Normalized defining polynomial

\( x^{36} - 2 x^{34} - 2 x^{33} + 8 x^{31} + 4 x^{30} - 8 x^{29} - 8 x^{28} - 8 x^{27} + 16 x^{26} + \cdots + 262144 \) x^36 - 2*x^34 - 2*x^33 + 8*x^31 + 4*x^30 - 8*x^29 - 8*x^28 - 8*x^27 + 16*x^26 - 16*x^24 + 64*x^23 - 64*x^22 - 32*x^21 - 64*x^20 - 64*x^19 + 576*x^18 - 128*x^17 - 256*x^16 - 256*x^15 - 1024*x^14 + 2048*x^13 - 1024*x^12 + 4096*x^10 - 4096*x^9 - 8192*x^8 - 16384*x^7 + 16384*x^6 + 65536*x^5 - 65536*x^3 - 131072*x^2 + 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:		\(2209286039084312039484359423586799049408706824590560826753024\) 2209286039084312039484359423586799049408706824590560826753024 \(\medspace = 2^{24}\cdot 7^{30}\cdot 31^{6}\cdot 37^{12}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(47.45\)	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:		$2^{2/3}7^{5/6}31^{1/2}37^{1/2}\approx 272.0926055185523$
Ramified primes:		\(2\), \(7\), \(31\), \(37\) 2, 7, 31, 37	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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{4}a^{6}$, $\frac{1}{4}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{8}a^{9}$, $\frac{1}{8}a^{10}$, $\frac{1}{8}a^{11}$, $\frac{1}{16}a^{12}$, $\frac{1}{16}a^{13}$, $\frac{1}{48}a^{14}-\frac{1}{12}a^{7}-\frac{1}{3}$, $\frac{1}{96}a^{15}+\frac{1}{12}a^{8}+\frac{1}{3}a$, $\frac{1}{96}a^{16}-\frac{1}{24}a^{9}+\frac{1}{3}a^{2}$, $\frac{1}{96}a^{17}-\frac{1}{24}a^{10}-\frac{1}{6}a^{3}$, $\frac{1}{192}a^{18}+\frac{1}{24}a^{11}+\frac{1}{6}a^{4}$, $\frac{1}{192}a^{19}-\frac{1}{48}a^{12}+\frac{1}{6}a^{5}$, $\frac{1}{384}a^{20}-\frac{1}{192}a^{17}-\frac{1}{96}a^{14}+\frac{1}{48}a^{13}-\frac{1}{16}a^{11}-\frac{1}{24}a^{10}-\frac{1}{8}a^{8}-\frac{1}{12}a^{7}+\frac{1}{12}a^{6}-\frac{1}{4}a^{5}-\frac{1}{6}a^{3}-\frac{1}{2}a^{2}-\frac{1}{3}$, $\frac{1}{768}a^{21}-\frac{1}{384}a^{19}-\frac{1}{384}a^{18}-\frac{1}{192}a^{15}-\frac{1}{96}a^{14}-\frac{1}{32}a^{13}+\frac{1}{96}a^{12}-\frac{1}{48}a^{11}-\frac{1}{16}a^{9}+\frac{1}{12}a^{8}-\frac{1}{8}a^{6}-\frac{1}{12}a^{5}-\frac{1}{12}