37}a^{25}-\frac{15\cdots 71}{66\cdots 59}a^{24}-\frac{25\cdots 74}{22\cdots 37}a^{23}+\frac{59\cdots 08}{22\cdots 37}a^{22}+\frac{32\cdots 87}{22\cdots 37}a^{21}-\frac{34\cdots 80}{22\cdots 37}a^{20}-\frac{20\cdots 98}{22\cdots 37}a^{19}+\frac{11\cdots 55}{22\cdots 37}a^{18}+\frac{77\cdots 99}{22\cdots 37}a^{17}-\frac{25\cdots 35}{22\cdots 37}a^{16}-\frac{19\cdots 00}{24\cdots 01}a^{15}+\frac{76\cdots 24}{46\cdots 13}a^{14}+\frac{38\cdots 53}{32\cdots 91}a^{13}-\frac{54\cdots 05}{32\cdots 91}a^{12}-\frac{55\cdots 36}{46\cdots 13}a^{11}+\frac{56\cdots 82}{46\cdots 13}a^{10}+\frac{36\cdots 30}{46\cdots 13}a^{9}-\frac{56\cdots 23}{94\cdots 37}a^{8}-\frac{21\cdots 01}{66\cdots 59}a^{7}+\frac{23\cdots 49}{13\cdots 91}a^{6}+\frac{10\cdots 15}{13\cdots 91}a^{5}-\frac{42\cdots 22}{15\cdots 09}a^{4}-\frac{14\cdots 94}{13\cdots 91}a^{3}+\frac{14\cdots 12}{13\cdots 91}a^{2}+\frac{75\cdots 57}{13\cdots 91}a+\frac{78\cdots 87}{13\cdots 91}$, $\frac{15\cdots 74}{22\cdots 37}a^{26}+\frac{52\cdots 23}{22\cdots 37}a^{25}-\frac{44\cdots 33}{22\cdots 37}a^{24}-\frac{28\cdots 52}{32\cdots 91}a^{23}+\frac{49\cdots 14}{22\cdots 37}a^{22}+\frac{37\cdots 18}{32\cdots 91}a^{21}-\frac{29\cdots 75}{22\cdots 37}a^{20}-\frac{16\cdots 25}{22\cdots 37}a^{19}+\frac{10\cdots 10}{22\cdots 37}a^{18}+\frac{62\cdots 91}{22\cdots 37}a^{17}-\frac{22\cdots 99}{22\cdots 37}a^{16}-\frac{22\cdots 89}{34\cdots 43}a^{15}+\frac{48\cdots 52}{32\cdots 91}a^{14}+\frac{45\cdots 11}{46\cdots 13}a^{13}-\frac{51\cdots 37}{32\cdots 91}a^{12}-\frac{46\cdots 22}{46\cdots 13}a^{11}+\frac{54\cdots 09}{46\cdots 13}a^{10}+\frac{43\cdots 42}{66\cdots 59}a^{9}-\frac{39\cdots 43}{66\cdots 59}a^{8}-\frac{26\cdots 47}{94\cdots 37}a^{7}+\frac{11\cdots 65}{66\cdots 59}a^{6}+\frac{66\cdots 05}{94\cdots 37}a^{5}-\frac{43\cdots 54}{15\cdots 09}a^{4}-\frac{90\cdots 94}{94\cdots 37}a^{3}+\frac{16\cdots 80}{13\cdots 91}a^{2}+\frac{69\cdots 20}{13\cdots 91}a+\frac{71\cdots 01}{13\cdots 91}$, $\frac{10\cdots 87}{22\cdots 37}a^{26}-\frac{32\cdots 89}{22\cdots 37}a^{25}+\frac{29\cdots 55}{22\cdots 37}a^{24}+\frac{12\cdots 79}{22\cdots 37}a^{23}-\frac{33\cdots 52}{22\cdots 37}a^{22}-\frac{16\cdots 23}{22\cdots 37}a^{21}+\frac{19\cdots 44}{22\cdots 37}a^{20}+\frac{11\cdots 13}{22\cdots 37}a^{19}-\frac{68\cdots 07}{22\cdots 37}a^{18}-\frac{41\cdots 29}{22\cdots 37}a^{17}+\frac{22\cdots 61}{32\cdots 91}a^{16}+\frac{10\cdots 13}{24\cdots 01}a^{15}-\frac{34\cdots 26}{32\cdots 91}a^{14}-\frac{21\cdots 51}{32\cdots 91}a^{13}+\frac{36\cdots 59}{32\cdots 91}a^{12}+\frac{31\cdots 25}{46\cdots 13}a^{11}-\frac{38\cdots 68}{46\cdots 13}a^{10}-\frac{29\cdots 