Normalized defining polynomial

\( x^{27} - 9 x^{25} + 36 x^{23} - 18 x^{22} - 144 x^{21} - 54 x^{20} + 423 x^{19} + 42 x^{18} - 405 x^{17} + \cdots - 64 \) x^27 - 9*x^25 + 36*x^23 - 18*x^22 - 144*x^21 - 54*x^20 + 423*x^19 + 42*x^18 - 405*x^17 - 1062*x^16 + 1188*x^15 + 1872*x^14 - 3258*x^13 - 1980*x^12 + 5724*x^11 + 504*x^10 - 9564*x^9 + 7128*x^8 + 3744*x^7 - 8640*x^6 + 4248*x^5 + 1008*x^4 - 1440*x^3 + 288*x - 64

Invariants

Degree:		$27$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[3, 12]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(638641997009196294829147338632324972544\) 638641997009196294829147338632324972544 \(\medspace = 2^{56}\cdot 3^{46}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(27.37\)	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:		$2^{17/6}3^{31/18}\approx 47.27452225037032$
Ramified primes:		\(2\), \(3\) 2, 3	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q\)
$\card{ \Aut(K/\Q) }$:		$1$	sage: K.automorphisms()   magma: Automorphisms(K);   oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}$, $\frac{1}{6}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}+\frac{1}{3}a$, $\frac{1}{6}a^{11}-\frac{1}{6}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}+\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{6}a^{12}-\frac{1}{2}a^{4}+\frac{1}{3}a^{3}$, $\frac{1}{6}a^{13}-\frac{1}{2}a^{5}+\frac{1}{3}a^{4}$, $\frac{1}{36}a^{14}-\frac{1}{18}a^{13}-\frac{1}{18}a^{12}+\frac{1}{18}a^{11}+\frac{1}{18}a^{10}-\frac{1}{9}a^{9}-\frac{1}{6}a^{8}-\frac{1}{6}a^{7}-\frac{5}{12}a^{6}-\frac{5}{18}a^{5}+\frac{1}{18}a^{4}-\frac{4}{9}a^{3}-\frac{7}{18}a^{2}+\frac{1}{9}a-\frac{2}{9}$, $\frac{1}{36}a^{15}-\frac{1}{18}a^{12}+\frac{1}{9}a^{9}-\frac{1}{2}a^{8}-\frac{1}{4}a^{7}-\frac{1}{9}a^{6}-\frac{1}{2}a^{5}-\frac{5}{18}a^{3}-\frac{4}{9}$, $\frac{1}{36}a^{16}-\frac{1}{18}a^{13}-\frac{1}{18}a^{10}-\frac{1}{6}a^{9}+\frac{1}{4}a^{8}-\frac{1}{9}a^{7}+\frac{2}{9}a^{4}+\frac{2}{9}a-\frac{1}{3}$, $\frac{1}{36}a^{17}+\frac{1}{18}a^{13}+\frac{1}{18}a^{12}+\frac{1}{18}a^{11}-\frac{1}{18}a^{10}+\frac{1}{36}a^{9}-\frac{4}{9}a^{8}-\frac{1}{3}a^{7}+\frac{1}{6}a^{6}+\frac{1}{6}a^{5}-\frac{1}{18}a^{4}+\frac{4}{9}a^{3}+\frac{4}{9}a^{2}-\frac{1}{9}a-\frac{4}{9}$, $\frac{1}{108}a^{18}+\frac{1}{18}a^{12}-\frac{1}{12}a^{10}-\frac{7}{54}a^{9}-\frac{1}{2}a^{8}-\frac{1}{6}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{1}{9}a^{3}+\frac{1}{3}a-\frac{8}{27}$, $\frac{1}{216}a^{19}-\frac{1}{216}a^{18}+\frac{1}{36}a^{13}+\frac{1}{18}a^{12}-\frac{1}{24}a^{11}-\frac{5}{216}a^{10}+\frac{4}{27}a^{9}+\frac{1}{4}a^{8}+\frac{5}{12}a^{7}-\frac{1}{6}a^{6}+\frac{1}{18}a^{4}+\frac{1}{9}a^{3}+\frac{1}{6}a^{2}+\frac{5}{27}a-\frac{5}{27}$, $\frac{1}{216}a^{20}-\frac{1}{216}a^{18}-\frac{1}{36}a^{13}+\frac{5}{72}a^{12}+\frac{5}{108}a^{11}+\frac{5}{72}a^{10}+\frac{1}{108}a^{9}-\frac{1}{6}a^{8}-\frac{1}{12}a^{7}+\frac{1}{4}a^{6}+\frac{1}{3}a^{5}-\frac{2}{9}a^{4}-\frac{5}{18}a^{3}+\frac{2}{27}a^{2}-\frac{1}{9}a+\frac{1}{27}$, $\frac{1}{432}a^{21}-\frac{1}{432}a^{19}-\frac{1}{72}a^{17}-\frac{1}{72}a^{16}-\frac{1}{72}a^{15}-\frac{1}{72}a^{14}+\frac{5}{144}a^{13}-\frac{13}{216}a^{12}-\frac{11}{144}a^{11}-\frac{5}{216}a^{10}-\frac{11}{72}a^{9}-\frac{4}{9}a^{8}+\frac{2}{9}a^{7}-\frac{1}{9}a^{6}-\frac{7}{36}a^{5}+\frac{5}{18}a^{4}+\frac{31}{108}a^{3}+\frac{1}{18}a^{2}-\frac{11}{54}a+\frac{4}{9}$, $\frac{1}{432}a^{22}-\frac{1}{432}a^{20}-\frac{1}{216}a^{18}-\frac{1}{72}a^{17}-\frac{1}{72}a^{16}-\frac{1}{72}a^{15}+\frac{1}{144}a^{14}-\frac{1}{216}a^{13}+\frac{5}{144}a^{12}-\frac{17}{216}a^{11}+\frac{1}{24}a^{10}-\frac{7}{54}a^{9}-\frac{1}{9}a^{8}+\frac{1}{18}a^{7}+\frac{1}{18}a^{6}+\frac{1}{18}a^{5}-\frac{29}{108}a^{4}-\frac{7}{18}a^{3}+\frac{5}{27}a^{2}+\frac{1}{3}a-\frac{11}{27}$, $\frac{