Normalized defining polynomial

\( x^{36} - x^{35} + x^{34} - x^{33} + x^{32} - 5 x^{31} + x^{30} - 24 x^{29} + 20 x^{28} - 16 x^{27} + \cdots + 262144 \) x^36 - x^35 + x^34 - x^33 + x^32 - 5*x^31 + x^30 - 24*x^29 + 20*x^28 - 16*x^27 + 28*x^26 + 8*x^25 + 80*x^24 + 32*x^23 + 208*x^22 - 192*x^21 - 64*x^20 - 320*x^19 - 448*x^18 - 640*x^17 - 256*x^16 - 1536*x^15 + 3328*x^14 + 1024*x^13 + 5120*x^12 + 1024*x^11 + 7168*x^10 - 8192*x^9 + 20480*x^8 - 49152*x^7 + 4096*x^6 - 40960*x^5 + 16384*x^4 - 32768*x^3 + 65536*x^2 - 131072*x + 262144

Invariants

Degree:		$36$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[0, 18]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(7864785536926430215681870035583404646232472578456198834028544\) 7864785536926430215681870035583404646232472578456198834028544 \(\medspace = 2^{24}\cdot 7^{30}\cdot 229^{12}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(49.15\)	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:		$2^{3/2}7^{5/6}229^{1/2}\approx 216.62625977783622$
Ramified primes:		\(2\), \(7\), \(229\) 2, 7, 229	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q\)
$\card{ \Aut(K/\Q) }$:		$12$	sage: K.automorphisms()   magma: Automorphisms(K);   oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:		unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}+\frac{1}{4}a^{4}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{11}-\frac{1}{8}a^{10}+\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{3}{8}a^{7}-\frac{1}{8}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{13}-\frac{3}{8}a^{6}$, $\frac{1}{48}a^{14}-\frac{1}{16}a^{13}-\frac{1}{16}a^{12}+\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{3}{16}a^{9}-\frac{1}{16}a^{8}-\frac{5}{12}a^{7}+\frac{1}{8}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{3}$, $\frac{1}{48}a^{15}-\frac{11}{48}a^{8}-\frac{1}{3}a$, $\frac{1}{96}a^{16}-\frac{1}{96}a^{15}-\frac{1}{96}a^{14}-\frac{1}{32}a^{13}+\frac{1}{32}a^{12}+\frac{3}{32}a^{11}+\frac{1}{32}a^{10}-\frac{1}{48}a^{9}-\frac{5}{48}a^{8}+\frac{11}{24}a^{7}+\frac{3}{8}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{6}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{96}a^{17}-\frac{11}{96}a^{10}+\frac{1}{3}a^{3}$, $\frac{1}{192}a^{18}-\frac{1}{192}a^{17}-\frac{1}{192}a^{16}+\frac{1}{192}a^{15}-\frac{1}{192}a^{14}-\frac{1}{64}a^{13}-\frac{3}{64}a^{12}+\frac{5}{96}a^{11}-\frac{11}{96}a^{10}+\frac{1}{24}a^{9}-\frac{5}{48}a^{8}+\frac{5}{12}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{6}a^{4}-\frac{1}{6}a^{3}-\frac{1}{6}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{384}a^{19}-\frac{1}{384}a^{18}-\frac{1}{384}a^{17}+\frac{1}{384}a^{16}-\frac{1}{384}a^{15}-\frac{1}{128}a^{14}+\frac{5}{128}a^{13}-\frac{7}{192}a^{12}-\frac{23}{192}a^{11}+\frac{1}{12}a^{10}+\frac{13}{96}a^{9}-\frac{11}{48}a^{8}+\frac{1}{16}a^{7}+\frac{1}{4}a^{6}+\frac{11}{24}a^{5}+\frac{1}{6}a^{4}+\frac{5}{12}a^{3}+\frac{1}{3}a^{2}+\frac{1}{6}a$, $\frac{1}{384}a^{20}-\frac{1}{192}a^{17}-\frac{1}{192}a^{16}-\frac{1}{192}a^{15}+\frac{1}{192}a^{14}+\frac{19}{384}a^{13}-\frac{1}{64}a^{12}+\frac{5}{64}a^{11}+\frac{1}{24}a^{10}-\frac{23}{96}a^{9}+\frac{1}{6}a^{8}+\frac{13}{48}a^{7}-\frac{7}{24}a^{6}+\frac{1}{8}a^{5}-\frac{1}{2}a^{4}-\frac{5}{12}a^{3}+\frac{1}{3}a^{2}-\frac{1}{6}a-\frac{1}{3}$, $\frac{1}{768}a^{21}-\frac{1}{768}a^{20}-\frac{1}{768}a^{19}+\frac{1}{768}a^{18}-\frac{1}{768}a^{17}-\frac{1}{256}a^{16}-\frac{1}{768}a^{15}+\frac{1}{384}a^{14}+\frac{1}{384}a^{13}-\frac{1}{48}a^{12}-\frac{23}{192}a^{11}+\frac{7}{96}a^{10}+\frac{3}{32}a^{9}+\frac{1}{24}a^{8}+\frac{1}{16}a^{7}-\frac{5}{12}a^{6}+\frac{11}{24}a^{5}+\frac{5}{12}a^{4}-\frac{5}{12}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{768}a^{22}+\frac{5}{768}a^{15}-\frac{3}{16}a^{8}+\frac{1}{6}a$, $\frac{1}{1536}a^{23}-\frac{1}{1536}a^{22}-\frac{1}{1536}a^{21}+\frac{1}{1536}a^{20}-\frac{1}{1536}a^{19}-\frac{1}{512}a^{18}-\frac{1}{1536}a^{17}+\frac{1}{768}a^{16}+\frac{1}{768}a^{15}-\frac{1}{96}a^{14}-\frac{23}{384}a^{13}+\frac{7}{192}a^{12}+\frac{3}{64}a^{11}+\frac{1}{48}a^{10}+\frac{1}{32}a^{9}-\frac{5}{24}a^{8}-\frac{13}{48}a^{7}-\frac{7}{24}a^{6}+\frac{7}{24}a^{5}-\frac{1}{2}a^{4}+\frac{1}{6}a^{3}-\frac{1}{6}a^{2}$, $\frac{1}{1536}a^{24}+\frac{5}{1536}a^{17}-\frac{3}{32}a^{10}-\frac{5}{12}a^{3}$, $\frac{1}{3072}a^{25}-\frac{1}{3072}a^{24}-\frac{1}{3072}a^{23}+\frac{1}{3072}a^{22}-\frac{1}{3072}a^{21}-\frac{1}{1024}a^{20}-\frac{1}{3072}a^{19}+\frac{1}{1536}a^{18}+\frac{1}{1536}a^{17}-\frac{1}{192}a^{16}-\frac{7}{768}a^{15}-\frac{1}{384}a^{14}-\frac{5}{128}a^{13}-\frac{5}{96}a^{12}+\frac{5}{64}a^{11}+\frac{1}{12}a^{10}-\frac{7}{96}a^{9}-\frac{3}{16}a^{8}-\frac{1}{16}a^{7}-\frac{3}{8}a^{6}+\frac{1}{3}a^{5}+\frac{5}{12}a^{4}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3072}a^{26}-\frac{1}{1024}a^{19}-\frac{1}{384}a^{18}-\frac{1}{384}a^{17}+\frac{1}{384}a^{16}-\frac{1}{384}a^{15}-\frac{1}{128}a^{14}+\frac{5}{128}a^{13}-\frac{5}{192}a^{12}-\frac{23}{192}a^{11}+\frac{1}{12}a^{10}+\frac{13}{96}a^{9}-\frac{11}{48}a^{8}+\frac{1}{16}a^{7}+\frac{1}{4}a^{6}-\frac{5}{12}a^{5}+\frac{1}{6}a^{4}+\frac{5}{12}a^{3}+\frac{1}{3}a^{2}+\frac{1}{6}a$, $\frac{1}{6144