40}{66\cdots 59}a^{9}+\frac{26\cdots 04}{66\cdots 59}a^{8}+\frac{12\cdots 18}{66\cdots 59}a^{7}-\frac{79\cdots 20}{66\cdots 59}a^{6}-\frac{64\cdots 65}{13\cdots 91}a^{5}+\frac{29\cdots 98}{15\cdots 09}a^{4}+\frac{60\cdots 01}{94\cdots 37}a^{3}-\frac{10\cdots 88}{13\cdots 91}a^{2}-\frac{46\cdots 79}{13\cdots 91}a-\frac{49\cdots 37}{13\cdots 91}$, $\frac{20\cdots 69}{22\cdots 37}a^{26}+\frac{64\cdots 28}{22\cdots 37}a^{25}-\frac{57\cdots 49}{22\cdots 37}a^{24}-\frac{25\cdots 28}{22\cdots 37}a^{23}+\frac{64\cdots 81}{22\cdots 37}a^{22}+\frac{33\cdots 69}{22\cdots 37}a^{21}-\frac{37\cdots 82}{22\cdots 37}a^{20}-\frac{21\cdots 36}{22\cdots 37}a^{19}+\frac{13\cdots 81}{22\cdots 37}a^{18}+\frac{80\cdots 15}{22\cdots 37}a^{17}-\frac{30\cdots 20}{22\cdots 37}a^{16}-\frac{20\cdots 19}{24\cdots 01}a^{15}+\frac{65\cdots 30}{32\cdots 91}a^{14}+\frac{41\cdots 39}{32\cdots 91}a^{13}-\frac{69\cdots 19}{32\cdots 91}a^{12}-\frac{60\cdots 69}{46\cdots 13}a^{11}+\frac{73\cdots 04}{46\cdots 13}a^{10}+\frac{40\cdots 43}{46\cdots 13}a^{9}-\frac{51\cdots 32}{66\cdots 59}a^{8}-\frac{24\cdots 08}{66\cdots 59}a^{7}+\frac{15\cdots 32}{66\cdots 59}a^{6}+\frac{12\cdots 44}{13\cdots 91}a^{5}-\frac{55\cdots 48}{15\cdots 09}a^{4}-\frac{11\cdots 19}{94\cdots 37}a^{3}+\frac{19\cdots 86}{13\cdots 91}a^{2}+\frac{89\cdots 12}{13\cdots 91}a+\frac{94\cdots 12}{13\cdots 91}$, $\frac{19\cdots 75}{22\cdots 37}a^{26}+\frac{69\cdots 38}{22\cdots 37}a^{25}-\frac{11\cdots 14}{46\cdots 13}a^{24}-\frac{26\cdots 78}{22\cdots 37}a^{23}+\frac{61\cdots 07}{22\cdots 37}a^{22}+\frac{33\cdots 40}{22\cdots 37}a^{21}-\frac{35\cdots 89}{22\cdots 37}a^{20}-\frac{21\cdots 65}{22\cdots 37}a^{19}+\frac{12\cdots 78}{22\cdots 37}a^{18}+\frac{79\cdots 37}{22\cdots 37}a^{17}-\frac{26\cdots 58}{22\cdots 37}a^{16}-\frac{19\cdots 86}{24\cdots 01}a^{15}+\frac{55\cdots 18}{32\cdots 91}a^{14}+\frac{39\cdots 11}{32\cdots 91}a^{13}-\frac{82\cdots 79}{46\cdots 13}a^{12}-\frac{57\cdots 24}{46\cdots 13}a^{11}+\frac{59\cdots 62}{46\cdots 13}a^{10}+\frac{37\cdots 98}{46\cdots 13}a^{9}-\frac{41\cdots 13}{66\cdots 59}a^{8}-\frac{22\cdots 39}{66\cdots 59}a^{7}+\frac{12\cdots 96}{66\cdots 59}a^{6}+\frac{79\cdots 79}{94\cdots 37}a^{5}-\frac{44\cdots 68}{15\cdots 09}a^{4}-\frac{10\cdots 79}{94\cdots 37}a^{3}+\frac{15\cdots 90}{13\cdots 91}a^{2}+\frac{78\cdots 40}{13\cdots 91}a+\frac{84\cdots 74}{13\cdots 91}$, $\frac{55\cdots 07}{22\cdots 37}a^{26}-\frac{18\cdots 81}{22\cdots 37}a^{25}+\frac{15\cdots 29}{22\cdots 37}a^{24}+\frac{71\cdots 36}{22\cdots 37}a^{23}-\frac{17\cdots 61}{22\cdots 37}a^{22}-\frac{92\cdots 41}{22\cdots 37}a^{21}+\frac{10\cdots 72}{22\cdots 