1}{432}a^{23}-\frac{1}{432}a^{19}-\frac{1}{144}a^{15}+\frac{1}{108}a^{14}+\frac{1}{24}a^{13}+\frac{1}{36}a^{12}+\frac{5}{144}a^{11}-\frac{7}{108}a^{10}+\frac{1}{24}a^{9}-\frac{1}{2}a^{8}+\frac{1}{12}a^{7}+\frac{7}{36}a^{6}+\frac{23}{54}a^{5}+\frac{1}{6}a^{4}-\frac{7}{36}a^{3}-\frac{7}{18}a^{2}+\frac{7}{54}a$, $\frac{1}{432}a^{24}-\frac{1}{432}a^{20}-\frac{1}{144}a^{16}+\frac{1}{108}a^{15}-\frac{1}{72}a^{14}-\frac{1}{36}a^{13}-\frac{1}{48}a^{12}-\frac{1}{108}a^{11}-\frac{5}{72}a^{10}-\frac{1}{9}a^{9}+\frac{5}{12}a^{8}+\frac{1}{36}a^{7}+\frac{7}{27}a^{6}-\frac{5}{18}a^{5}-\frac{5}{36}a^{4}+\frac{1}{6}a^{3}+\frac{13}{54}a^{2}-\frac{2}{9}a-\frac{2}{9}$, $\frac{1}{864}a^{25}-\frac{1}{864}a^{23}-\frac{1}{432}a^{20}+\frac{1}{432}a^{18}+\frac{1}{288}a^{17}+\frac{5}{432}a^{16}+\frac{1}{288}a^{15}+\frac{1}{432}a^{14}-\frac{1}{18}a^{13}-\frac{1}{18}a^{12}+\frac{23}{432}a^{11}+\frac{1}{72}a^{10}+\frac{7}{108}a^{9}-\frac{5}{36}a^{8}+\frac{103}{216}a^{7}-\frac{1}{18}a^{6}+\frac{7}{27}a^{5}+\frac{7}{18}a^{4}-\frac{7}{36}a^{3}+\frac{2}{27}a^{2}-\frac{7}{18}a+\frac{1}{27}$, $\frac{1}{55\!\cdots\!32}a^{26}+\frac{43\!\cdots\!53}{13\!\cdots\!08}a^{25}+\frac{19\!\cdots\!33}{55\!\cdots\!32}a^{24}-\frac{11\!\cdots\!19}{15\!\cdots\!12}a^{23}+\frac{17\!\cdots\!33}{27\!\cdots\!16}a^{22}+\frac{41\!\cdots\!91}{92\!\cdots\!72}a^{21}+\frac{28\!\cdots\!83}{13\!\cdots\!08}a^{20}+\frac{60\!\cdots\!85}{27\!\cdots\!16}a^{19}-\frac{14\!\cdots\!81}{55\!\cdots\!32}a^{18}+\frac{35\!\cdots\!41}{27\!\cdots\!16}a^{17}+\frac{30\!\cdots\!13}{55\!\cdots\!32}a^{16}-\frac{32\!\cdots\!51}{27\!\cdots\!16}a^{15}+\frac{12\!\cdots\!83}{92\!\cdots\!72}a^{14}+\frac{16\!\cdots\!49}{69\!\cdots\!04}a^{13}+\frac{64\!\cdots\!97}{30\!\cdots\!24}a^{12}+\frac{23\!\cdots\!33}{34\!\cdots\!52}a^{11}-\frac{36\!\cdots\!15}{69\!\cdots\!04}a^{10}-\frac{35\!\cdots\!35}{69\!\cdots\!04}a^{9}+\frac{74\!\cdots\!91}{44\!\cdots\!68}a^{8}+\frac{11\!\cdots\!99}{69\!\cdots\!04}a^{7}-\frac{15\!\cdots\!83}{69\!\cdots\!04}a^{6}-\frac{15\!\cdots\!55}{57\!\cdots\!42}a^{5}-\frac{89\!\cdots\!51}{69\!\cdots\!04}a^{4}-\frac{10\!\cdots\!89}{28\!\cdots\!71}a^{3}-\frac{87\!\cdots\!93}{34\!\cdots\!52}a^{2}+\frac{43\!\cdots\!55}{86\!\cdots\!13}a-\frac{21\!\cdots\!41}{86\!\cdots\!13}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, 1/3*a^9 - 1/3, 1/6*a^10 - 1/2*a^8 - 1/2*a^6 - 1/2*a^4 + 1/3*a, 1/6*a^11 - 1/6*a^9 - 1/2*a^7 - 1/2*a^5 + 1/3*a^2 - 1/3, 1/6*a^12 - 1/2*a^4 + 1/3*a^3, 1/6*a^13 - 1/2*a^5 + 1/3*a^4, 1/36*a^14 - 1/18*a^13 - 1/18*a^12 + 1/18*a^11 + 1/18*a^10 - 1/9*a^9 - 1/6*a^8 - 1/6*a^7 - 5/12*a^6 - 5/18*a^5 + 1/18*a^4 - 4/9*a^3 - 7/18*a^2 + 1/9*a - 2/9, 1/36*a^15 - 1/18*a^12 + 1/9*a^9 - 1/2*a^8 - 1/4*a^7 - 1/9*a^6 - 1/2*a^5 - 5/18*a^3 - 4/9, 1/36*a^16 - 1/18*a^13 - 1/18*a^10 - 1/6*a^9 + 1/4*a^8 - 1/9*a^7 + 2/9*a^4 + 2/9*a - 1/3, 1/36*a^17 + 1/18*a^13 + 1/18*a^12 + 1/18*a^11 - 1/18*a^10 + 1/36*a^9 - 4/9*a^8 - 1/3*a^7 + 1/6*a^6 + 1/6*a^5 - 1/18*a^4 + 4/9*a^3 + 4/9*a^2 - 1/9*a - 4/9, 1/108*a^18 + 1/18*a^12 - 1/12*a^10 - 7/54*a^9 - 1/2*a^8 - 1/6*a^6 - 1/2*a^5 - 1/2*a^4 + 1/9*a^3 + 1/3*a - 8/27, 1/216*a^19 - 1/216*a^18 + 1/36*a^13 + 1/18*a^12 - 1/24*a^11 - 5/216*a^10 + 4/27*a^9 + 1/4*a^8 + 5/12*a^7 - 1/6*a^6 + 1/18*a^4 + 1/9*a^3 + 1/6*a^2 + 5/27*a - 5/27, 1/216*a^20 - 1/216*a^18 - 1/36*a^13 + 5/72*a^12 + 5/108*a^11 + 5/72*a^10 + 1/108*a^9 - 1/6*a^8 - 1/12*a^7 + 1/4*a^6 + 1/3*a^5 - 2/9*a^4 - 5/18*a^3 + 2/27*a^2 - 1/9*a + 1/27, 1/432*a^21 - 1/432*a^19 - 1/72*a^17 - 1/72*a^16 - 1/72*a^15 - 1/72*a^14 + 5/144*a^13 - 13/216*a^12 - 11/144*a^11 - 5/216*a^10 - 11/72*a^9 - 4/9*a^8 + 2/9*a^7 - 1/9*a^6 - 7/36*a^5 + 5/18*a^4 + 31/108*a^3 + 1/18*a^2 - 11/54*a + 4/9, 1/432*a^22 - 1/432*a^20 - 1/216*a^18 - 1/72*a^17 - 1/72*a^16 - 1/72*a^15 + 1/144*a^14 - 1/216*a^13 + 5/144*a^12 - 17/216*a^11 + 1/24*a^10 - 7/54*a^9 - 1/9*a^8 + 1/18*a^7 + 1/18*a^6 + 1/18*a^5 - 29/108*a^4 - 7/18*a^3 + 5/27*a^2 + 1/3*a - 11/27, 1/432*a^23 - 1/432*a^19 - 1/144*a^15 + 1/108*a^14 + 1/24*a^13 + 1/36*a^12 + 5/144*a^11 - 7/108*a^10 + 1/24*a^9 - 1/2*a^8 + 1/12*a^7 + 7/36*a^6 + 23/54*a^5 + 1/6*a^4 - 7/36*a^3 - 7/18*a^2 + 7/54*a, 1/432*a^24 - 1/432*a^20 - 1/144*a^16 + 1/108*a^15 - 1/72*a^14 - 1/36*a^13 - 1/48*a^12 - 1/108*a^11 - 5/72*a^10 - 1/9*a^9 + 5/12*a^8 + 1/36*a^7 + 7/27*a^6 - 5/18*a^5 - 5/36*a^4 + 1/6*a^3 + 13/54*a^2 - 2/9*a - 2/9, 1/864*a^25 - 1/864*a^23 - 1/432*a^20 + 1/432*a^18 + 1/288*a^17 + 5/432*a^16 + 1/288*a^15 + 1/432*a^14 - 1/18*a^13 - 1/18*a^12 + 23/432*a^11 + 1/72*a^10 + 7/108*a^9 - 5/36*a^8 + 103/216*a^7 - 1/18*a^6 + 7/27*a^5 + 7/18*a^4 - 7/36*a^3 + 2/27*a^2 - 7/18*a + 1/27, 1/554432789686209656692032*a^26 + 43753545060189282953/138608197421552414173008*a^25 + 191331198034867755533/554432789686209656692032*a^24 - 11268095870177193319/15400910824616934908112*a^23 + 178412502755138030633/277216394843104828346016*a^22 + 41515309389063503591/92405464947701609448672*a^21 + 284599280701949087383/138608197421552414173008*a^20 + 603372288146478553285/277216394843104828346016*a^19 - 1409180163315353325281/554432789686209656692032*a^18 + 3599689999818265843841/277216394843104828346016*a^17 + 3031548714980782163813/554432789686209656692032*a^16 - 3211412487332130521651/277216394843104828346016*a^15 + 1247709594414617442683/92405464947701609448672*a^14 + 1619677843711745407049/69304098710776207086504*a^13 + 642135599384004938497/30801821649233869816224*a^12 + 231961004223274083433/34652049355388103543252*a^11 - 3680275162785768422015/69304098710776207086504*a^10 - 3510047560432869551635/69304098710776207086504*a^9 + 749226255684764452591/4471232174888787553968*a^8 + 11642281215305277758299/69304098710776207086504*a^7 - 15386654390255281365383/69304098710776207086504*a^6 - 1508954746358286613355/5775341559231350590542*a^5 - 8900343302552788293151/69304098710776207086504*a^4 - 1071840195575537866189/2887670779615675295271*a^3 - 8713483333290565105193/34652049355388103543252*a^2 + 434070679015412420255/8663012338847025885813*a - 2148828193355759947841/8663012338847025885813

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$14$	sage: UK.rank()   gp: K.fu   magma: UnitRank(K);   oscar: rank(UK)
Torsion generator:		\( -1 \) -1  (order $2$)	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:		$\frac{51\!\cdots\!39}{55\!\cdots\!32}a^{26}+\frac{22\!\cdots\!17}{69\!\cdots\!04}a^{25}-\frac{45\!\cdots\!33}{55\!\cdots\!32}a^{24}-\frac{44\!\cdots\!03}{15\!\cdots\!12}a^{23}+\frac{90\!\cdots\!93}{27\!\cdots\!16}a^{22}-\frac{54\!\cdots\!85}{92\!\cdots\!72}a^{21}-\frac{23\!\cdots\!01}{17\!\cdots\!26}a^{20}-\frac{27\!\cdots\!05}{27\!\cdots\!16}a^{19}+\frac{19\!\cdots\!37}{55\!\cdots\!32}a^{18}+\frac{43\!\cdots\!97}{27\!\cdots\!16}a^{17}-\frac{17\!\cdots\!29}{55\!\cdots\!32}a^{16}-\frac{30\!\cdots\!17}{27\!\cdots\!16}a^{15}+\frac{68\!\cdots\!75}{92\!\cdots\!72}a^{14}+\frac{27\!\cdots\!83}{13\!\cdots\!08}a^{13}-\frac{22\!\cdots\!05}{92\!\cdots\!72}a^{12}-\frac{91\!\cdots\!53}{34\!\cdots\!52}a^{11}+\frac{31\!\cdots\!53}{69\!\cdots\!04}a^{10}+\frac{13\!\cdots\!37}{69\!\cdots\!04}a^{9}-\frac{37\!\cdots\!63}{44\!\cdots\!68}a^{8}+\frac{26\!\cdots\!99}{69\!\cdots\!04}a^{7}+\frac{33\!\cdots\!71}{69\!\cdots\!04}a^{6}-\frac{75\!\cdots\!45}{11\!\cdots\!84}a^{5}+\frac{12\!\cdots\!69}{69\!\cdots\!04}a^{4}+\frac{89\!\cdots\!01}{57\!\cdots\!42}a^{3}-\frac{27\!\cdots\!85}{34\!\cdots\!52}a^{2}-\frac{25\!\cdots\!96}{86\!\cdots\!13}a+\frac{15\!\cdots\!96}{86\!\cdots\!13}$, $\frac{16\!\cdots\!39}{55\!\cdots\!32}a^{26}-\frac{16\!\cdots\!88}{96\!\cdots\!57}a^{25}+\frac{16\!\cdots\!25}{61\!\cdots\!48}a^{24}+\frac{22\!