}a^{27}-\frac{1}{6144}a^{26}-\frac{1}{6144}a^{25}+\frac{1}{6144}a^{24}-\frac{1}{6144}a^{23}-\frac{1}{2048}a^{22}-\frac{1}{6144}a^{21}+\frac{1}{3072}a^{20}+\frac{1}{3072}a^{19}-\frac{1}{384}a^{18}-\frac{7}{1536}a^{17}-\frac{1}{768}a^{16}+\frac{1}{768}a^{15}-\frac{1}{192}a^{14}-\frac{3}{128}a^{13}-\frac{1}{48}a^{12}+\frac{5}{192}a^{11}+\frac{3}{32}a^{10}+\frac{1}{32}a^{9}-\frac{11}{48}a^{8}-\frac{1}{2}a^{7}+\frac{1}{12}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{6}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{18432}a^{28}+\frac{5}{18432}a^{21}-\frac{1}{1152}a^{14}-\frac{1}{16}a^{13}-\frac{1}{16}a^{12}+\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{3}{16}a^{9}-\frac{1}{16}a^{8}+\frac{65}{144}a^{7}+\frac{1}{8}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{2}{9}$, $\frac{1}{36864}a^{29}-\frac{1}{36864}a^{28}-\frac{1}{12288}a^{27}+\frac{1}{12288}a^{26}-\frac{1}{12288}a^{25}-\frac{1}{4096}a^{24}-\frac{1}{12288}a^{23}-\frac{1}{9216}a^{22}+\frac{1}{2304}a^{21}-\frac{1}{768}a^{20}+\frac{1}{3072}a^{19}+\frac{1}{512}a^{18}+\frac{5}{1536}a^{17}-\frac{1}{192}a^{16}-\frac{19}{2304}a^{15}-\frac{1}{288}a^{14}+\frac{7}{192}a^{13}-\frac{7}{192}a^{12}-\frac{5}{96}a^{11}-\frac{1}{96}a^{10}+\frac{17}{96}a^{9}-\frac{13}{72}a^{8}+\frac{19}{72}a^{7}-\frac{1}{2}a^{6}-\frac{1}{8}a^{5}+\frac{5}{12}a^{4}+\frac{5}{12}a^{3}-\frac{1}{2}a^{2}-\frac{7}{18}a+\frac{2}{9}$, $\frac{1}{36864}a^{30}+\frac{5}{36864}a^{23}-\frac{1}{2304}a^{16}-\frac{1}{96}a^{15}-\frac{1}{96}a^{14}-\frac{1}{32}a^{13}+\frac{1}{32}a^{12}+\frac{3}{32}a^{11}+\frac{1}{32}a^{10}-\frac{25}{288}a^{9}-\frac{5}{48}a^{8}+\frac{11}{24}a^{7}+\frac{3}{8}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{9}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{73728}a^{31}-\frac{1}{73728}a^{30}-\frac{1}{73728}a^{29}+\frac{1}{73728}a^{28}+\frac{1}{24576}a^{27}+\frac{1}{8192}a^{26}+\frac{1}{24576}a^{25}+\frac{7}{36864}a^{24}+\frac{11}{36864}a^{23}-\frac{5}{18432}a^{22}-\frac{5}{9216}a^{21}-\frac{1}{3072}a^{20}+\frac{1}{3072}a^{19}-\frac{1}{512}a^{18}+\frac{1}{2304}a^{17}-\frac{5}{1152}a^{16}+\frac{5}{2304}a^{15}-\frac{1}{1152}a^{14}-\frac{11}{192}a^{13}-\frac{11}{192}a^{12}-\frac{19}{192}a^{11}-\frac{7}{144}a^{10}+\frac{41}{288}a^{9}-\frac{17}{144}a^{8}-\frac{7}{36}a^{7}+\frac{3}{8}a^{6}-\frac{11}{24}a^{5}-\frac{1}{3}a^{4}+\frac{7}{18}a^{3}+\frac{1}{9}a^{2}-\frac{7}{18}a+\frac{2}{9}$, $\frac{1}{147456}a^{32}-\frac{1}{147456}a^{31}+\frac{1}{147456}a^{30}-\frac{1}{147456}a^{29}+\frac{1}{147456}a^{28}+\frac{1}{49152}a^{27}+\frac{1}{16384}a^{26}-\frac{1}{9216}a^{25}-\frac{5}{36864}a^{24}-\frac{5}{18432}a^{23}-\frac{11}{36864}a^{22}-\frac{1}{9216}a^{21}+\frac{1}{3072}a^{20}-\frac{1}{1024}a^{19}+\frac{23}{9216}a^{18}+\frac{23}{4608}a^{17}-\frac{5}{1152}a^{16}+\frac{1}{576}a^{15}-\frac{1}{2304}a^{14}+\frac{7}{384}a^{13}-\frac{1}{32}a^{12}+\frac{1}{144}a^{11}+\frac{29}{576}a^{10}-\frac{2}{9}a^{9}+\frac{23}{144}a^{8}-\frac{13}{72}a^{7}-\frac{3}{16}a^{6}+\frac{1}{8}a^{5}-\frac{17}{36}a^{4}+\frac{11}{36}a^{3}-\frac{5}{36}a^{2}+\frac{2}{9}a-\frac{2}{9}$, $\frac{1}{294912}a^{33}-\frac{1}{294912}a^{32}+\frac{1}{294912}a^{31}-\frac{1}{294912}a^{30}+\frac{1}{294912}a^{29}-\frac{5}{294912}a^{28}-\frac{5}{98304}a^{27}-\frac{5}{36864}a^{26}+\frac{1}{73728}a^{25}-\frac{1}{4608}a^{24}+\frac{19}{73728}a^{23}+\frac{19}{36864}a^{22}+\frac{1}{2304}a^{21}+\frac{1}{3072}a^{20}-\frac{13}{18432}a^{19}+\frac{1}{1152}a^{18}+\frac{5}{1152}a^{17}+\frac{1}{4608}a^{16}-\frac{13}{4608}a^{15}+\frac{7}{2304}a^{14}-\frac{1}{16}a^{13}+\frac{35}{576}a^{12}+\frac{131}{1152}a^{11}+\frac{5}{144}a^{10}-\frac{5}{36}a^{9}-\frac{41}{288}a^{8}+\frac{115}{288}a^{7}-\frac{7}{24}a^{6}-\frac{23}{72}a^{5}+\frac{1}{36}a^{4}-\frac{23}{72}a^{3}+\frac{7}{36}a^{2}+\frac{7}{18}a+\frac{2}{9}$, $\frac{1}{589824}a^{34}-\frac{1}{589824}a^{33}+\frac{1}{589824}a^{32}-\frac{1}{589824}a^{31}+\frac{1}{589824}a^{30}-\frac{5}{589824}a^{29}+\frac{1}{589824}a^{28}+\frac{1}{73728}a^{27}+\frac{13}{147456}a^{26}+\frac{5}{36864}a^{25}-\frac{17}{147456}a^{24}+\frac{13}{73728}a^{23}+\frac{23}{36864}a^{22}+\frac{1}{4608}a^{21}+\frac{41}{36864}a^{20}-\frac{11}{9216}a^{19}+\frac{23}{9216}a^{18}+\frac{37}{9216}a^{17}+\frac{5}{9216}a^{16}-\frac{11}{4608}a^{15}+\frac{11}{2304}a^{14}+\frac{17}{288}a^{13}+\frac{65}{2304}a^{12}+\frac{29}{288}a^{11}+\frac{23}{576}a^{10}-\frac{89}{576}a^{9}-\frac{77}{576}a^{8}+\frac{13}{36}a^{7}-\frac{47}{144}a^{6}+\frac{19}{72}a^{5}-\frac{35}{144}a^{4}-\frac{29}{72}a^{3}+\frac{13}{36}a^{2}+\frac{5}{18}a+\frac{4}{9}$, $\frac{1}{1179648}a^{35}-\frac{1}{1179648}a^{34}+\frac{1}{1179648}a^{33}-\frac{1}{1179648}a^{32}+\frac{1}{1179648}a^{31}-\frac{5}{1179648}a^{30}+\frac{1}{1179648}a^{29}-\frac{1}{49152}a^{28}-\frac{11}{294912}a^{27}+\frac{11}{73728}a^{26}-\frac{41}{294912}a^{25}+\frac{25}{147456}a^{24}-\frac{7}{73728}a^{23}-\frac{17}{36864}a^{22}+\frac{1}{73728}a^{21}-\frac{17}{18432}a^{20}-\frac{13}{18432}a^{19}-\frac{5}{18432}a^{18}-\frac{43}{18432}a^{17}-\frac{41}{9216}a^{16}+\frac{17}{4608}a^{15}+\frac{1}{768}a^{14}+\frac{125}{4608}a^{13}-\frac{35}{1152}a^{12}-\frac{115}{1152}a^{11}+\frac{13}{1152}a^{10}-\frac{113}{1152}a^{9}-\frac{1}{144}a^{8}+\frac{131}{288}a^{7}+\frac{23}{72}a^{6}+\frac{25}{288}a^{5}+\frac{7}{144}a^{4}+\frac{31}{72}a^{3}+\frac{11}{36}a^{2}-\frac{4}{9}a-\frac{4}{9}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 