37}a^{20}+\frac{59\cdots 65}{22\cdots 37}a^{19}-\frac{36\cdots 71}{22\cdots 37}a^{18}-\frac{22\cdots 82}{22\cdots 37}a^{17}+\frac{81\cdots 58}{22\cdots 37}a^{16}+\frac{56\cdots 56}{24\cdots 01}a^{15}-\frac{17\cdots 13}{32\cdots 91}a^{14}-\frac{11\cdots 23}{32\cdots 91}a^{13}+\frac{18\cdots 74}{32\cdots 91}a^{12}+\frac{23\cdots 48}{66\cdots 59}a^{11}-\frac{19\cdots 51}{46\cdots 13}a^{10}-\frac{11\cdots 03}{46\cdots 13}a^{9}+\frac{13\cdots 46}{66\cdots 59}a^{8}+\frac{66\cdots 83}{66\cdots 59}a^{7}-\frac{41\cdots 04}{66\cdots 59}a^{6}-\frac{23\cdots 84}{94\cdots 37}a^{5}+\frac{15\cdots 76}{15\cdots 09}a^{4}+\frac{31\cdots 10}{94\cdots 37}a^{3}-\frac{53\cdots 97}{13\cdots 91}a^{2}-\frac{24\cdots 58}{13\cdots 91}a-\frac{25\cdots 00}{13\cdots 91}$, $\frac{41\cdots 96}{22\cdots 37}a^{26}-\frac{14\cdots 55}{22\cdots 37}a^{25}+\frac{17\cdots 52}{32\cdots 91}a^{24}+\frac{54\cdots 45}{22\cdots 37}a^{23}-\frac{13\cdots 40}{22\cdots 37}a^{22}-\frac{70\cdots 10}{22\cdots 37}a^{21}+\frac{75\cdots 64}{22\cdots 37}a^{20}+\frac{44\cdots 42}{22\cdots 37}a^{19}-\frac{25\cdots 46}{22\cdots 37}a^{18}-\frac{16\cdots 97}{22\cdots 37}a^{17}+\frac{57\cdots 22}{22\cdots 37}a^{16}+\frac{41\cdots 03}{24\cdots 01}a^{15}-\frac{12\cdots 21}{32\cdots 91}a^{14}-\frac{83\cdots 96}{32\cdots 91}a^{13}+\frac{12\cdots 09}{32\cdots 91}a^{12}+\frac{11\cdots 07}{46\cdots 13}a^{11}-\frac{12\cdots 68}{46\cdots 13}a^{10}-\frac{11\cdots 65}{66\cdots 59}a^{9}+\frac{90\cdots 75}{66\cdots 59}a^{8}+\frac{66\cdots 14}{94\cdots 37}a^{7}-\frac{26\cdots 67}{66\cdots 59}a^{6}-\frac{16\cdots 83}{94\cdots 37}a^{5}+\frac{96\cdots 83}{15\cdots 09}a^{4}+\frac{22\cdots 38}{94\cdots 37}a^{3}-\frac{33\cdots 03}{13\cdots 91}a^{2}-\frac{16\cdots 17}{13\cdots 91}a-\frac{17\cdots 52}{13\cdots 91}$, $\frac{20\cdots 46}{22\cdots 37}a^{26}-\frac{90\cdots 70}{32\cdots 91}a^{25}+\frac{83\cdots 73}{32\cdots 91}a^{24}+\frac{25\cdots 12}{22\cdots 37}a^{23}-\frac{65\cdots 94}{22\cdots 37}a^{22}-\frac{32\cdots 23}{22\cdots 37}a^{21}+\frac{38\cdots 96}{22\cdots 37}a^{20}+\frac{21\cdots 39}{22\cdots 37}a^{19}-\frac{28\cdots 78}{46\cdots 13}a^{18}-\frac{81\cdots 23}{22\cdots 37}a^{17}+\frac{31\cdots 44}{22\cdots 37}a^{16}+\frac{20\cdots 24}{24\cdots 01}a^{15}-\frac{69\cdots 27}{32\cdots 91}a^{14}-\frac{42\cdots 64}{32\cdots 91}a^{13}+\frac{10\cdots 02}{46\cdots 13}a^{12}+\frac{62\cdots 51}{46\cdots 13}a^{11}-\frac{11\cdots 07}{66\cdots 59}a^{10}-\frac{41\cdots 36}{46\cdots 13}a^{9}+\frac{55\cdots 41}{66\cdots 59}a^{8}+\frac{25\cdots 86}{66\cdots 59}a^{7}-\frac{16\cdots 78}{66\cdots 59}a^{6}-\frac{90\cdots 80}{94\cdots 37}a^{5}+\frac{61\cdots 74}{15\cdots 