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{768}a^{22}-\frac{1}{384}a^{19}-\frac{1}{192}a^{17}-\frac{1}{192}a^{16}-\frac{1}{32}a^{13}-\frac{1}{48}a^{12}-\frac{1}{16}a^{11}+\frac{1}{48}a^{10}-\frac{1}{24}a^{9}-\frac{1}{24}a^{8}-\frac{1}{8}a^{7}+\frac{1}{6}a^{5}-\frac{1}{4}a^{4}-\frac{1}{6}a^{3}-\frac{1}{6}a^{2}-\frac{1}{3}a$, $\frac{1}{1536}a^{23}-\frac{1}{768}a^{20}-\frac{1}{384}a^{19}-\frac{1}{384}a^{18}+\frac{1}{384}a^{17}-\frac{1}{192}a^{16}+\frac{1}{192}a^{14}+\frac{1}{48}a^{13}+\frac{1}{96}a^{12}+\frac{1}{96}a^{11}-\frac{1}{24}a^{10}-\frac{1}{16}a^{9}+\frac{1}{16}a^{8}+\frac{1}{24}a^{7}+\frac{1}{12}a^{6}-\frac{5}{24}a^{5}+\frac{1}{6}a^{4}+\frac{1}{12}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3072}a^{24}-\frac{1}{1536}a^{22}-\frac{1}{1536}a^{21}-\frac{1}{384}a^{19}+\frac{1}{768}a^{18}-\frac{1}{384}a^{17}-\frac{1}{384}a^{16}-\frac{1}{384}a^{15}+\frac{1}{192}a^{14}-\frac{1}{192}a^{12}-\frac{1}{48}a^{11}-\frac{1}{16}a^{10}-\frac{5}{96}a^{9}+\frac{1}{16}a^{8}+\frac{1}{16}a^{7}-\frac{1}{16}a^{6}-\frac{5}{24}a^{5}-\frac{1}{12}a^{4}+\frac{1}{12}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3072}a^{25}-\frac{1}{1536}a^{22}-\frac{1}{768}a^{20}-\frac{1}{768}a^{19}-\frac{1}{192}a^{17}+\frac{1}{384}a^{16}-\frac{1}{192}a^{15}-\frac{1}{192}a^{14}-\frac{5}{192}a^{13}-\frac{1}{96}a^{12}+\frac{5}{96}a^{11}-\frac{1}{96}a^{10}-\frac{1}{24}a^{9}-\frac{1}{12}a^{8}-\frac{5}{48}a^{7}-\frac{1}{24}a^{6}-\frac{1}{24}a^{5}-\frac{1}{12}a^{4}+\frac{1}{12}a^{3}-\frac{1}{6}a^{2}+\frac{1}{3}$, $\frac{1}{6144}a^{26}-\frac{1}{3072}a^{23}-\frac{1}{1536}a^{22}-\frac{1}{1536}a^{21}+\frac{1}{1536}a^{20}-\frac{1}{768}a^{19}-\frac{1}{384}a^{18}-\frac{1}{256}a^{17}+\frac{1}{384}a^{15}-\frac{1}{128}a^{14}+\frac{1}{48}a^{13}+\frac{1}{64}a^{12}-\frac{1}{192}a^{11}-\frac{1}{32}a^{10}+\frac{1}{24}a^{9}+\frac{7}{96}a^{8}-\frac{1}{24}a^{7}+\frac{1}{48}a^{6}+\frac{1}{6}a^{5}+\frac{1}{6}a^{4}+\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{12288}a^{27}-\frac{1}{6144}a^{25}-\frac{1}{6144}a^{24}-\frac{1}{1536}a^{22}+\frac{1}{3072}a^{21}-\frac{1}{1536}a^{20