1/2*a^8 - 1/2*a^7 - 1/2*a^6 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3 - 1/2*a^2, 1/2*a^9 - 1/2*a^2, 1/4*a^10 - 1/4*a^9 - 1/4*a^8 + 1/4*a^7 - 1/4*a^6 + 1/4*a^5 - 1/4*a^4 - 1/2*a^3 - 1/2*a^2, 1/4*a^11 + 1/4*a^4, 1/8*a^12 - 1/8*a^11 - 1/8*a^10 + 1/8*a^9 - 1/8*a^8 - 3/8*a^7 - 1/8*a^6 + 1/4*a^5 + 1/4*a^4 - 1/2*a^2, 1/8*a^13 - 3/8*a^6, 1/48*a^14 - 1/16*a^13 - 1/16*a^12 + 1/16*a^11 - 1/16*a^10 - 3/16*a^9 - 1/16*a^8 - 5/12*a^7 + 1/8*a^6 + 1/4*a^4 - 1/2*a^3 - 1/2*a^2 - 1/3, 1/48*a^15 - 11/48*a^8 - 1/3*a, 1/96*a^16 - 1/96*a^15 - 1/96*a^14 - 1/32*a^13 + 1/32*a^12 + 3/32*a^11 + 1/32*a^10 - 1/48*a^9 - 5/48*a^8 + 11/24*a^7 + 3/8*a^6 - 1/4*a^5 + 1/4*a^4 - 1/2*a^3 - 1/6*a^2 - 1/3*a - 1/3, 1/96*a^17 - 11/96*a^10 + 1/3*a^3, 1/192*a^18 - 1/192*a^17 - 1/192*a^16 + 1/192*a^15 - 1/192*a^14 - 1/64*a^13 - 3/64*a^12 + 5/96*a^11 - 11/96*a^10 + 1/24*a^9 - 5/48*a^8 + 5/12*a^7 - 1/4*a^6 - 1/4*a^5 + 1/6*a^4 - 1/6*a^3 - 1/6*a^2 - 1/3*a + 1/3, 1/384*a^19 - 1/384*a^18 - 1/384*a^17 + 1/384*a^16 - 1/384*a^15 - 1/128*a^14 + 5/128*a^13 - 7/192*a^12 - 23/192*a^11 + 1/12*a^10 + 13/96*a^9 - 11/48*a^8 + 1/16*a^7 + 1/4*a^6 + 11/24*a^5 + 1/6*a^4 + 5/12*a^3 + 1/3*a^2 + 1/6*a, 1/384*a^20 - 1/192*a^17 - 1/192*a^16 - 1/192*a^15 + 1/192*a^14 + 19/384*a^13 - 1/64*a^12 + 5/64*a^11 + 1/24*a^10 - 23/96*a^9 + 1/6*a^8 + 13/48*a^7 - 7/24*a^6 + 1/8*a^5 - 1/2*a^4 - 5/12*a^3 + 1/3*a^2 - 1/6*a - 1/3, 1/768*a^21 - 1/768*a^20 - 1/768*a^19 + 1/768*a^18 - 1/768*a^17 - 1/256*a^16 - 1/768*a^15 + 1/384*a^14 + 1/384*a^13 - 1/48*a^12 - 23/192*a^11 + 7/96*a^10 + 3/32*a^9 + 1/24*a^8 + 1/16*a^7 - 5/12*a^6 + 11/24*a^5 + 5/12*a^4 - 5/12*a^3 + 1/3*a - 1/3, 1/768*a^22 + 5/768*a^15 - 3/16*a^8 + 1/6*a, 1/1536*a^23 - 1/1536*a^22 - 1/1536*a^21 + 1/1536*a^20 - 1/1536*a^19 - 1/512*a^18 - 1/1536*a^17 + 1/768*a^16 + 1/768*a^15 - 1/96*a^14 - 23/384*a^13 + 7/192*a^12 + 3/64*a^11 + 1/48*a^10 + 1/32*a^9 - 5/24*a^8 - 13/48*a^7 - 7/24*a^6 + 7/24*a^5 - 1/2*a^4 + 1/6*a^3 - 1/6*a^2, 1/1536*a^24 + 5/1536*a^17 - 3/32*a^10 - 5/12*a^3, 1/3072*a^25 - 1/3072*a^24 - 1/3072*a^23 + 1/3072*a^22 - 1/3072*a^21 - 1/1024*a^20 - 1/3072*a^19 + 1/1536*a^18 + 1/1536*a^17 - 1/192*a^16 - 7/768*a^15 - 1/384*a^14 - 5/128*a^13 - 5/96*a^12 + 5/64*a^11 + 1/12*a^10 - 7/96*a^9 - 3/16*a^8 - 1/16*a^7 - 3/8*a^6 + 1/3*a^5 + 5/12*a^4 - 1/3*a + 1/3, 1/3072*a^26 - 1/1024*a^19 - 1/384*a^18 - 1/384*a^17 + 1/384*a^16 - 1/384*a^15 - 1/128*a^14 + 5/128*a^13 - 5/192*a^12 - 23/192*a^11 + 1/12*a^10 + 13/96*a^9 - 11/48*a^8 + 1/16*a^7 + 1/4*a^6 - 5/12*a^5 + 1/6*a^4 + 5/12*a^3 + 1/3*a^2 + 1/6*a, 1/6144*a^27 - 1/6144*a^26 - 1/6144*a^25 + 1/6144*a^24 - 1/6144*a^23 - 1/2048*a^22 - 1/6144*a^21 + 1/3072*a^20 + 1/3072*a^19 - 1/384*a^18 - 7/1536*a^17 - 1/768*a^16 + 1/768*a^15 - 1/192*a^14 - 3/128*a^13 - 1/48*a^12 + 5/192*a^11 + 3/32*a^10 + 1/32*a^9 - 11/48*a^8 - 1/2*a^7 + 1/12*a^6 + 1/4*a^5 - 1/2*a^4 - 1/6*a^3 - 1/3*a^2 - 1/3*a - 1/3, 1/18432*a^28 + 5/18432*a^21 - 1/1152*a^14 - 1/16*a^13 - 1/16*a^12 + 1/16*a^11 - 1/16*a^10 - 3/16*a^9 - 1/16*a^8 + 65/144*a^7 + 1/8*a^6 + 1/4*a^4 - 1/2*a^3 - 1/2*a^2 + 2/9, 1/36864*a^29 - 1/36864*a^28 - 1/12288*a^27 + 1/12288*a^26 - 1/12288*a^25 - 1/4096*a^24 - 1/12288*a^23 - 1/9216*a^22 + 1/2304*a^21 - 1/768*a^20 + 1/3072*a^19 + 1/512*a^18 + 5/1536*a^17 - 1/192*a^16 - 19/2304*a^15 - 1/288*a^14 + 7/192*a^13 - 7/192*a^12 - 5/96*a^11 - 1/96*a^10 + 17/96*a^9 - 13/72*a^8 + 19/72*a^7 - 1/2*a^6 - 1/8*a^5 + 5/12*a^4 + 5/12*a^3 - 1/2*a^2 - 7/18*a + 2/9, 1/36864*a^30 + 5/36864*a^23 - 1/2304*a^16 - 1/96*a^15 - 1/96*a^14 - 1/32*a^13 + 1/32*a^12 + 3/32*a^11 + 1/32*a^10 - 25/288*a^9 - 5/48*a^8 + 11/24*a^7 + 3/8*a^6 - 1/4*a^5 + 1/4*a^4 - 1/2*a^3 + 1/9*a^2 - 1/3*a - 1/3, 1/73728*a^31 - 1/73728*a^30 - 1/73728*a^29 + 1/73728*a^28 + 1/24576*a^27 + 1/8192*a^26 + 1/24576*a^25 + 7/36864*a^24 + 11/36864*a^23 - 5/18432*a^22 - 5/9216*a^21 - 1/3072*a^20 + 1/3072*a^19 - 1/512*a^18 + 1/2304*a^17 - 5/1152*a^16 + 5/2304*a^15 - 1/1152*a^14 - 11/192*a^13 - 11/192*a^12 - 19/192*a^11 - 7/144*a^10 + 41/288*a^9 - 17/144*a^8 - 7/36*a^7 + 3/8*a^6 - 11/24*a^5 - 1/3*a^4 + 7/18*a^3 + 1/9*a^2 - 7/18*a + 2/9, 1/147456*a^32 - 1/147456*a^31 + 1/147456*a^30 - 1/147456*a^29 + 1/147456*a^28 + 1/49152*a^27 + 1/16384*a^26 - 1/9216*a^25 - 5/36864*a^24 - 5/18432*a^23 - 11/36864*a^22 - 1/9216*a^21 + 1/3072*a^20 - 1/1024*a^19 + 23/9216*a^18 + 23/4608*a^17 - 5/1152*a^16 + 1/576*a^15 - 1/2304*a^14 + 7/384*a^13 - 1/32*a^12 + 1/144*a^11 + 29/576*a^10 - 2/9*a^9 + 23/144*a^8 - 13/72*a^7 - 3/16*a^6 + 1/8*a^5 - 17/36*a^4 + 11/36*a^3 - 5/36*a^2 + 2/9*a - 2/9, 1/294912*a^33 - 1/294912*a^32 + 1/294912*a^31 - 1/294912*a^30 + 1/294912*a^29 - 5/294912*a^28 - 5/98304*a^27 - 5/36864*a^26 + 1/73728*a^25 - 1/4608*a^24 + 19/73728*a^23 + 19/36864*a^22 + 1/2304*a^21 + 1/3072*a^20 - 13/18432*a^19 + 1/1152*a^18 + 5/1152*a^17 + 1/4608*a^16 - 13/4608*a^15 + 7/2304*a^14 - 1/16*a^13 + 35/576*a^12 + 131/1152*a^11 + 5/144*a^10 - 5/36*a^9 - 41/288*a^8 + 115/288*a^7 - 7/24*a^6 - 23/72*a^5 + 1/36*a^4 - 23/72*a^3 + 7/36*a^2 + 