\cdots\!51}{15\!\cdots\!12}a^{23}-\frac{27\!\cdots\!61}{27\!\cdots\!16}a^{22}-\frac{70\!\cdots\!01}{27\!\cdots\!16}a^{21}+\frac{15\!\cdots\!05}{34\!\cdots\!52}a^{20}+\frac{37\!\cdots\!83}{92\!\cdots\!72}a^{19}-\frac{58\!\cdots\!05}{55\!\cdots\!32}a^{18}-\frac{19\!\cdots\!73}{27\!\cdots\!16}a^{17}+\frac{15\!\cdots\!91}{18\!\cdots\!44}a^{16}+\frac{33\!\cdots\!03}{92\!\cdots\!72}a^{15}-\frac{13\!\cdots\!59}{92\!\cdots\!72}a^{14}-\frac{90\!\cdots\!47}{13\!\cdots\!08}a^{13}+\frac{17\!\cdots\!83}{27\!\cdots\!16}a^{12}+\frac{65\!\cdots\!83}{69\!\cdots\!04}a^{11}-\frac{34\!\cdots\!24}{28\!\cdots\!71}a^{10}-\frac{57\!\cdots\!05}{69\!\cdots\!04}a^{9}+\frac{10\!\cdots\!27}{44\!\cdots\!68}a^{8}-\frac{17\!\cdots\!49}{23\!\cdots\!68}a^{7}-\frac{36\!\cdots\!55}{23\!\cdots\!68}a^{6}+\frac{19\!\cdots\!65}{11\!\cdots\!84}a^{5}-\frac{21\!\cdots\!81}{69\!\cdots\!04}a^{4}-\frac{41\!\cdots\!27}{86\!\cdots\!13}a^{3}+\frac{55\!\cdots\!97}{34\!\cdots\!52}a^{2}+\frac{26\!\cdots\!88}{28\!\cdots\!71}a-\frac{28\!\cdots\!32}{86\!\cdots\!13}$, $\frac{59\!\cdots\!91}{55\!\cdots\!32}a^{26}-\frac{55\!\cdots\!41}{69\!\cdots\!04}a^{25}+\frac{50\!\cdots\!53}{55\!\cdots\!32}a^{24}+\frac{96\!\cdots\!27}{13\!\cdots\!08}a^{23}-\frac{94\!\cdots\!07}{27\!\cdots\!16}a^{22}-\frac{65\!\cdots\!35}{92\!\cdots\!72}a^{21}+\frac{26\!\cdots\!99}{17\!\cdots\!26}a^{20}+\frac{15\!\cdots\!51}{92\!\cdots\!72}a^{19}-\frac{18\!\cdots\!29}{55\!\cdots\!32}a^{18}-\frac{86\!\cdots\!97}{27\!\cdots\!16}a^{17}+\frac{12\!\cdots\!81}{55\!\cdots\!32}a^{16}+\frac{36\!\cdots\!97}{27\!\cdots\!16}a^{15}-\frac{77\!\cdots\!55}{27\!\cdots\!16}a^{14}-\frac{32\!\cdots\!45}{13\!\cdots\!08}a^{13}+\frac{53\!\cdots\!07}{30\!\cdots\!24}a^{12}+\frac{25\!\cdots\!97}{69\!\cdots\!04}a^{11}-\frac{40\!\cdots\!89}{11\!\cdots\!84}a^{10}-\frac{24\!\cdots\!29}{69\!\cdots\!04}a^{9}+\frac{34\!\cdots\!11}{44\!\cdots\!68}a^{8}-\frac{10\!\cdots\!87}{69\!\cdots\!04}a^{7}-\frac{38\!\cdots\!31}{69\!\cdots\!04}a^{6}+\frac{17\!\cdots\!27}{34\!\cdots\!52}a^{5}-\frac{25\!\cdots\!69}{69\!\cdots\!04}a^{4}-\frac{48\!\cdots\!73}{28\!\cdots\!71}a^{3}+\frac{11\!\cdots\!45}{34\!\cdots\!52}a^{2}+\frac{89\!\cdots\!44}{28\!\cdots\!71}a-\frac{69\!\cdots\!14}{86\!\cdots\!13}$, $\frac{93\!\cdots\!81}{18\!\cdots\!44}a^{26}+\frac{44\!\cdots\!59}{34\!\cdots\!52}a^{25}-\frac{25\!\cdots\!69}{55\!\cdots\!32}a^{24}-\frac{17\!\cdots\!01}{13\!\cdots\!08}a^{23}+\frac{49\!\cdots\!23}{27\!\cdots\!16}a^{22}-\frac{10\!\cdots\!17}{27\!\cdots\!16}a^{21}-\frac{10\!\cdots\!05}{13\!\cdots\!08}a^{20}-\frac{13\!\cdots\!43}{27\!\cdots\!16}a^{19}+\frac{36\!\cdots\!11}{18\!\cdots\!44}a^{18}+\frac{25\!\cdots\!81}{30\!\cdots\!24}a^{17}-\frac{90\!\cdots\!57}{55\!\cdots\!32}a^{16}-\frac{16\!\cdots\!61}{27\!\cdots\!16}a^{15}+\frac{11\!\cdots\!47}{27\!\cdots\!16}a^{14}+\frac{72\!\cdots\!99}{69\!\cdots\!04}a^{13}-\frac{35\!\cdots\!61}{27\!\cdots\!16}a^{12}-\frac{17\!\cdots\!55}{13\!\cdots\!08}a^{11}+\frac{16\!\cdots\!23}{69\!\cdots\!04}a^{10}+\frac{25\!\cdots\!22}{28\!\cdots\!71}a^{9}-\frac{64\!\cdots\!05}{14\!\cdots\!56}a^{8}+\frac{16\!\cdots\!33}{69\!\cdots\!04}a^{7}+\frac{15\!\cdots\!85}{69\!\cdots\!04}a^{6}-\frac{29\!\cdots\!75}{86\!\cdots\!13}a^{5}+\frac{91\!\cdots\!99}{69\!\cdots\!04}a^{4}+\frac{18\!\cdots\!53}{34\!\cdots\!52}a^{3}-\frac{13\!\cdots\!53}{34\!\cdots\!52}a^{2}-\frac{25\!\cdots\!19}{17\!\cdots\!26}a+\frac{75\!\cdots\!97}{96\!\cdots\!57}$, $\frac{25\!\cdots\!37}{61\!\cdots\!48}a^{26}-\frac{28\!\cdots\!53}{92\!\cdots\!72}a^{25}+\frac{66\!\cdots\!25}{18\!\cdots\!44}a^{24}+\frac{75\!\cdots\!65}{27\!\cdots\!16}a^{23}-\frac{38\!\cdots\!47}{27\!\cdots\!16}a^{22}-\frac{91\!\cdots\!49}{27\!\cdots\!16}a^{21}+\frac{21\!\cdots\!87}{34\!\cdots\!52}a^{20}+\frac{18\!\cdots\!01}{27\!\cdots\!16}a^{19}-\frac{78\!