09}a^{4}+\frac{12\cdots 91}{94\cdots 37}a^{3}-\frac{22\cdots 00}{13\cdots 91}a^{2}-\frac{94\cdots 87}{13\cdots 91}a-\frac{10\cdots 56}{13\cdots 91}$, $\frac{30\cdots 27}{22\cdots 37}a^{26}+\frac{10\cdots 82}{22\cdots 37}a^{25}-\frac{87\cdots 47}{22\cdots 37}a^{24}-\frac{39\cdots 23}{22\cdots 37}a^{23}+\frac{97\cdots 88}{22\cdots 37}a^{22}+\frac{51\cdots 35}{22\cdots 37}a^{21}-\frac{56\cdots 12}{22\cdots 37}a^{20}-\frac{33\cdots 39}{22\cdots 37}a^{19}+\frac{19\cdots 44}{22\cdots 37}a^{18}+\frac{17\cdots 63}{32\cdots 91}a^{17}-\frac{44\cdots 83}{22\cdots 37}a^{16}-\frac{31\cdots 22}{24\cdots 01}a^{15}+\frac{96\cdots 03}{32\cdots 91}a^{14}+\frac{63\cdots 30}{32\cdots 91}a^{13}-\frac{10\cdots 70}{32\cdots 91}a^{12}-\frac{91\cdots 58}{46\cdots 13}a^{11}+\frac{10\cdots 18}{46\cdots 13}a^{10}+\frac{60\cdots 55}{46\cdots 13}a^{9}-\frac{74\cdots 03}{66\cdots 59}a^{8}-\frac{36\cdots 54}{66\cdots 59}a^{7}+\frac{22\cdots 96}{66\cdots 59}a^{6}+\frac{13\cdots 52}{94\cdots 37}a^{5}-\frac{81\cdots 43}{15\cdots 09}a^{4}-\frac{17\cdots 18}{94\cdots 37}a^{3}+\frac{28\cdots 14}{13\cdots 91}a^{2}+\frac{13\cdots 66}{13\cdots 91}a+\frac{13\cdots 23}{13\cdots 91}$, $\frac{11\cdots 57}{32\cdots 91}a^{26}-\frac{26\cdots 09}{22\cdots 37}a^{25}+\frac{23\cdots 41}{22\cdots 37}a^{24}+\frac{10\cdots 12}{22\cdots 37}a^{23}-\frac{26\cdots 47}{22\cdots 37}a^{22}-\frac{19\cdots 58}{32\cdots 91}a^{21}+\frac{15\cdots 32}{22\cdots 37}a^{20}+\frac{12\cdots 16}{32\cdots 91}a^{19}-\frac{53\cdots 12}{22\cdots 37}a^{18}-\frac{32\cdots 40}{22\cdots 37}a^{17}+\frac{12\cdots 30}{22\cdots 37}a^{16}+\frac{82\cdots 35}{24\cdots 01}a^{15}-\frac{37\cdots 95}{46\cdots 13}a^{14}-\frac{17\cdots 11}{32\cdots 91}a^{13}+\frac{28\cdots 11}{32\cdots 91}a^{12}+\frac{24\cdots 87}{46\cdots 13}a^{11}-\frac{42\cdots 81}{66\cdots 59}a^{10}-\frac{16\cdots 57}{46\cdots 13}a^{9}+\frac{29\cdots 53}{94\cdots 37}a^{8}+\frac{14\cdots 80}{94\cdots 37}a^{7}-\frac{61\cdots 93}{66\cdots 59}a^{6}-\frac{35\cdots 38}{94\cdots 37}a^{5}+\frac{22\cdots 10}{15\cdots 09}a^{4}+\frac{47\cdots 02}{94\cdots 37}a^{3}-\frac{79\cdots 39}{13\cdots 91}a^{2}-\frac{36\cdots 49}{13\cdots 91}a-\frac{38\cdots 81}{13\cdots 91}$, $\frac{66\cdots 79}{22\cdots 37}a^{26}+\frac{85\cdots 62}{22\cdots 37}a^{25}-\frac{22\cdots 19}{22\cdots 37}a^{24}-\frac{25\cdots 48}{22\cdots 37}a^{23}+\frac{26\cdots 31}{22\cdots 37}a^{22}+\frac{29\cdots 83}{22\cdots 37}a^{21}-\frac{17\cdots 83}{22\cdots 37}a^{20}-\frac{18\cdots 30}{22\cdots 37}a^{19}+\frac{69\cdots 50}{22\cdots 37}a^{18}+\frac{13\cdots 87}{46\cdots 13}a^{17}-\frac{19\cdots 63}{22\cdots 37}a^{16}-\frac{24\cdots 00}{34\cdots 