7/18*a + 2/9, 1/589824*a^34 - 1/589824*a^33 + 1/589824*a^32 - 1/589824*a^31 + 1/589824*a^30 - 5/589824*a^29 + 1/589824*a^28 + 1/73728*a^27 + 13/147456*a^26 + 5/36864*a^25 - 17/147456*a^24 + 13/73728*a^23 + 23/36864*a^22 + 1/4608*a^21 + 41/36864*a^20 - 11/9216*a^19 + 23/9216*a^18 + 37/9216*a^17 + 5/9216*a^16 - 11/4608*a^15 + 11/2304*a^14 + 17/288*a^13 + 65/2304*a^12 + 29/288*a^11 + 23/576*a^10 - 89/576*a^9 - 77/576*a^8 + 13/36*a^7 - 47/144*a^6 + 19/72*a^5 - 35/144*a^4 - 29/72*a^3 + 13/36*a^2 + 5/18*a + 4/9, 1/1179648*a^35 - 1/1179648*a^34 + 1/1179648*a^33 - 1/1179648*a^32 + 1/1179648*a^31 - 5/1179648*a^30 + 1/1179648*a^29 - 1/49152*a^28 - 11/294912*a^27 + 11/73728*a^26 - 41/294912*a^25 + 25/147456*a^24 - 7/73728*a^23 - 17/36864*a^22 + 1/73728*a^21 - 17/18432*a^20 - 13/18432*a^19 - 5/18432*a^18 - 43/18432*a^17 - 41/9216*a^16 + 17/4608*a^15 + 1/768*a^14 + 125/4608*a^13 - 35/1152*a^12 - 115/1152*a^11 + 13/1152*a^10 - 113/1152*a^9 - 1/144*a^8 + 131/288*a^7 + 23/72*a^6 + 25/288*a^5 + 7/144*a^4 + 31/72*a^3 + 11/36*a^2 - 4/9*a - 4/9

Class group and class number

$C_{3}\times C_{3}\times C_{18}$, which has order $162$ (assuming GRH)

Unit group

Rank:		$17$	sage: UK.rank()   gp: K.fu   magma: UnitRank(K);   oscar: rank(UK)
Torsion generator:		\( -\frac{13}{147456} a^{32} - \frac{13}{147456} a^{31} - \frac{19}{147456} a^{30} - \frac{13}{147456} a^{29} + \frac{13}{147456} a^{28} + \frac{13}{49152} a^{27} + \frac{13}{16384} a^{26} + \frac{143}{73728} a^{25} + \frac{91}{36864} a^{24} + \frac{29}{18432} a^{23} + \frac{13}{36864} a^{22} - \frac{13}{2304} a^{21} - \frac{13}{1536} a^{20} - \frac{13}{768} a^{19} - \frac{221}{9216} a^{18} - \frac{91}{4608} a^{17} + \frac{5}{1152} a^{16} + \frac{91}{2304} a^{15} + \frac{221}{2304} a^{14} + \frac{13}{96} a^{13} + \frac{13}{96} a^{12} + \frac{13}{72} a^{11} - \frac{13}{576} a^{10} - \frac{83}{288} a^{9} - \frac{91}{144} a^{8} - \frac{143}{144} a^{7} - \frac{13}{16} a^{6} - \frac{13}{24} a^{5} - \frac{13}{36} a^{4} + \frac{13}{18} a^{3} + \frac{107}{36} a^{2} + \frac{26}{9} a + \frac{52}{9} \) -(13)/(147456)*a^(32) - (13)/(147456)*a^(31) - (19)/(147456)*a^(30) - (13)/(147456)*a^(29) + (13)/(147456)*a^(28) + (13)/(49152)*a^(27) + (13)/(16384)*a^(26) + (143)/(73728)*a^(25) + (91)/(36864)*a^(24) + (29)/(18432)*a^(23) + (13)/(36864)*a^(22) - (13)/(2304)*a^(21) - (13)/(1536)*a^(20) - (13)/(768)*a^(19) - (221)/(9216)*a^(18) - (91)/(4608)*a^(17) + (5)/(1152)*a^(16) + (91)/(2304)*a^(15) + (221)/(2304)*a^(14) + (13)/(96)*a^(13) + (13)/(96)*a^(12) + (13)/(72)*a^(11) - (13)/(576)*a^(10) - (83)/(288)*a^(9) - (91)/(144)*a^(8) - (143)/(144)*a^(7) - (13)/(16)*a^(6) - (13)/(24)*a^(5) - (13)/(36)*a^(4) + (13)/(18)*a^(3) + (107)/(36)*a^(2) + (26)/(9)*a + (52)/(9)  (order $14$)	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:		$\frac{13}{196608}a^{34}+\frac{13}{196608}a^{33}+\frac{7}{196608}a^{32}-\frac{13}{589824}a^{31}-\frac{115}{589824}a^{30}-\frac{169}{589824}a^{29}-\frac{299}{589824}a^{28}-\frac{39}{32768}a^{27}-\frac{13}{12288}a^{26}+\frac{7}{12288}a^{25}+\frac{325}{147456}a^{24}+\frac{107}{18432}a^{23}+\frac{247}{36864}a^{22}+\frac{65}{9216}a^{21}+\frac{39}{4096}a^{20}-\frac{13}{6144}a^{19}-\frac{37}{1536}a^{18}-\frac{455}{9216}a^{17}-\frac{623}{9216}a^{16}-\frac{143}{2304}a^{15}-\frac{13}{2304}a^{14}+\frac{39}{256}a^{12}+\frac{127}{384}a^{11}+\frac{65}{144}a^{10}+\frac{263}{576}a^{9}-\frac{13}{576}a^{8}-\frac{169}{288}a^{7}-\frac{13}{24}a^{6}-\frac{13}{12}a^{5}-\frac{107}{48}a^{4}-\frac{13}{9}a^{3}-\frac{49}{36}a^{2}+\frac{26}{9}a+\frac{52}{9}$, $\frac{17}{294912}a^{35}+\frac{5}{589824}a^{34}+\frac{3}{65536}a^{33}-\frac{59}{589824}a^{32}+\frac{59}{589824}a^{31}-\frac{65}{196608}a^{30}+\frac{53}{196608}a^{29}-\frac{841}{589824}a^{28}+\frac{1}{2304}a^{27}-\frac{47}{49152}a^{26}+\frac{169}{73728}a^{25}-\frac{107}{147456}a^{24}+\frac{35}{8192}a^{23}-\frac{19}{12288}a^{22}+\frac{49}{4608}a^{21}-\frac{65}{36864}a^{20}-\frac{5}{1024}a^{19}-\frac{145}{9216}a^{18}-\frac{113}{9216}a^{17}-\frac{55}{3072}a^{16}+\frac{19}{1536}a^{15}-\frac{179}{2304}a^{14}+\frac{35}{576}a^{13}+\frac{13}{768}a^{12}+\frac{145}{576}a^{11}-\frac{91}{576}a^{10}+\frac{71}{192}a^{9}-\frac{149}{192}a^{8}+\frac{55}{36}a^{7}-\frac{181}{144}a^{6}+\frac{23}{24}a^{5}-\frac{365}{144}a^{4}+\frac{193}{72}a^{3}-\frac{31}{12}a^{2}+\frac{31}{6}a-\frac{28}{3}$, $\frac{1}{6144}a^{35}-\frac{23}{9216}a^{28}+\frac{463}{18432}a^{21}-\frac{83}{288}a^{14}+\frac{517}{144}a^{7}-\frac{263}{9}$, $\frac{49}{1179648}a^{35}+\frac{49}{1179648}a^{34}+\frac{37}{393216}a^{33}+\frac{49}{1179648}a^{32}-\frac{49}{1179648}a^{31}-\frac{49}{393216}a^{30}-\frac{49}{131072}a^{29}-\frac{539}{589824}a^{28}-\frac{343}{294912}a^{27}-\frac{17}{16384}a^{26}-\frac{49}{294912}a^{25}+\frac{49}{18432}a^{24}+\frac{49}{12288}a^{23}+\frac{49}{6144}a^{22}+\frac{833}{73728}a^{21}+\frac{343}{36864}a^{20}+\frac{13}{6144}a^{19}-\frac{343}{18432}a^{18}-\frac{833}{18432}a^{17}-\frac{49}{768}a^{16}-\frac{49}{768}a^{15}-\frac{49}{576}a^{14}+\frac{49}{4608}a^{13}+\frac{77}{768}a^{12}+\frac{343}{1152}a^{11}+\frac{539}{1152}a^{10}+\frac{49}{128}a^{9}+\frac{49}{192}a^{8}+\frac{49}{288}a^{7}-\frac{49}{144}a^{6}-\frac{35