\cdots\!03}{55\!\cdots\!32}a^{18}-\frac{31\!\cdots\!15}{23\!\cdots\!68}a^{17}+\frac{70\!\cdots\!79}{61\!\cdots\!48}a^{16}+\frac{24\!\cdots\!21}{46\!\cdots\!36}a^{15}-\frac{34\!\cdots\!57}{27\!\cdots\!16}a^{14}-\frac{34\!\cdots\!97}{34\!\cdots\!52}a^{13}+\frac{19\!\cdots\!43}{27\!\cdots\!16}a^{12}+\frac{54\!\cdots\!01}{34\!\cdots\!52}a^{11}-\frac{10\!\cdots\!71}{69\!\cdots\!04}a^{10}-\frac{56\!\cdots\!99}{34\!\cdots\!52}a^{9}+\frac{49\!\cdots\!67}{14\!\cdots\!56}a^{8}-\frac{34\!\cdots\!03}{11\!\cdots\!84}a^{7}-\frac{66\!\cdots\!07}{23\!\cdots\!68}a^{6}+\frac{37\!\cdots\!99}{17\!\cdots\!26}a^{5}+\frac{19\!\cdots\!63}{69\!\cdots\!04}a^{4}-\frac{20\!\cdots\!39}{17\!\cdots\!26}a^{3}+\frac{10\!\cdots\!93}{34\!\cdots\!52}a^{2}+\frac{17\!\cdots\!86}{86\!\cdots\!13}a-\frac{70\!\cdots\!36}{86\!\cdots\!13}$, $\frac{15\!\cdots\!43}{13\!\cdots\!08}a^{26}+\frac{30\!\cdots\!09}{30\!\cdots\!24}a^{25}-\frac{16\!\cdots\!51}{17\!\cdots\!26}a^{24}-\frac{23\!\cdots\!79}{27\!\cdots\!16}a^{23}+\frac{47\!\cdots\!41}{13\!\cdots\!08}a^{22}+\frac{19\!\cdots\!95}{17\!\cdots\!26}a^{21}-\frac{21\!\cdots\!11}{13\!\cdots\!08}a^{20}-\frac{27\!\cdots\!91}{13\!\cdots\!08}a^{19}+\frac{18\!\cdots\!25}{57\!\cdots\!42}a^{18}+\frac{98\!\cdots\!35}{27\!\cdots\!16}a^{17}-\frac{85\!\cdots\!95}{46\!\cdots\!36}a^{16}-\frac{38\!\cdots\!99}{27\!\cdots\!16}a^{15}+\frac{80\!\cdots\!11}{69\!\cdots\!04}a^{14}+\frac{16\!\cdots\!83}{69\!\cdots\!04}a^{13}-\frac{54\!\cdots\!35}{34\!\cdots\!52}a^{12}-\frac{34\!\cdots\!70}{86\!\cdots\!13}a^{11}+\frac{22\!\cdots\!63}{69\!\cdots\!04}a^{10}+\frac{99\!\cdots\!97}{25\!\cdots\!52}a^{9}-\frac{87\!\cdots\!31}{11\!\cdots\!92}a^{8}+\frac{15\!\cdots\!37}{23\!\cdots\!68}a^{7}+\frac{20\!\cdots\!33}{34\!\cdots\!52}a^{6}-\frac{41\!\cdots\!84}{86\!\cdots\!13}a^{5}-\frac{17\!\cdots\!99}{17\!\cdots\!26}a^{4}+\frac{30\!\cdots\!51}{17\!\cdots\!26}a^{3}-\frac{22\!\cdots\!04}{86\!\cdots\!13}a^{2}-\frac{26\!\cdots\!64}{86\!\cdots\!13}a+\frac{15\!\cdots\!17}{28\!\cdots\!71}$, $\frac{21\!\cdots\!35}{69\!\cdots\!04}a^{26}+\frac{54\!\cdots\!59}{30\!\cdots\!24}a^{25}-\frac{12\!\cdots\!07}{46\!\cdots\!36}a^{24}-\frac{43\!\cdots\!29}{27\!\cdots\!16}a^{23}+\frac{75\!\cdots\!35}{69\!\cdots\!04}a^{22}+\frac{46\!\cdots\!39}{13\!\cdots\!08}a^{21}-\frac{53\!\cdots\!67}{11\!\cdots\!84}a^{20}-\frac{24\!\cdots\!25}{57\!\cdots\!42}a^{19}+\frac{53\!\cdots\!17}{46\!\cdots\!36}a^{18}+\frac{21\!\cdots\!21}{27\!\cdots\!16}a^{17}-\frac{63\!\cdots\!25}{64\!\cdots\!38}a^{16}-\frac{39\!\cdots\!79}{10\!\cdots\!08}a^{15}+\frac{22\!\cdots\!77}{13\!\cdots\!08}a^{14}+\frac{10\!\cdots\!15}{13\!\cdots\!08}a^{13}-\frac{92\!\cdots\!91}{13\!\cdots\!08}a^{12}-\frac{48\!\cdots\!13}{46\!\cdots\!36}a^{11}+\frac{15\!\cdots\!57}{11\!\cdots\!84}a^{10}+\frac{26\!\cdots\!05}{28\!\cdots\!71}a^{9}-\frac{30\!\cdots\!89}{11\!\cdots\!92}a^{8}+\frac{63\!\cdots\!13}{77\!\cdots\!56}a^{7}+\frac{22\!\cdots\!93}{11\!\cdots\!84}a^{6}-\frac{71\!\cdots\!97}{34\!\cdots\!52}a^{5}+\frac{85\!\cdots\!61}{34\!\cdots\!52}a^{4}+\frac{27\!\cdots\!75}{34\!\cdots\!52}a^{3}-\frac{10\!\cdots\!63}{28\!\cdots\!71}a^{2}-\frac{54\!\cdots\!89}{57\!\cdots\!42}a+\frac{21\!\cdots\!72}{32\!\cdots\!19}$, $\frac{66\!\cdots\!61}{46\!\cdots\!36}a^{26}+\frac{98\!\cdots\!77}{92\!\cdots\!72}a^{25}-\frac{47\!\cdots\!21}{38\!\cdots\!28}a^{24}-\frac{84\!\cdots\!67}{92\!\cdots\!72}a^{23}+\frac{31\!\cdots\!69}{69\!\cdots\!04}a^{22}+\frac{10\!\cdots\!23}{11\!\cdots\!84}a^{21}-\frac{15\!\cdots\!31}{77\!\cdots\!56}a^{20}-\frac{31\!\cdots\!17}{13\!\cdots\!08}a^{19}+\frac{31\!\cdots\!87}{69\!\cdots\!04}a^{18}+\frac{37\!\cdots\!83}{92\!\cdots\!72}a^{17}-\frac{14\!\cdots\!79}{46\!\cdots\!36}a^{16}-\frac{55\!\cdots\!93}{30\!\cdots\!24}a^{15}+\frac{18\!\cdots\!97}{46\!\cdots\!36}a^{14}+\frac{10\!