}+\frac{1}{512}a^{19}-\frac{1}{1536}a^{18}-\frac{1}{256}a^{17}-\frac{1}{192}a^{16}-\frac{1}{768}a^{15}+\frac{1}{192}a^{14}-\frac{1}{64}a^{13}-\frac{3}{128}a^{12}+\frac{1}{64}a^{11}-\frac{5}{192}a^{10}+\frac{1}{192}a^{9}-\frac{5}{96}a^{8}-\frac{1}{16}a^{7}-\frac{5}{48}a^{6}-\frac{1}{4}a^{5}+\frac{1}{12}a^{4}-\frac{1}{12}a^{3}-\frac{1}{6}a^{2}+\frac{1}{3}$, $\frac{1}{36864}a^{28}-\frac{1}{6144}a^{25}-\frac{1}{3072}a^{23}-\frac{1}{3072}a^{22}+\frac{1}{2304}a^{21}-\frac{1}{768}a^{20}+\frac{1}{1536}a^{19}+\frac{1}{768}a^{18}-\frac{1}{768}a^{17}+\frac{1}{256}a^{16}-\frac{1}{384}a^{15}-\frac{11}{1152}a^{14}-\frac{1}{384}a^{13}+\frac{1}{48}a^{12}-\frac{5}{192}a^{10}+\frac{1}{96}a^{9}+\frac{11}{96}a^{8}-\frac{1}{36}a^{7}+\frac{1}{48}a^{6}-\frac{1}{24}a^{5}+\frac{1}{12}a^{4}+\frac{1}{12}a^{3}-\frac{1}{6}a^{2}+\frac{1}{9}$, $\frac{1}{73728}a^{29}-\frac{1}{12288}a^{26}-\frac{1}{6144}a^{25}-\frac{1}{6144}a^{24}+\frac{1}{6144}a^{23}+\frac{5}{9216}a^{22}-\frac{1}{1536}a^{21}-\frac{1}{1024}a^{20}-\frac{1}{384}a^{19}+\frac{1}{1536}a^{18}-\frac{1}{512}a^{17}+\frac{7}{2304}a^{15}+\frac{7}{768}a^{14}-\frac{1}{128}a^{13}-\frac{1}{96}a^{12}-\frac{17}{384}a^{11}-\frac{1}{96}a^{10}+\frac{5}{192}a^{9}+\frac{11}{288}a^{8}-\frac{1}{4}a^{4}-\frac{1}{12}a^{3}-\frac{1}{6}a^{2}+\frac{2}{9}a+\frac{1}{3}$, $\frac{1}{147456}a^{30}-\frac{1}{73728}a^{28}-\frac{1}{24576}a^{27}+\frac{1}{12288}a^{24}-\frac{1}{18432}a^{23}-\frac{1}{2048}a^{22}+\frac{5}{18432}a^{21}-\frac{1}{1024}a^{20}+\frac{1}{1536}a^{19}+\frac{1}{1024}a^{18}-\frac{7}{2304}a^{16}-\frac{5}{1536}a^{15}+\frac{23}{2304}a^{14}+\frac{13}{768}a^{13}-\frac{23}{768}a^{12}+\frac{5}{128}a^{11}+\frac{5}{96}a^{10}-\frac{11}{288}a^{9}-\frac{7}{96}a^{8}-\frac{5}{72}a^{7}-\frac{1}{12}a^{6}-\frac{1}{24}a^{5}+\frac{1}{6}a^{3}+\frac{1}{9}a^{2}+\frac{1}{3}a+\frac{1}{9}$, $\frac{1}{147456}a^{31}-\frac{1}{73728}a^{28}-\frac{1}{12288}a^{26}+\frac{1}{12288}a^{25}+\frac{1}{9216}a^{24}+\frac{1}{2048}a^{22}-\frac{5}{9216}a^{21}+\frac{1}{1024}a^{20}-\frac{7}{3072}a^{19}-\frac{1}{512}a^{18}+\frac{19}{4608}a^{17}+\frac{1}{1