}{32}a^{5}-\frac{49}{36}a^{4}-\frac{49}{18}a^{3}$, $\frac{5}{36864}a^{35}+\frac{73}{589824}a^{34}+\frac{109}{589824}a^{33}+\frac{65}{196608}a^{32}+\frac{101}{589824}a^{31}+\frac{3}{65536}a^{30}-\frac{39}{65536}a^{29}-\frac{623}{196608}a^{28}-\frac{1061}{294912}a^{27}-\frac{661}{147456}a^{26}-\frac{21}{4096}a^{25}-\frac{113}{147456}a^{24}+\frac{27}{4096}a^{23}+\frac{55}{3072}a^{22}+\frac{913}{18432}a^{21}+\frac{1721}{36864}a^{20}+\frac{817}{18432}a^{19}+\frac{15}{1024}a^{18}-\frac{539}{9216}a^{17}-\frac{503}{3072}a^{16}-\frac{205}{768}a^{15}-\frac{67}{128}a^{14}-\frac{25}{72}a^{13}-\frac{167}{2304}a^{12}+\frac{39}{128}a^{11}+\frac{683}{576}a^{10}+\frac{325}{192}a^{9}+\frac{141}{64}a^{8}+\frac{1127}{288}a^{7}+\frac{175}{144}a^{6}-\frac{103}{72}a^{5}-\frac{145}{48}a^{4}-\frac{283}{36}a^{3}-\frac{25}{3}a^{2}-\frac{19}{3}a-\frac{199}{9}$, $\frac{65}{589824}a^{35}+\frac{5}{147456}a^{34}-\frac{5}{98304}a^{33}-\frac{17}{294912}a^{32}-\frac{29}{98304}a^{31}-\frac{35}{294912}a^{30}-\frac{121}{294912}a^{29}-\frac{251}{196608}a^{28}-\frac{55}{294912}a^{27}+\frac{107}{49152}a^{26}+\frac{289}{147456}a^{25}+\frac{275}{49152}a^{24}+\frac{25}{9216}a^{23}+\frac{37}{18432}a^{22}+\frac{299}{36864}a^{21}-\frac{491}{36864}a^{20}-\frac{185}{6144}a^{19}-\frac{187}{4608}a^{18}-\frac{59}{1536}a^{17}-\frac{11}{9216}a^{16}+\frac{73}{1152}a^{15}-\frac{1}{32}a^{14}+\frac{419}{2304}a^{13}+\frac{247}{768}a^{12}+\frac{205}{1152}a^{11}+\frac{5}{24}a^{10}-\frac{91}{144}a^{9}-\frac{383}{576}a^{8}+\frac{95}{288}a^{7}-\frac{97}{144}a^{6}-\frac{103}{48}a^{5}+\frac{167}{144}a^{4}-\frac{7}{12}a^{3}+\frac{47}{9}a^{2}+\frac{77}{18}a-\frac{97}{9}$, $\frac{107}{1179648}a^{35}-\frac{29}{1179648}a^{34}-\frac{95}{1179648}a^{33}-\frac{137}{1179648}a^{32}-\frac{295}{1179648}a^{31}-\frac{37}{1179648}a^{30}-\frac{127}{1179648}a^{29}-\frac{145}{196608}a^{28}+\frac{121}{147456}a^{27}+\frac{203}{73728}a^{26}+\frac{581}{294912}a^{25}+\frac{149}{36864}a^{24}-\frac{83}{73728}a^{23}-\frac{31}{9216}a^{22}-\frac{1}{73728}a^{21}-\frac{815}{36864}a^{20}-\frac{287}{9216}a^{19}-\frac{415}{18432}a^{18}-\frac{131}{18432}a^{17}+\frac{217}{4608}a^{16}+\frac{511}{4608}a^{15}+\frac{1}{96}a^{14}+\frac{1099}{4608}a^{13}+\frac{497}{2304}a^{12}-\frac{11}{288}a^{11}-\frac{205}{1152}a^{10}-\frac{1153}{1152}a^{9}-\frac{485}{576}a^{8}+\frac{65}{144}a^{7}-\frac{25}{36}a^{6}-\frac{317}{288}a^{5}+\frac{37}{18}a^{4}+\frac{83}{72}a^{3}+\frac{67}{9}a^{2}+\frac{53}{18}a-\frac{101}{9}$, $\frac{77}{393216}a^{35}+\frac{31}{393216}a^{34}+\frac{187}{1179648}a^{33}+\frac{341}{1179648}a^{32}+\frac{203}{1179648}a^{31}+\frac{121}{1179648}a^{30}-\frac{517}{1179648}a^{29}-\frac{1163}{294912}a^{28}-\frac{281}{98304}a^{27}-\frac{283}{73728}a^{26}-\frac{1487}{294912}a^{25}-\frac{187}{147456}a^{24}+\frac{305}{73728}a^{23}+\frac{533}{36864}a^{22}+\frac{4043}{73728}a^{21}+\frac{21}{512}a^{20}+\frac{761}{18432}a^{19}+\frac{433}{18432}a^{18}-\frac{725}{18432}a^{17}-\frac{1193}{9216}a^{16}-\frac{1067}{4608}a^{15}-\frac{1385}{2304}a^{14}-\frac{515}{1536}a^{13}-\frac{35}{288}a^{12}+\frac{191}{1152}a^{11}+\frac{1091}{1152}a^{10}+\frac{1621}{1152}a^{9}+\frac{605}{288}a^{8}+\frac{1555}{288}a^{7}+\frac{4}{3}a^{6}-\frac{281}{288}a^{5}-\frac{305}{144}a^{4}-\frac{499}{72}a^{3}-\frac{247}{36}a^{2}-\frac{68}{9}a-\frac{316}{9}$, $\frac{25}{1179648}a^{35}-\frac{97}{1179648}a^{34}+\frac{109}{1179648}a^{33}-\frac{95}{393216}a^{32}-\frac{35}{1179648}a^{31}-\frac{353}{1179648}a^{30}-\frac{227}{1179648}a^{29}+\frac{37}{294912}a^{28}+\frac{85}{73728}a^{27}+\frac{31}{36864}a^{26}+\frac{91}{32768}a^{25}+\frac{721}{147456}a^{24}-\frac{11}{36864}a^{23}+\frac{49}{9216}a^{22}-\frac{115}{8192}a^{21}-\frac{263}{18432}a^{20}-\frac{59}{2304}a^{19}-\frac{227}{6144}a^{18}-\frac{487}{18432}a^{17}+\frac{49}{9216}a^{16}+\frac{65}{1152}a^{15}+\frac{35}{288}a^{14}+\frac{1553}{4608}a^{13}+\frac{61}{1152}a^{12}+\frac{5}{12}a^{11}-\frac{227}{1152}a^{10}-\frac{389}{1152}a^{9}-\frac{233}{288}a^{8}-\frac{19}{12}a^{7}-\frac{71}{36}a^{6}-\frac{95}{288}a^{5}-\frac{1}{16}a^{4}-\frac{25}{36}a^{3}+\frac{151}{18}a^{2}-\frac{53}{18}a+\frac{145}{9}$, $\frac{173}{1179648}a^{35}-\frac{61}{1179648}a^{34}-\frac{11}{1179648}a^{33}-\frac{325}{1179648}a^{32}-\frac{11}{1179648}a^{31}-\frac{425}{1179648}a^{30}+\frac{167}{393216}a^{29}-\frac{79}{36864}a^{28}+\frac{559}{294912}a^{27}+\frac{83}{73728}a^{26}+\frac{1327}{294912}a^{25}+\frac{133}{147456}a^{24}+\frac{59}{73728}a^{23}-\frac{85}{12288}a^{22}+\frac{697}{73728}a^{21}-\frac{395}{18432}a^{20}-\frac{457}{18432}a^{19}-\frac{485}{18432}a^{18}+\frac{41}{18432}a^{17}+\frac{379}{9216}a^{16}+\frac{61}{512}a^{15}-\frac{239}{2304}a^{14}+\frac{1145}{4608}a^{13}+\frac{79}{1152}a^{12}+\frac{269}{1152}a^{11}-\frac{623}{1152}a^{10}-\frac{413}{1152}a^{9}-\frac{79}{48}a^{8}+\frac{743}{288}a^{7}-\frac{71}{36}a^{6}+\frac{313}{288}a^{5}-\frac{149}{144}a^{4}+\frac{313}{72}a^{3}+\frac{107}{36}a^{2}+\frac{13}{2}a-22$, $\frac{89}{1179648}a^{35}+\frac{179}{1179648}a^{34}+\frac{139}{393216}a^{33}-\frac{67}{393216}a^{32}-\frac{167}{1179648}a^{31}-\frac{1469}{1179648}a^{30}-\frac{1463}{1179648}a^{29}-\frac{131}{49152}a^{28}-\frac{115}{36864}a^{27}-\frac{145}{49152}a^{26}+\frac