\cdots\!89}{34\!\cdots\!52}a^{13}-\frac{10\!\cdots\!63}{46\!\cdots\!36}a^{12}-\frac{37\!\cdots\!37}{77\!\cdots\!56}a^{11}+\frac{32\!\cdots\!35}{69\!\cdots\!04}a^{10}+\frac{31\!\cdots\!51}{69\!\cdots\!04}a^{9}-\frac{19\!\cdots\!51}{18\!\cdots\!82}a^{8}+\frac{15\!\cdots\!69}{77\!\cdots\!56}a^{7}+\frac{28\!\cdots\!45}{38\!\cdots\!28}a^{6}-\frac{19\!\cdots\!96}{28\!\cdots\!71}a^{5}+\frac{18\!\cdots\!05}{34\!\cdots\!52}a^{4}+\frac{69\!\cdots\!91}{32\!\cdots\!19}a^{3}-\frac{30\!\cdots\!47}{64\!\cdots\!38}a^{2}-\frac{35\!\cdots\!66}{86\!\cdots\!13}a+\frac{95\!\cdots\!97}{86\!\cdots\!13}$, $\frac{14\!\cdots\!17}{13\!\cdots\!08}a^{26}+\frac{73\!\cdots\!53}{15\!\cdots\!12}a^{25}-\frac{68\!\cdots\!53}{77\!\cdots\!56}a^{24}-\frac{19\!\cdots\!97}{46\!\cdots\!36}a^{23}+\frac{39\!\cdots\!45}{11\!\cdots\!84}a^{22}-\frac{36\!\cdots\!87}{13\!\cdots\!08}a^{21}-\frac{20\!\cdots\!19}{13\!\cdots\!08}a^{20}-\frac{56\!\cdots\!71}{46\!\cdots\!36}a^{19}+\frac{50\!\cdots\!17}{13\!\cdots\!08}a^{18}+\frac{28\!\cdots\!55}{13\!\cdots\!08}a^{17}-\frac{70\!\cdots\!81}{23\!\cdots\!68}a^{16}-\frac{55\!\cdots\!37}{46\!\cdots\!36}a^{15}+\frac{16\!\cdots\!85}{25\!\cdots\!52}a^{14}+\frac{99\!\cdots\!51}{46\!\cdots\!36}a^{13}-\frac{32\!\cdots\!89}{13\!\cdots\!08}a^{12}-\frac{41\!\cdots\!21}{13\!\cdots\!08}a^{11}+\frac{12\!\cdots\!54}{28\!\cdots\!71}a^{10}+\frac{16\!\cdots\!07}{69\!\cdots\!04}a^{9}-\frac{47\!\cdots\!57}{55\!\cdots\!46}a^{8}+\frac{99\!\cdots\!44}{28\!\cdots\!71}a^{7}+\frac{30\!\cdots\!99}{57\!\cdots\!42}a^{6}-\frac{74\!\cdots\!91}{11\!\cdots\!84}a^{5}+\frac{17\!\cdots\!87}{11\!\cdots\!84}a^{4}+\frac{57\!\cdots\!13}{34\!\cdots\!52}a^{3}-\frac{65\!\cdots\!55}{86\!\cdots\!13}a^{2}-\frac{58\!\cdots\!69}{19\!\cdots\!14}a+\frac{15\!\cdots\!98}{86\!\cdots\!13}$, $\frac{91\!\cdots\!41}{27\!\cdots\!16}a^{26}+\frac{55\!\cdots\!23}{27\!\cdots\!16}a^{25}-\frac{79\!\cdots\!99}{27\!\cdots\!16}a^{24}-\frac{16\!\cdots\!31}{92\!\cdots\!72}a^{23}+\frac{25\!\cdots\!59}{23\!\cdots\!68}a^{22}+\frac{27\!\cdots\!87}{34\!\cdots\!52}a^{21}-\frac{18\!\cdots\!37}{38\!\cdots\!28}a^{20}-\frac{64\!\cdots\!15}{13\!\cdots\!08}a^{19}+\frac{31\!\cdots\!97}{27\!\cdots\!16}a^{18}+\frac{23\!\cdots\!91}{27\!\cdots\!16}a^{17}-\frac{23\!\cdots\!17}{27\!\cdots\!16}a^{16}-\frac{11\!\cdots\!83}{27\!\cdots\!16}a^{15}+\frac{22\!\cdots\!73}{15\!\cdots\!12}a^{14}+\frac{33\!\cdots\!31}{46\!\cdots\!36}a^{13}-\frac{22\!\cdots\!79}{34\!\cdots\!52}a^{12}-\frac{49\!\cdots\!31}{46\!\cdots\!36}a^{11}+\frac{10\!\cdots\!33}{86\!\cdots\!13}a^{10}+\frac{16\!\cdots\!31}{17\!\cdots\!26}a^{9}-\frac{58\!\cdots\!11}{22\!\cdots\!84}a^{8}+\frac{51\!\cdots\!81}{69\!\cdots\!04}a^{7}+\frac{60\!\cdots\!19}{34\!\cdots\!52}a^{6}-\frac{20\!\cdots\!03}{11\!\cdots\!84}a^{5}+\frac{15\!\cdots\!69}{57\!\cdots\!42}a^{4}+\frac{18\!\cdots\!83}{34\!\cdots\!52}a^{3}-\frac{85\!\cdots\!85}{57\!\cdots\!42}a^{2}-\frac{17\!\cdots\!65}{17\!\cdots\!26}a+\frac{27\!\cdots\!04}{86\!\cdots\!13}$, $\frac{27\!\cdots\!89}{30\!\cdots\!24}a^{26}-\frac{13\!\cdots\!47}{28\!\cdots\!71}a^{25}+\frac{21\!\cdots\!87}{27\!\cdots\!16}a^{24}+\frac{28\!\cdots\!89}{69\!\cdots\!04}a^{23}-\frac{43\!\cdots\!63}{13\!\cdots\!08}a^{22}-\frac{54\!\cdots\!95}{69\!\cdots\!04}a^{21}+\frac{45\!\cdots\!31}{34\!\cdots\!52}a^{20}+\frac{82\!\cdots\!41}{69\!\cdots\!04}a^{19}-\frac{93\!\cdots\!67}{27\!\cdots\!16}a^{18}-\frac{10\!\cdots\!75}{46\!\cdots\!36}a^{17}+\frac{26\!\cdots\!45}{92\!\cdots\!72}a^{16}+\frac{16\!\cdots\!89}{13\!\cdots\!08}a^{15}-\frac{69\!\cdots\!77}{13\!\cdots\!08}a^{14}-\frac{29\!\cdots\!47}{13\!\cdots\!08}a^{13}+\frac{24\!\cdots\!59}{13\!\cdots\!08}a^{12}+\frac{43\!\cdots\!75}{13\!\cdots\!08}a^{11}-\frac{30\!\cdots\!08}{86\!\cdots\!13}a^{10}-\frac{20\!\cdots\!97}{69\!\cdots\!04}a^{9}+\frac{53\!\cdots\!85}{74\!