43}a^{15}+\frac{55\cdots 33}{32\cdots 91}a^{14}+\frac{51\cdots 62}{46\cdots 13}a^{13}-\frac{81\cdots 90}{32\cdots 91}a^{12}-\frac{54\cdots 50}{46\cdots 13}a^{11}+\frac{11\cdots 44}{46\cdots 13}a^{10}+\frac{37\cdots 23}{46\cdots 13}a^{9}-\frac{10\cdots 99}{66\cdots 59}a^{8}-\frac{24\cdots 21}{66\cdots 59}a^{7}+\frac{57\cdots 61}{94\cdots 37}a^{6}+\frac{95\cdots 65}{94\cdots 37}a^{5}-\frac{18\cdots 45}{15\cdots 09}a^{4}-\frac{14\cdots 69}{94\cdots 37}a^{3}+\frac{95\cdots 43}{13\cdots 91}a^{2}+\frac{12\cdots 00}{13\cdots 91}a+\frac{12\cdots 14}{13\cdots 91}$, $\frac{68\cdots 73}{22\cdots 37}a^{26}-\frac{26\cdots 66}{32\cdots 91}a^{25}+\frac{19\cdots 40}{22\cdots 37}a^{24}+\frac{75\cdots 77}{22\cdots 37}a^{23}-\frac{22\cdots 12}{22\cdots 37}a^{22}-\frac{10\cdots 78}{22\cdots 37}a^{21}+\frac{13\cdots 70}{22\cdots 37}a^{20}+\frac{96\cdots 53}{32\cdots 91}a^{19}-\frac{49\cdots 68}{22\cdots 37}a^{18}-\frac{37\cdots 48}{32\cdots 91}a^{17}+\frac{11\cdots 92}{22\cdots 37}a^{16}+\frac{67\cdots 36}{24\cdots 01}a^{15}-\frac{27\cdots 70}{32\cdots 91}a^{14}-\frac{14\cdots 02}{32\cdots 91}a^{13}+\frac{30\cdots 04}{32\cdots 91}a^{12}+\frac{21\cdots 59}{46\cdots 13}a^{11}-\frac{48\cdots 48}{66\cdots 59}a^{10}-\frac{11\cdots 98}{36\cdots 39}a^{9}+\frac{35\cdots 63}{94\cdots 37}a^{8}+\frac{90\cdots 25}{66\cdots 59}a^{7}-\frac{15\cdots 12}{13\cdots 91}a^{6}-\frac{33\cdots 09}{94\cdots 37}a^{5}+\frac{28\cdots 54}{15\cdots 09}a^{4}+\frac{67\cdots 25}{13\cdots 91}a^{3}-\frac{11\cdots 23}{13\cdots 91}a^{2}-\frac{37\cdots 24}{13\cdots 91}a-\frac{39\cdots 56}{13\cdots 91}$, $\frac{19\cdots 80}{22\cdots 37}a^{26}-\frac{58\cdots 98}{22\cdots 37}a^{25}+\frac{56\cdots 16}{22\cdots 37}a^{24}+\frac{23\cdots 67}{22\cdots 37}a^{23}-\frac{63\cdots 18}{22\cdots 37}a^{22}-\frac{30\cdots 66}{22\cdots 37}a^{21}+\frac{37\cdots 29}{22\cdots 37}a^{20}+\frac{20\cdots 21}{22\cdots 37}a^{19}-\frac{13\cdots 66}{22\cdots 37}a^{18}-\frac{76\cdots 27}{22\cdots 37}a^{17}+\frac{44\cdots 69}{32\cdots 91}a^{16}+\frac{19\cdots 67}{24\cdots 01}a^{15}-\frac{69\cdots 75}{32\cdots 91}a^{14}-\frac{82\cdots 96}{66\cdots 59}a^{13}+\frac{75\cdots 18}{32\cdots 91}a^{12}+\frac{59\cdots 00}{46\cdots 13}a^{11}-\frac{81\cdots 78}{46\cdots 13}a^{10}-\frac{39\cdots 22}{46\cdots 13}a^{9}+\frac{57\cdots 84}{66\cdots 59}a^{8}+\frac{23\cdots 03}{66\cdots 59}a^{7}-\frac{17\cdots 68}{66\cdots 59}a^{6}-\frac{85\cdots 64}{94\cdots 37}a^{5}+\frac{64\cdots 64}{15\cdots 09}a^{4}+\frac{16\cdots 61}{13\cdots 91}a^{3}-\frac{25\cdots 39}{13\cdots 91}a^{2}-\frac{88\cdots 69}{13\cdots 91}a-\frac{93\cdots 81}{13\cdots 91}$, $\frac{46\cdots 10}{22\cdots 37}a^{26}-\frac{15\cdots 