{199}{32768}a^{25}+\frac{2035}{147456}a^{24}+\frac{437}{18432}a^{23}+\frac{1081}{36864}a^{22}+\frac{1385}{73728}a^{21}+\frac{143}{9216}a^{20}-\frac{121}{3072}a^{19}-\frac{279}{2048}a^{18}-\frac{4087}{18432}a^{17}-\frac{2405}{9216}a^{16}-\frac{169}{1152}a^{15}+\frac{29}{768}a^{14}+\frac{1877}{4608}a^{13}+\frac{127}{192}a^{12}+\frac{59}{32}a^{11}+\frac{1663}{1152}a^{10}+\frac{1747}{1152}a^{9}-\frac{181}{144}a^{8}-\frac{103}{36}a^{7}-\frac{659}{144}a^{6}-\frac{413}{96}a^{5}-\frac{425}{48}a^{4}-\frac{95}{18}a^{3}+\frac{95}{36}a^{2}+\frac{193}{18}a+\frac{256}{9}$, $\frac{49}{294912}a^{35}+\frac{5}{36864}a^{34}-\frac{59}{294912}a^{33}-\frac{5}{32768}a^{32}-\frac{53}{98304}a^{31}-\frac{173}{294912}a^{30}-\frac{35}{294912}a^{29}-\frac{23}{8192}a^{28}+\frac{101}{294912}a^{27}+\frac{5}{1152}a^{26}+\frac{65}{8192}a^{25}+\frac{199}{24576}a^{24}+\frac{803}{73728}a^{23}-\frac{305}{36864}a^{22}+\frac{277}{18432}a^{21}-\frac{625}{18432}a^{20}-\frac{1357}{18432}a^{19}-\frac{35}{384}a^{18}-\frac{7}{96}a^{17}+\frac{1}{144}a^{16}+\frac{881}{4608}a^{15}+\frac{61}{768}a^{14}+\frac{391}{1152}a^{13}+\frac{937}{1152}a^{12}+\frac{151}{384}a^{11}+\frac{1}{32}a^{10}-\frac{311}{288}a^{9}-\frac{433}{144}a^{8}+\frac{227}{288}a^{7}-\frac{149}{72}a^{6}-\frac{25}{9}a^{5}-\frac{25}{24}a^{4}+\frac{47}{8}a^{3}+\frac{161}{36}a^{2}+\frac{427}{18}a-\frac{244}{9}$, $\frac{1}{196608}a^{35}-\frac{1}{12288}a^{34}-\frac{1}{16384}a^{33}-\frac{25}{147456}a^{32}-\frac{1}{147456}a^{31}-\frac{1}{73728}a^{30}+\frac{7}{18432}a^{29}+\frac{191}{589824}a^{28}+\frac{23}{12288}a^{27}+\frac{23}{16384}a^{26}+\frac{343}{147456}a^{25}-\frac{137}{147456}a^{24}-\frac{251}{73728}a^{23}-\frac{311}{36864}a^{22}-\frac{395}{36864}a^{21}-\frac{81}{4096}a^{20}-\frac{7}{512}a^{19}-\frac{1}{4608}a^{18}+\frac{41}{1152}a^{17}+\frac{689}{9216}a^{16}+\frac{503}{4608}a^{15}+\frac{199}{2304}a^{14}+\frac{113}{768}a^{13}-\frac{13}{256}a^{12}-\frac{79}{576}a^{11}-\frac{173}{288}a^{10}-\frac{89}{144}a^{9}-\frac{487}{576}a^{8}+\frac{17}{144}a^{7}-\frac{29}{48}a^{6}+\frac{19}{16}a^{5}+\frac{125}{144}a^{4}+\frac{259}{72}a^{3}+\frac{97}{36}a^{2}+\frac{31}{18}a-\frac{31}{9}$, $\frac{1}{12288}a^{35}-\frac{11}{49152}a^{34}+\frac{5}{18432}a^{33}-\frac{47}{147456}a^{32}+\frac{19}{147456}a^{31}-\frac{7}{147456}a^{30}-\frac{77}{147456}a^{29}-\frac{7}{147456}a^{28}+\frac{35}{24576}a^{27}-\frac{19}{147456}a^{26}+\frac{5}{36864}a^{25}+\frac{203}{36864}a^{24}-\frac{263}{36864}a^{23}+\frac{557}{36864}a^{22}-\frac{113}{9216}a^{21}-\frac{7}{768}a^{20}-\frac{37}{9216}a^{19}-\frac{145}{9216}a^{18}-\frac{37}{2304}a^{17}+\frac{7}{576}a^{16}-\frac{1}{144}a^{15}-\frac{119}{2304}a^{14}+\frac{53}{128}a^{13}-\frac{185}{576}a^{12}+\frac{277}{576}a^{11}-\frac{113}{576}a^{10}-\frac{43}{144}a^{9}+\frac{79}{144}a^{8}-\frac{97}{144}a^{7}-\frac{33}{16}a^{6}+\frac{7}{36}a^{5}+\frac{49}{36}a^{4}-\frac{215}{36}a^{3}+\frac{491}{36}a^{2}-\frac{349}{18}a+\frac{173}{9}$, $\frac{7}{147456}a^{35}-\frac{11}{589824}a^{34}-\frac{11}{589824}a^{33}-\frac{1}{65536}a^{32}-\frac{7}{589824}a^{31}+\frac{55}{589824}a^{30}+\frac{37}{589824}a^{29}-\frac{103}{196608}a^{28}+\frac{115}{294912}a^{27}+\frac{17}{36864}a^{26}+\frac{7}{24576}a^{25}-\frac{17}{147456}a^{24}-\frac{55}{36864}a^{23}-\frac{67}{36864}a^{22}+\frac{83}{18432}a^{21}-\frac{163}{36864}a^{20}-\frac{53}{18432}a^{19}+\frac{1}{1536}a^{18}+\frac{91}{9216}a^{17}+\frac{203}{9216}a^{16}+\frac{59}{2304}a^{15}-\frac{53}{768}a^{14}+\frac{1}{36}a^{13}-\frac{35}{2304}a^{12}-\frac{25}{384}a^{11}-\frac{19}{144}a^{10}-\frac{173}{576}a^{9}-\frac{71}{576}a^{8}+\frac{271}{288}a^{7}-\frac{1}{72}a^{6}+\frac{7}{36}a^{5}+\frac{5}{16}a^{4}+\frac{5}{9}a^{3}+\frac{37}{36}a^{2}-\frac{2}{9}a-\frac{80}{9}$, $\frac{1}{98304}a^{35}+\frac{13}{294912}a^{34}-\frac{1}{98304}a^{33}+\frac{1}{98304}a^{32}-\frac{43}{294912}a^{31}-\frac{25}{294912}a^{30}-\frac{25}{98304}a^{29}-\frac{41}{147456}a^{28}-\frac{65}{147456}a^{27}+\frac{7}{12288}a^{26}+\frac{23}{24576}a^{25}+\frac{107}{36864}a^{24}+\frac{55}{18432}a^{23}+\frac{17}{6144}a^{22}+\frac{1}{1152}a^{21}-\frac{5}{2304}a^{20}-\frac{19}{1536}a^{19}-\frac{35}{1536}a^{18}-\frac{149}{4608}a^{17}-\frac{55}{2304}a^{16}+\frac{5}{768}a^{15}+\frac{43}{1152}a^{14}+\frac{35}{576}a^{13}+\frac{19}{96}a^{12}+\frac{5}{32}a^{11}+\frac{77}{288}a^{10}-\frac{31}{288}a^{9}-\frac{1}{4}a^{8}-\frac{95}{144}a^{7}-\frac{29}{72}a^{6}-\frac{31}{24}a^{5}-\frac{5}{12}a^{4}-\frac{13}{18}a^{3}+\frac{13}{9}a^{2}+\frac{5}{2}a+\frac{31}{9}$, $\frac{755}{1179648}a^{35}+\frac{199}{1179648}a^{34}+\frac{1037}{1179648}a^{33}+\frac{233}{393216}a^{32}+\frac{1877}{1179648}a^{31}-\frac{737}{1179648}a^{30}+\frac{5}{131072}a^{29}-\frac{2975}{196608}a^{28}-\frac{1079}{147456}a^{27}-\frac{1541}{73728}a^{26}-\frac{1289}{98304}a^{25}-\frac{1247}{73728}a^{24}+\frac{1547}{73728}a^{23}+\frac{35}{768}a^{22}+\frac{14567}{73728}a^{21}+\frac{5635}{36864}a^{20}+\frac{1781}{9216}a^{19}+\frac{587}{6144}a^{18}-\frac{1847}{18432}a^{17}-\frac{565}{1152}a^{16}-\frac{1313}{1536}a^{15}-\frac{881}{384}a^{14}-\frac{5249}{4608}a^{13}-\frac{2777}{2304}a^{12}+\frac{83}{64}a^{11}+\frac{2447}{1152}a^{10}+\frac{8503}{1152}a^{9}+\frac{1123}{192}a^{8}+\frac{409}{18}a^{7}+\frac{43}{72}a^{6}+\frac{1415}{288}a^{5}-\frac{437}{24}a^{4}-\frac{1057}{72}a^{3}-\frac{370}{9}a^{2}-23a-\frac{1127}{9}$ 