\cdots\!28}a^{8}-\frac{71\!\cdots\!93}{38\!\cdots\!28}a^{7}-\frac{42\!\cdots\!91}{86\!\cdots\!13}a^{6}+\frac{16\!\cdots\!51}{34\!\cdots\!52}a^{5}-\frac{24\!\cdots\!79}{34\!\cdots\!52}a^{4}-\frac{43\!\cdots\!41}{34\!\cdots\!52}a^{3}+\frac{37\!\cdots\!15}{86\!\cdots\!13}a^{2}+\frac{39\!\cdots\!83}{17\!\cdots\!26}a-\frac{90\!\cdots\!82}{86\!\cdots\!13}$, $\frac{12\!\cdots\!57}{30\!\cdots\!24}a^{26}+\frac{30\!\cdots\!69}{92\!\cdots\!72}a^{25}-\frac{90\!\cdots\!59}{27\!\cdots\!16}a^{24}-\frac{79\!\cdots\!09}{27\!\cdots\!16}a^{23}+\frac{18\!\cdots\!43}{15\!\cdots\!12}a^{22}+\frac{53\!\cdots\!87}{13\!\cdots\!08}a^{21}-\frac{24\!\cdots\!89}{46\!\cdots\!36}a^{20}-\frac{11\!\cdots\!63}{17\!\cdots\!26}a^{19}+\frac{30\!\cdots\!49}{27\!\cdots\!16}a^{18}+\frac{12\!\cdots\!53}{10\!\cdots\!08}a^{17}-\frac{52\!\cdots\!51}{92\!\cdots\!72}a^{16}-\frac{13\!\cdots\!79}{27\!\cdots\!16}a^{15}+\frac{21\!\cdots\!91}{69\!\cdots\!04}a^{14}+\frac{95\!\cdots\!95}{11\!\cdots\!84}a^{13}-\frac{69\!\cdots\!65}{13\!\cdots\!08}a^{12}-\frac{75\!\cdots\!17}{57\!\cdots\!42}a^{11}+\frac{36\!\cdots\!35}{34\!\cdots\!52}a^{10}+\frac{88\!\cdots\!53}{69\!\cdots\!04}a^{9}-\frac{63\!\cdots\!89}{24\!\cdots\!76}a^{8}+\frac{26\!\cdots\!03}{77\!\cdots\!56}a^{7}+\frac{15\!\cdots\!60}{86\!\cdots\!13}a^{6}-\frac{13\!\cdots\!63}{86\!\cdots\!13}a^{5}+\frac{16\!\cdots\!79}{11\!\cdots\!84}a^{4}+\frac{85\!\cdots\!13}{17\!\cdots\!26}a^{3}-\frac{18\!\cdots\!31}{19\!\cdots\!14}a^{2}-\frac{78\!\cdots\!74}{86\!\cdots\!13}a+\frac{23\!\cdots\!07}{86\!\cdots\!13}$, $\frac{24\!\cdots\!21}{34\!\cdots\!52}a^{26}+\frac{56\!\cdots\!71}{13\!\cdots\!08}a^{25}-\frac{90\!\cdots\!19}{13\!\cdots\!08}a^{24}-\frac{60\!\cdots\!19}{13\!\cdots\!08}a^{23}+\frac{25\!\cdots\!99}{96\!\cdots\!57}a^{22}+\frac{37\!\cdots\!53}{46\!\cdots\!36}a^{21}-\frac{15\!\cdots\!57}{13\!\cdots\!08}a^{20}-\frac{15\!\cdots\!95}{13\!\cdots\!08}a^{19}+\frac{10\!\cdots\!35}{34\!\cdots\!52}a^{18}+\frac{43\!\cdots\!95}{13\!\cdots\!08}a^{17}-\frac{30\!\cdots\!03}{13\!\cdots\!08}a^{16}-\frac{15\!\cdots\!45}{13\!\cdots\!08}a^{15}+\frac{12\!\cdots\!95}{69\!\cdots\!04}a^{14}+\frac{32\!\cdots\!01}{15\!\cdots\!12}a^{13}-\frac{34\!\cdots\!99}{46\!\cdots\!36}a^{12}-\frac{42\!\cdots\!41}{13\!\cdots\!08}a^{11}+\frac{14\!\cdots\!05}{69\!\cdots\!04}a^{10}+\frac{24\!\cdots\!17}{69\!\cdots\!04}a^{9}-\frac{54\!\cdots\!79}{11\!\cdots\!92}a^{8}-\frac{10\!\cdots\!39}{34\!\cdots\!52}a^{7}+\frac{15\!\cdots\!19}{34\!\cdots\!52}a^{6}-\frac{71\!\cdots\!41}{34\!\cdots\!52}a^{5}-\frac{21\!\cdots\!65}{38\!\cdots\!28}a^{4}+\frac{34\!\cdots\!23}{38\!\cdots\!28}a^{3}+\frac{11\!\cdots\!80}{86\!\cdots\!13}a^{2}-\frac{46\!\cdots\!75}{17\!\cdots\!26}a-\frac{10\!\cdots\!36}{86\!\cdots\!13}$, $\frac{34\!\cdots\!93}{55\!\cdots\!32}a^{26}+\frac{56\!\cdots\!87}{17\!\cdots\!26}a^{25}-\frac{10\!\cdots\!21}{18\!\cdots\!44}a^{24}-\frac{99\!\cdots\!29}{34\!\cdots\!52}a^{23}+\frac{19\!\cdots\!35}{92\!\cdots\!72}a^{22}-\frac{12\!\cdots\!25}{27\!\cdots\!16}a^{21}-\frac{62\!\cdots\!29}{69\!\cdots\!04}a^{20}-\frac{22\!\cdots\!93}{27\!\cdots\!16}a^{19}+\frac{12\!\cdots\!67}{55\!\cdots\!32}a^{18}+\frac{40\!\cdots\!67}{27\!\cdots\!16}a^{17}-\frac{10\!\cdots\!75}{55\!\cdots\!32}a^{16}-\frac{78\!\cdots\!91}{10\!\cdots\!08}a^{15}+\frac{98\!\cdots\!17}{27\!\cdots\!16}a^{14}+\frac{21\!\cdots\!57}{15\!\cdots\!12}a^{13}-\frac{36\!\cdots\!93}{27\!\cdots\!16}a^{12}-\frac{27\!\cdots\!41}{13\!\cdots\!08}a^{11}+\frac{45\!\cdots\!73}{17\!\cdots\!26}a^{10}+\frac{61\!\cdots\!13}{34\!\cdots\!52}a^{9}-\frac{23\!\cdots\!53}{44\!\cdots\!68}a^{8}+\frac{11\!\cdots\!21}{69\!\cdots\!04}a^{7}+\frac{79\!\cdots\!77}{23\!\cdots\!68}a^{6}-\frac{12\!\cdots\!75}{34\!\cdots\!52}a^{5}+\frac{13\!\cdots\!77}{23\!\cdots\!68}a^{4}+\frac{37\!\cdots\!39}{34\!\cdots\!52}a^{3}-\frac{11\!