536}a^{16}-\frac{1}{384}a^{15}-\frac{11}{1152}a^{14}-\frac{1}{256}a^{13}-\frac{1}{128}a^{12}-\frac{13}{384}a^{11}-\frac{1}{576}a^{10}+\frac{7}{192}a^{9}-\frac{1}{12}a^{8}+\frac{7}{72}a^{7}-\frac{1}{12}a^{5}+\frac{1}{6}a^{4}+\frac{1}{9}a^{3}-\frac{1}{6}a^{2}-\frac{1}{3}a+\frac{4}{9}$, $\frac{1}{294912}a^{32}-\frac{1}{147456}a^{29}-\frac{1}{73728}a^{28}-\frac{1}{24576}a^{27}-\frac{1}{24576}a^{26}+\frac{5}{36864}a^{25}+\frac{1}{4096}a^{23}-\frac{1}{2304}a^{22}+\frac{11}{18432}a^{21}-\frac{1}{6144}a^{20}-\frac{1}{1536}a^{19}-\frac{11}{9216}a^{18}+\frac{13}{3072}a^{17}-\frac{5}{1536}a^{16}-\frac{11}{2304}a^{15}+\frac{19}{4608}a^{14}+\frac{3}{128}a^{13}-\frac{5}{256}a^{12}-\frac{1}{72}a^{11}-\frac{1}{192}a^{10}+\frac{1}{192}a^{9}+\frac{5}{72}a^{8}+\frac{11}{144}a^{7}+\frac{1}{48}a^{6}-\frac{1}{24}a^{5}-\frac{7}{36}a^{4}+\frac{1}{6}a^{3}+\frac{1}{3}a^{2}-\frac{1}{9}a-\frac{2}{9}$, $\frac{1}{589824}a^{33}-\frac{1}{294912}a^{31}-\frac{1}{294912}a^{30}-\frac{1}{73728}a^{28}-\frac{1}{49152}a^{27}-\frac{1}{73728}a^{26}-\frac{1}{8192}a^{25}+\frac{11}{73728}a^{24}-\frac{11}{36864}a^{23}-\frac{1}{3072}a^{22}+\frac{19}{36864}a^{21}-\frac{1}{1024}a^{20}-\frac{1}{9216}a^{19}-\frac{5}{2048}a^{18}+\frac{23}{9216}a^{17}+\frac{11}{9216}a^{16}-\frac{1}{3072}a^{15}-\frac{5}{4608}a^{14}-\frac{1}{768}a^{13}+\frac{53}{2304}a^{12}-\frac{1}{16}a^{11}+\frac{5}{144}a^{10}-\frac{17}{288}a^{9}-\frac{1}{8}a^{8}+\frac{1}{72}a^{7}-\frac{1}{24}a^{6}-\frac{1}{72}a^{5}-\frac{1}{4}a^{4}+\frac{7}{36}a^{3}-\frac{7}{18}a^{2}-\frac{1}{3}a+\frac{4}{9}$, $\frac{1}{589824}a^{34}-\frac{1}{294912}a^{31}-\frac{1}{147456}a^{29}-\frac{1}{147456}a^{28}+\frac{1}{36864}a^{27}-\frac{1}{12288}a^{26}+\frac{1}{8192}a^{25}+\frac{1}{36864}a^{24}+\frac{1}{12288}a^{23}-\frac{13}{36864}a^{22}+\frac{7}{18432}a^{21}+\frac{1}{18432}a^{20}+\frac{13}{6144}a^{19}-\frac{7}{2304}a^{17}-\frac{7}{3072}a^{16}-\frac{19}{4608}a^{15}+\frac{47}{4608}a^{14}+\frac{35}{2304}a^{13}-\frac{7}{256}a^{12}+\frac{1}{128}a^{11}-\frac{1}{576}a^{10}-\frac{7}{192}a^{9}+\frac{1}{9}a^{8}-\frac{17}{144}a^{7}+\frac{1}