13/196608*a^34 + 13/196608*a^33 + 7/196608*a^32 - 13/589824*a^31 - 115/589824*a^30 - 169/589824*a^29 - 299/589824*a^28 - 39/32768*a^27 - 13/12288*a^26 + 7/12288*a^25 + 325/147456*a^24 + 107/18432*a^23 + 247/36864*a^22 + 65/9216*a^21 + 39/4096*a^20 - 13/6144*a^19 - 37/1536*a^18 - 455/9216*a^17 - 623/9216*a^16 - 143/2304*a^15 - 13/2304*a^14 + 39/256*a^12 + 127/384*a^11 + 65/144*a^10 + 263/576*a^9 - 13/576*a^8 - 169/288*a^7 - 13/24*a^6 - 13/12*a^5 - 107/48*a^4 - 13/9*a^3 - 49/36*a^2 + 26/9*a + 52/9, 17/294912*a^35 + 5/589824*a^34 + 3/65536*a^33 - 59/589824*a^32 + 59/589824*a^31 - 65/196608*a^30 + 53/196608*a^29 - 841/589824*a^28 + 1/2304*a^27 - 47/49152*a^26 + 169/73728*a^25 - 107/147456*a^24 + 35/8192*a^23 - 19/12288*a^22 + 49/4608*a^21 - 65/36864*a^20 - 5/1024*a^19 - 145/9216*a^18 - 113/9216*a^17 - 55/3072*a^16 + 19/1536*a^15 - 179/2304*a^14 + 35/576*a^13 + 13/768*a^12 + 145/576*a^11 - 91/576*a^10 + 71/192*a^9 - 149/192*a^8 + 55/36*a^7 - 181/144*a^6 + 23/24*a^5 - 365/144*a^4 + 193/72*a^3 - 31/12*a^2 + 31/6*a - 28/3, 1/6144*a^35 - 23/9216*a^28 + 463/18432*a^21 - 83/288*a^14 + 517/144*a^7 - 263/9, 49/1179648*a^35 + 49/1179648*a^34 + 37/393216*a^33 + 49/1179648*a^32 - 49/1179648*a^31 - 49/393216*a^30 - 49/131072*a^29 - 539/589824*a^28 - 343/294912*a^27 - 17/16384*a^26 - 49/294912*a^25 + 49/18432*a^24 + 49/12288*a^23 + 49/6144*a^22 + 833/73728*a^21 + 343/36864*a^20 + 13/6144*a^19 - 343/18432*a^18 - 833/18432*a^17 - 49/768*a^16 - 49/768*a^15 - 49/576*a^14 + 49/4608*a^13 + 77/768*a^12 + 343/1152*a^11 + 539/1152*a^10 + 49/128*a^9 + 49/192*a^8 + 49/288*a^7 - 49/144*a^6 - 35/32*a^5 - 49/36*a^4 - 49/18*a^3, 5/36864*a^35 + 73/589824*a^34 + 109/589824*a^33 + 65/196608*a^32 + 101/589824*a^31 + 3/65536*a^30 - 39/65536*a^29 - 623/196608*a^28 - 1061/294912*a^27 - 661/147456*a^26 - 21/4096*a^25 - 113/147456*a^24 + 27/4096*a^23 + 55/3072*a^22 + 913/18432*a^21 + 1721/36864*a^20 + 817/18432*a^19 + 15/1024*a^18 - 539/9216*a^17 - 503/3072*a^16 - 205/768*a^15 - 67/128*a^14 - 25/72*a^13 - 167/2304*a^12 + 39/128*a^11 + 683/576*a^10 + 325/192*a^9 + 141/64*a^8 + 1127/288*a^7 + 175/144*a^6 - 103/72*a^5 - 145/48*a^4 - 283/36*a^3 - 25/3*a^2 - 19/3*a - 199/9, 65/589824*a^35 + 5/147456*a^34 - 5/98304*a^33 - 17/294912*a^32 - 29/98304*a^31 - 35/294912*a^30 - 121/294912*a^29 - 251/196608*a^28 - 55/294912*a^27 + 107/49152*a^26 + 289/147456*a^25 + 275/49152*a^24 + 25/9216*a^23 + 37/18432*a^22 + 299/36864*a^21 - 491/36864*a^20 - 185/6144*a^19 - 187/4608*a^18 - 59/1536*a^17 - 11/9216*a^16 + 73/1152*a^15 - 1/32*a^14 + 419/2304*a^13 + 247/768*a^12 + 205/1152*a^11 + 5/24*a^10 - 91/144*a^9 - 383/576*a^8 + 95/288*a^7 - 97/144*a^6 - 103/48*a^5 + 167/144*a^4 - 7/12*a^3 + 47/9*a^2 + 77/18*a - 97/9, 107/1179648*a^35 - 29/1179648*a^34 - 95/1179648*a^33 - 137/1179648*a^32 - 295/1179648*a^31 - 37/1179648*a^30 - 127/1179648*a^29 - 145/196608*a^28 + 121/147456*a^27 + 203/73728*a^26 + 581/294912*a^25 + 149/36864*a^24 - 83/73728*a^23 - 31/9216*a^22 - 1/73728*a^21 - 815/36864*a^20 - 287/9216*a^19 - 415/18432*a^18 - 131/18432*a^17 + 217/4608*a^16 + 511/4608*a^15 + 1/96*a^14 + 1099/4608*a^13 + 497/2304*a^12 - 11/288*a^11 - 205/1152*a^10 - 1153/1152*a^9 - 485/576*a^8 + 65/144*a^7 - 25/36*a^6 - 317/288*a^5 + 37/18*a^4 + 83/72*a^3 + 67/9*a^2 + 53/18*a - 101/9, 77/393216*a^35 + 31/393216*a^34 + 187/1179648*a^33 + 341/1179648*a^32 + 203/1179648*a^31 + 121/1179648*a^30 - 517/1179648*a^29 - 1163/294912*a^28 - 281/98304*a^27 - 283/73728*a^26 - 1487/294912*a^25 - 187/147456*a^24 + 305/73728*a^23 + 533/36864*a^22 + 4043/73728*a^21 + 21/512*a^20 + 761/18432*a^19 + 433/18432*a^18 - 725/18432*a^17 - 1193/9216*a^16 - 1067/4608*a^15 - 1385/2304*a^14 - 515/1536*a^13 - 35/288*a^12 + 191/1152*a^11 + 1091/1152*a^10 + 1621/1152*a^9 + 605/288*a^8 + 1555/288*a^7 + 4/3*a^6 - 281/288*a^5 - 305/144*a^4 - 499/72*a^3 - 247/36*a^2 - 68/9*a - 316/9, 25/1179648*a^35 - 97/1179648*a^34 + 109/1179648*a^33 - 95/393216*a^32 - 35/1179648*a^31 - 353/1179648*a^30 - 227/1179648*a^29 + 37/294912*a^28 + 85/73728*a^27 + 31/36864*a^26 + 91/32768*a^25 + 721/147456*a^24 - 11/36864*a^23 + 49/9216*a^22 - 115/8192*a^21 - 263/18432*a^20 - 59/2304*a^19 - 227/6144*a^18 - 487/18432*a^17 + 49/9216*a^16 + 65/1152*a^15 + 35/288*a^14 + 1553/4608*a^13 + 61/1152*a^12 + 5/12*a^11 - 227/1152*a^10 - 389/1152*a^9 - 233/288*a^8 - 19/12*a^7 - 71/36*a^6 - 95/288*a^5 - 1/16*a^4 - 25/36*a^3 + 151/18*a^2 - 53/18*a + 145/9, 173/1179648*a^35 - 61/1179648*a^34 - 11/1179648*a^33 - 325/1179648*a^32 - 11/1179648*a^31 - 425/1179648*a^30 + 167/393216*a^29 - 79/36864*a^28 + 559/294912*a^27 + 83/73728*a^26 + 1327/294912*a^25 + 133/147456*a^24 + 59/73728*a^23 - 85/12288*a^22 + 697/73728*a^21 - 395/18432*a^20 - 457/18432*a^19 - 485/18432*a^18 + 41/18432*a^17 + 379/9216*a^16 + 61/512*a^15 - 239/2304*a^14 + 1145/4608*a^13 + 79/1152*a^12 + 269/1152*a^11 - 623/1152*a^10 - 413/1152*a^9 - 79/48*a^8 + 743/288*a^7 - 71/36*a^6 + 313/288*a^5 - 149/144*a^4 + 313/72*a^3 + 107/36*a^2 + 13/2*a - 22, 89/1179648*a^35 + 179/1179648*a^34 + 139/393216*a^33 - 67/393216*a^32 - 