12}{22\cdots 37}a^{25}+\frac{13\cdots 92}{22\cdots 37}a^{24}+\frac{59\cdots 40}{22\cdots 37}a^{23}-\frac{14\cdots 21}{22\cdots 37}a^{22}-\frac{77\cdots 06}{22\cdots 37}a^{21}+\frac{48\cdots 13}{12\cdots 77}a^{20}+\frac{50\cdots 20}{22\cdots 37}a^{19}-\frac{30\cdots 46}{22\cdots 37}a^{18}-\frac{18\cdots 51}{22\cdots 37}a^{17}+\frac{68\cdots 53}{22\cdots 37}a^{16}+\frac{47\cdots 21}{24\cdots 01}a^{15}-\frac{14\cdots 52}{32\cdots 91}a^{14}-\frac{97\cdots 37}{32\cdots 91}a^{13}+\frac{15\cdots 99}{32\cdots 91}a^{12}+\frac{14\cdots 10}{46\cdots 13}a^{11}-\frac{16\cdots 59}{46\cdots 13}a^{10}-\frac{92\cdots 14}{46\cdots 13}a^{9}+\frac{11\cdots 10}{66\cdots 59}a^{8}+\frac{55\cdots 99}{66\cdots 59}a^{7}-\frac{48\cdots 36}{94\cdots 37}a^{6}-\frac{19\cdots 13}{94\cdots 37}a^{5}+\frac{12\cdots 16}{15\cdots 09}a^{4}+\frac{26\cdots 55}{94\cdots 37}a^{3}-\frac{43\cdots 57}{13\cdots 91}a^{2}-\frac{20\cdots 44}{13\cdots 91}a-\frac{21\cdots 18}{13\cdots 91}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 19270877866665787000000 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  \( 27 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{27}\cdot(2\pi)^{0}\cdot 19270877866665787000000 \cdot 1}{2\cdot\sqrt{48101612257836942453739988389275689559050655489929124517167441}}\cr\approx \mathstrut & 0.186466822236493 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^27 - 297*x^25 - 333*x^24 + 36072*x^23 + 61137*x^22 - 2408229*x^21 - 4602204*x^20 + 100240344*x^19 + 188516023*x^18 - 2781715041*x^17 - 4662330939*x^16 + 53254728639*x^15 + 72347832963*x^14 - 710593217208*x^13 - 702999577791*x^12 + 6541339948164*x^11 + 4112713341072*x^10 - 40351928688652*x^9 - 13288812632739*x^8 + 159103682965773*x^7 + 20338350996222*x^6 - 374458043565018*x^5 - 13448413508607*x^4 + 467216524743780*x^3 + 21089835153009*x^2 - 231674121736407*x - 25404456460733) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^27 - 297*x^25 - 333*x^24 + 36072*x^23 + 61137*x^22 - 2408229*x^21 - 4602204*x^20 + 100240344*x^19 + 188516023*x^18 - 2781715041*x^17 - 4662330939*x^16 + 53254728639*x^15 + 72347832963*x^14 - 710593217208*x^13 - 702999577791*x^12 + 6541339948164*x^11 + 4112713341072*x^10 - 40351928688652*x^9 - 13288812632739*x^8 + 159103682965773*x^7 + 20338350996222*x^6 - 374458043565018*x^5 - 13448413508607*x^4 + 467216524743780*x^3 + 21089835153009*x^2 - 231674121736407*x - 25404456460733, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 297*x^25 - 333*x^24 + 36072*x^23 + 61137*x^22 - 2408229*x^21 - 4602204*x^20 + 100240344*x^19 + 188516023*x^18 - 2781715041*x^17 - 4662330939*x^16 + 53254728639*x^15 + 