{36}a^{6}-\frac{1}{12}a^{5}-\frac{5}{36}a^{3}-\frac{1}{9}a+\frac{2}{9}$, $\frac{1}{1179648}a^{35}-\frac{1}{589824}a^{32}-\frac{1}{294912}a^{31}-\frac{1}{294912}a^{30}+\frac{1}{294912}a^{29}-\frac{1}{147456}a^{28}-\frac{1}{24576}a^{27}-\frac{1}{49152}a^{26}-\frac{1}{9216}a^{25}+\frac{5}{73728}a^{24}-\frac{19}{73728}a^{23}+\frac{1}{4608}a^{22}+\frac{7}{36864}a^{21}-\frac{11}{12288}a^{20}-\frac{5}{6144}a^{19}+\frac{1}{576}a^{18}-\frac{41}{18432}a^{17}+\frac{19}{4608}a^{16}+\frac{25}{9216}a^{15}-\frac{1}{1152}a^{14}-\frac{1}{128}a^{12}+\frac{1}{576}a^{11}-\frac{17}{288}a^{10}-\frac{5}{288}a^{9}+\frac{5}{72}a^{8}-\frac{13}{144}a^{7}+\frac{1}{48}a^{6}-\frac{5}{24}a^{5}-\frac{1}{36}a^{4}-\frac{5}{36}a^{3}-\frac{1}{18}a^{2}+\frac{2}{9}a-\frac{1}{3}$ 1, a, a^2, 1/2*a^3, 1/2*a^4, 1/2*a^5, 1/4*a^6, 1/4*a^7, 1/4*a^8, 1/8*a^9, 1/8*a^10, 1/8*a^11, 1/16*a^12, 1/16*a^13, 1/48*a^14 - 1/12*a^7 - 1/3, 1/96*a^15 + 1/12*a^8 + 1/3*a, 1/96*a^16 - 1/24*a^9 + 1/3*a^2, 1/96*a^17 - 1/24*a^10 - 1/6*a^3, 1/192*a^18 + 1/24*a^11 + 1/6*a^4, 1/192*a^19 - 1/48*a^12 + 1/6*a^5, 1/384*a^20 - 1/192*a^17 - 1/96*a^14 + 1/48*a^13 - 1/16*a^11 - 1/24*a^10 - 1/8*a^8 - 1/12*a^7 + 1/12*a^6 - 1/4*a^5 - 1/6*a^3 - 1/2*a^2 - 1/3, 1/768*a^21 - 1/384*a^19 - 1/384*a^18 - 1/192*a^15 - 1/96*a^14 - 1/32*a^13 + 1/96*a^12 - 1/48*a^11 - 1/16*a^9 + 1/12*a^8 - 1/8*a^6 - 1/12*a^5 - 1/12*a^4 - 1/4*a^3 - 1/2*a^2 + 1/3*a + 1/3, 1/768*a^22 - 1/384*a^19 - 1/192*a^17 - 1/192*a^16 - 1/32*a^13 - 1/48*a^12 - 1/16*a^11 + 1/48*a^10 - 1/24*a^9 - 1/24*a^8 - 1/8*a^7 + 1/6*a^5 - 1/4*a^4 - 1/6*a^3 - 1/6*a^2 - 1/3*a, 1/1536*a^23 - 1/768*a^20 - 1/384*a^19 - 1/384*a^18 + 1/384*a^17 - 1/192*a^16 + 1/192*a^14 + 1/48*a^13 + 1/96*a^12 + 1/96*a^11 - 1/24*a^10 - 1/16*a^9 + 1/16*a^8 + 1/24*a^7 + 1/12*a^6 - 5/24*a^5 + 1/6*a^4 + 1/12*a^3 - 1/3*a^2 - 1/3, 1/3072*a^24 - 1/1536*a^22 - 1/1536*a^21 - 1/384*a^19 + 1/768*a^18 - 1/384*a^17 - 1/384*a^16 - 1/384*a^15 + 1/192*a^14 - 1/192*a^12 - 1/48*a^11 - 1/16*a^10 - 5/96*a^9 + 1/16*a^8 + 1/16*a^7 - 1/16*a^6 - 5/24*a^5 - 1/12*a^4 + 1/12*a^3 - 1/3*a^2 + 1/3*a + 1/3, 