167/1179648*a^31 - 1469/1179648*a^30 - 1463/1179648*a^29 - 131/49152*a^28 - 115/36864*a^27 - 145/49152*a^26 + 199/32768*a^25 + 2035/147456*a^24 + 437/18432*a^23 + 1081/36864*a^22 + 1385/73728*a^21 + 143/9216*a^20 - 121/3072*a^19 - 279/2048*a^18 - 4087/18432*a^17 - 2405/9216*a^16 - 169/1152*a^15 + 29/768*a^14 + 1877/4608*a^13 + 127/192*a^12 + 59/32*a^11 + 1663/1152*a^10 + 1747/1152*a^9 - 181/144*a^8 - 103/36*a^7 - 659/144*a^6 - 413/96*a^5 - 425/48*a^4 - 95/18*a^3 + 95/36*a^2 + 193/18*a + 256/9, 49/294912*a^35 + 5/36864*a^34 - 59/294912*a^33 - 5/32768*a^32 - 53/98304*a^31 - 173/294912*a^30 - 35/294912*a^29 - 23/8192*a^28 + 101/294912*a^27 + 5/1152*a^26 + 65/8192*a^25 + 199/24576*a^24 + 803/73728*a^23 - 305/36864*a^22 + 277/18432*a^21 - 625/18432*a^20 - 1357/18432*a^19 - 35/384*a^18 - 7/96*a^17 + 1/144*a^16 + 881/4608*a^15 + 61/768*a^14 + 391/1152*a^13 + 937/1152*a^12 + 151/384*a^11 + 1/32*a^10 - 311/288*a^9 - 433/144*a^8 + 227/288*a^7 - 149/72*a^6 - 25/9*a^5 - 25/24*a^4 + 47/8*a^3 + 161/36*a^2 + 427/18*a - 244/9, 1/196608*a^35 - 1/12288*a^34 - 1/16384*a^33 - 25/147456*a^32 - 1/147456*a^31 - 1/73728*a^30 + 7/18432*a^29 + 191/589824*a^28 + 23/12288*a^27 + 23/16384*a^26 + 343/147456*a^25 - 137/147456*a^24 - 251/73728*a^23 - 311/36864*a^22 - 395/36864*a^21 - 81/4096*a^20 - 7/512*a^19 - 1/4608*a^18 + 41/1152*a^17 + 689/9216*a^16 + 503/4608*a^15 + 199/2304*a^14 + 113/768*a^13 - 13/256*a^12 - 79/576*a^11 - 173/288*a^10 - 89/144*a^9 - 487/576*a^8 + 17/144*a^7 - 29/48*a^6 + 19/16*a^5 + 125/144*a^4 + 259/72*a^3 + 97/36*a^2 + 31/18*a - 31/9, 1/12288*a^35 - 11/49152*a^34 + 5/18432*a^33 - 47/147456*a^32 + 19/147456*a^31 - 7/147456*a^30 - 77/147456*a^29 - 7/147456*a^28 + 35/24576*a^27 - 19/147456*a^26 + 5/36864*a^25 + 203/36864*a^24 - 263/36864*a^23 + 557/36864*a^22 - 113/9216*a^21 - 7/768*a^20 - 37/9216*a^19 - 145/9216*a^18 - 37/2304*a^17 + 7/576*a^16 - 1/144*a^15 - 119/2304*a^14 + 53/128*a^13 - 185/576*a^12 + 277/576*a^11 - 113/576*a^10 - 43/144*a^9 + 79/144*a^8 - 97/144*a^7 - 33/16*a^6 + 7/36*a^5 + 49/36*a^4 - 215/36*a^3 + 491/36*a^2 - 349/18*a + 173/9, 7/147456*a^35 - 11/589824*a^34 - 11/589824*a^33 - 1/65536*a^32 - 7/589824*a^31 + 55/589824*a^30 + 37/589824*a^29 - 103/196608*a^28 + 115/294912*a^27 + 17/36864*a^26 + 7/24576*a^25 - 17/147456*a^24 - 55/36864*a^23 - 67/36864*a^22 + 83/18432*a^21 - 163/36864*a^20 - 53/18432*a^19 + 1/1536*a^18 + 91/9216*a^17 + 203/9216*a^16 + 59/2304*a^15 - 53/768*a^14 + 1/36*a^13 - 35/2304*a^12 - 25/384*a^11 - 19/144*a^10 - 173/576*a^9 - 71/576*a^8 + 271/288*a^7 - 1/72*a^6 + 7/36*a^5 + 5/16*a^4 + 5/9*a^3 + 37/36*a^2 - 2/9*a - 80/9, 1/98304*a^35 + 13/294912*a^34 - 1/98304*a^33 + 1/98304*a^32 - 43/294912*a^31 - 25/294912*a^30 - 25/98304*a^29 - 41/147456*a^28 - 65/147456*a^27 + 7/12288*a^26 + 23/24576*a^25 + 107/36864*a^24 + 55/18432*a^23 + 17/6144*a^22 + 1/1152*a^21 - 5/2304*a^20 - 19/1536*a^19 - 35/1536*a^18 - 149/4608*a^17 - 55/2304*a^16 + 5/768*a^15 + 43/1152*a^14 + 35/576*a^13 + 19/96*a^12 + 5/32*a^11 + 77/288*a^10 - 31/288*a^9 - 1/4*a^8 - 95/144*a^7 - 29/72*a^6 - 31/24*a^5 - 5/12*a^4 - 13/18*a^3 + 13/9*a^2 + 5/2*a + 31/9, 755/1179648*a^35 + 199/1179648*a^34 + 1037/1179648*a^33 + 233/393216*a^32 + 1877/1179648*a^31 - 737/1179648*a^30 + 5/131072*a^29 - 2975/196608*a^28 - 1079/147456*a^27 - 1541/73728*a^26 - 1289/98304*a^25 - 1247/73728*a^24 + 1547/73728*a^23 + 35/768*a^22 + 14567/73728*a^21 + 5635/36864*a^20 + 1781/9216*a^19 + 587/6144*a^18 - 1847/18432*a^17 - 565/1152*a^16 - 1313/1536*a^15 - 881/384*a^14 - 5249/4608*a^13 - 2777/2304*a^12 + 83/64*a^11 + 2447/1152*a^10 + 8503/1152*a^9 + 1123/192*a^8 + 409/18*a^7 + 43/72*a^6 + 1415/288*a^5 - 437/24*a^4 - 1057/72*a^3 - 370/9*a^2 - 23*a - 1127/9 (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:		\( 290558369176692.94 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 290558369176692.94 \cdot 162}{14\cdot\sqrt{7864785536926430215681870035583404646232472578456198834028544}}\cr\approx \mathstrut & 0.279263744123773 \end{aligned}\] (assuming GRH)

Galois group

$C_2\times C_6\times S_4$ (as 36T330):

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	2.3.0.1	$x^{3} + x + 1$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	2.3.0.1	$x^{3} + x + 1$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	2.3.0.1	$x^{3} + x + 1$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	2.3.0.1	$x^{3} + x + 1$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	2.12.12.24	$x^{12} - 10 x^{11} + 72 x^{10} - 292 x^{9} + 1028 x^{8} - 2144 x^{7} + 5280 x^{6} - 5568 x^{5} + 12400 x^{4} - 12384 x^{3} + 24832 x^{2} - 14784 x + 11200$	$2$	$6$	$12$	$D_4 \times C_3$	$[2, 2]^{6}$
	2.12.12.24	$x^{12} - 10 x^{11} + 72 x^{10} - 292 x^{9} + 1028 x^{8} - 2144 x^{7} + 5280 x^{6} - 5568 x^{5} + 12400 x^{4} - 12384 x^{3} + 24832 x^{2} - 14784 x + 11200$	$2$	$6$	$12$	$D_4 \times C_3$	$[2, 2]^{6}$
\(7\) 7	7.6.5.5	$x^{6} + 7$	$6$	$1$	$5$	$C_6$	$[\ ]_{6}$
	7.6.5.5	$x^{6} + 7$	$6$	$1$	$5$	$C_6$	$[\ ]_{6}$
		Deg $24$	$6$	$4$	$20$
\(229\) 229		Deg $6$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
		Deg $6$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
		Deg $12$	$2$	$6$	$6$
		Deg $12$	$2$	$6$	$6$