72347832963*x^14 - 710593217208*x^13 - 702999577791*x^12 + 6541339948164*x^11 + 4112713341072*x^10 - 40351928688652*x^9 - 13288812632739*x^8 + 159103682965773*x^7 + 20338350996222*x^6 - 374458043565018*x^5 - 13448413508607*x^4 + 467216524743780*x^3 + 21089835153009*x^2 - 231674121736407*x - 25404456460733); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^27 - 297*x^25 - 333*x^24 + 36072*x^23 + 61137*x^22 - 2408229*x^21 - 4602204*x^20 + 100240344*x^19 + 188516023*x^18 - 2781715041*x^17 - 4662330939*x^16 + 53254728639*x^15 + 72347832963*x^14 - 710593217208*x^13 - 702999577791*x^12 + 6541339948164*x^11 + 4112713341072*x^10 - 40351928688652*x^9 - 13288812632739*x^8 + 159103682965773*x^7 + 20338350996222*x^6 - 374458043565018*x^5 - 13448413508607*x^4 + 467216524743780*x^3 + 21089835153009*x^2 - 231674121736407*x - 25404456460733); 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

$C_9.C_3^2$ (as 27T28):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 81
The 33 conjugacy class representatives for $C_9.C_3^2$
Character table for $C_9.C_3^2$

Intermediate fields

3.3.13689.1, 3.3.169.1, \(\Q(\zeta_{9})^+\), 3.3.13689.2, 9.9.2565164201769.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 27 siblings: data not computed
Minimal sibling: 27.27.1172432259184537536650001252175069616579556093422107974506601.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.9.0.1}{9} }^{3}$ R ${\href{/padicField/5.9.0.1}{9} }^{3}$ R ${\href{/padicField/11.9.0.1}{9} }^{3}$ R ${\href{/padicField/17.3.0.1}{3} }^{9}$ ${\href{/padicField/19.3.0.1}{3} }^{9}$ ${\href{/padicField/23.9.0.1}{9} }^{3}$ ${\href{/padicField/29.9.0.1}{9} }^{3}$ ${\href{/padicField/31.9.0.1}{9} }^{3}$ ${\href{/padicField/37.3.0.1}{3} }^{9}$ ${\href{/padicField/41.9.0.1}{9} }^{3}$ ${\href{/padicField/43.9.0.1}{9} }^{3}$ ${\href{/padicField/47.9.0.1}{9} }^{3}$ ${\href{/padicField/53.3.0.1}{3} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{9}$ ${\href{/padicField/59.9.0.1}{9} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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
\(3\) Copy content Toggle raw display Deg $27$$9$$3$$66$
\(7\) Copy content Toggle raw display 7.9.0.1$x^{9} + 6 x^{4} + x^{3} + 6 x + 4$$1$$9$$0$$C_9$$$[\ ]^{9}$$
7.9.6.3$x^{9} + 18 x^{8} + 108 x^{7} + 228 x^{6} + 144 x^{5} + 432 x^{4} + 48 x^{3} + 288 x^{2} + 7 x + 64$$3$$3$$6$$C_9$$$[\ ]_{3}^{3}$$
7.9.6.2$x^{9} + 18 x^{8} + 108 x^{7} + 228 x^{6} + 144 x^{5} + 432 x^{4} + 48 x^{3} + 295 x^{2} + 64$$3$$3$$6$$C_9$$$[\ ]_{3}^{3}$$
\(13\) Copy content Toggle raw display Deg $27$$3$$9$$18$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)