\cdots\!09}{34\!\cdots\!52}a^{2}-\frac{37\!\cdots\!67}{17\!\cdots\!26}a+\frac{65\!\cdots\!39}{86\!\cdots\!13}$ 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13442105034161245307602937/69304098710776207086504*a^9 - 3730403554470060914211263/4471232174888787553968*a^8 + 26888770801954543644186299/69304098710776207086504*a^7 + 33483562489424358855356171/69304098710776207086504*a^6 - 7510754672899974428328445/11550683118462701181084*a^5 + 12149163794012300942574169/69304098710776207086504*a^4 + 898326219329146200640001/5775341559231350590542*a^3 - 2746600117639855664011285/34652049355388103543252*a^2 - 253883546742533468019896/8663012338847025885813*a + 151404361365193741751296/8663012338847025885813, -167779408812960324763039/554432789686209656692032*a^26 - 165403217960042370388/962556926538558431757*a^25 + 161742533448602174245525/61603643298467739632448*a^24 + 22953208821294350120351/15400910824616934908112*a^23 - 2784649072968023408343361/277216394843104828346016*a^22 - 70060288246614286847401/277216394843104828346016*a^21 + 1504845172830572332591505/34652049355388103543252*a^20 + 3788067952562913797218483/92405464947701609448672*a^19 - 58048583490220893540770905/554432789686209656692032*a^18 - 19978004136668278683414773/277216394843104828346016*a^17 + 15091226236328153181753791/184810929895403218897344*a^16 + 33971371711189447541514803/92405464947701609448672*a^15 - 13960940624271947933522159/92405464947701609448672*a^14 - 90365776609375111323108647/138608197421552414173008*a^13 + 170872960398179015099485883/277216394843104828346016*a^12 + 65796428750176741096678183/69304098710776207086504*a^11 - 3446561626619806196931724/2887670779615675295271*a^10 - 57609587175941703530847605/69304098710776207086504*a^9 + 10834408825442460851142227/4471232174888787553968*a^8 - 17989937853090027045361549/23101366236925402362168*a^7 - 36403114694093629827374255/23101366236925402362168*a^6 + 19822834309146770955987265/11550683118462701181084*a^5 - 21287503224742595840983081/69304098710776207086504*a^4 - 4134014020097900084603327/8663012338847025885813*a^3 + 5546340940764532064946197/34652049355388103543252*a^2 + 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2300906587075068902070253/4471232174888787553968*a^8 + 11499359589855777007816921/69304098710776207086504*a^7 + 7948030942835061760878577/23101366236925402362168*a^6 - 12556784897170364991154675/34652049355388103543252*a^5 + 1314439248413442112141477/23101366236925402362168*a^4 + 3764916871731632454533239/34652049355388103543252*a^3 - 1162218942862952640034109/34652049355388103543252*a^2 - 372325232081208430051667/17326024677694051771626*a + 65129125937851415726039/8663012338847025885813 (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:		\( 19428990967.59157 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{12}\cdot 19428990967.59157 \cdot 1}{2\cdot\sqrt{638641997009196294829147338632324972544}}\cr\approx \mathstrut & 11.6423304215491 \end{aligned}\] (assuming GRH)

Galois group

$S_3\wr S_3$ (as 27T298):

Intermediate fields

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

Sibling fields

Frobenius cycle types

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

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
\(2\) 2	$\Q_{2}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
	2.4.10.3	$x^{4} + 4 x^{3} + 8 x^{2} + 2$	$4$	$1$	$10$	$D_{4}$	$[2, 3, 7/2]$
	2.4.9.1	$x^{4} + 10 x^{2} + 2$	$4$	$1$	$9$	$D_{4}$	$[2, 3, 7/2]$
	2.6.8.1	$x^{6} + 2 x^{3} + 2$	$6$	$1$	$8$	$D_{6}$	$[2]_{3}^{2}$
	2.12.29.25	$x^{12} + 10 x^{6} + 2$	$12$	$1$	$29$	12T28	$[2, 3, 7/2]_{3}^{2}$
\(3\) 3	3.9.15.30	$x^{9} + 3 x^{8} + 6 x^{7} + 6 x^{6} + 6 x^{3} + 12$	$9$	$1$	$15$	$S_3^2$	$[3/2, 2]_{2}^{2}$
		Deg $18$	$18$	$1$	$31$