1/3072*a^25 - 1/1536*a^22 - 1/768*a^20 - 1/768*a^19 - 1/192*a^17 + 1/384*a^16 - 1/192*a^15 - 1/192*a^14 - 5/192*a^13 - 1/96*a^12 + 5/96*a^11 - 1/96*a^10 - 1/24*a^9 - 1/12*a^8 - 5/48*a^7 - 1/24*a^6 - 1/24*a^5 - 1/12*a^4 + 1/12*a^3 - 1/6*a^2 + 1/3, 1/6144*a^26 - 1/3072*a^23 - 1/1536*a^22 - 1/1536*a^21 + 1/1536*a^20 - 1/768*a^19 - 1/384*a^18 - 1/256*a^17 + 1/384*a^15 - 1/128*a^14 + 1/48*a^13 + 1/64*a^12 - 1/192*a^11 - 1/32*a^10 + 1/24*a^9 + 7/96*a^8 - 1/24*a^7 + 1/48*a^6 + 1/6*a^5 + 1/6*a^4 + 1/3*a^2 - 1/3, 1/12288*a^27 - 1/6144*a^25 - 1/6144*a^24 - 1/1536*a^22 + 1/3072*a^21 - 1/1536*a^20 + 1/512*a^19 - 1/1536*a^18 - 1/256*a^17 - 1/192*a^16 - 1/768*a^15 + 1/192*a^14 - 1/64*a^13 - 3/128*a^12 + 1/64*a^11 - 5/192*a^10 + 1/192*a^9 - 5/96*a^8 - 1/16*a^7 - 5/48*a^6 - 1/4*a^5 + 1/12*a^4 - 1/12*a^3 - 1/6*a^2 + 1/3, 1/36864*a^28 - 1/6144*a^25 - 1/3072*a^23 - 1/3072*a^22 + 1/2304*a^21 - 1/768*a^20 + 1/1536*a^19 + 1/768*a^18 - 1/768*a^17 + 1/256*a^16 - 1/384*a^15 - 11/1152*a^14 - 1/384*a^13 + 1/48*a^12 - 5/192*a^10 + 1/96*a^9 + 11/96*a^8 - 1/36*a^7 + 1/48*a^6 - 1/24*a^5 + 1/12*a^4 + 1/12*a^3 - 1/6*a^2 + 1/9, 1/73728*a^29 - 1/12288*a^26 - 1/6144*a^25 - 1/6144*a^24 + 1/6144*a^23 + 5/9216*a^22 - 1/1536*a^21 - 1/1024*a^20 - 1/384*a^19 + 1/1536*a^18 - 1/512*a^17 + 7/2304*a^15 + 7/768*a^14 - 1/128*a^13 - 1/96*a^12 - 17/384*a^11 - 1/96*a^10 + 5/192*a^9 + 11/288*a^8 - 1/4*a^4 - 1/12*a^3 - 1/6*a^2 + 2/9*a + 1/3, 1/147456*a^30 - 1/73728*a^28 - 1/24576*a^27 + 1/12288*a^24 - 1/18432*a^23 - 1/2048*a^22 + 5/18432*a^21 - 1/1024*a^20 + 1/1536*a^19 + 1/1024*a^18 - 7/2304*a^16 - 5/1536*a^15 + 23/2304*a^14 + 13/768*a^13 - 23/768*a^12 + 5/128*a^11 + 5/96*a^10 - 11/288*a^9 - 7/96*a^8 - 5/72*a^7 - 1/12*a^6 - 1/24*a^5 + 1/6*a^3 + 1/9*a^2 + 1/3*a + 1/9, 1/147456*a^31 - 1/73728*a^28 - 1/12288*a^26 + 1/12288*a^25 + 1/9216*a^24 + 1/2048*a^22 - 5/9216*a^21 + 1/1024*a^20 - 7/3072*a^19 - 1/512*a^18 + 19/4608*a^17 + 1/1536*a^16 - 1/384*a^15 - 11/1152*a^14 - 1/256*a^13 - 1/128*a^12 - 13/384*a^11 - 1/576*a^10 + 7/192*a^9 - 1/12*a^8 + 7/72*a^7 - 1/12*a^5 + 1/6*a^4 + 1/9*a^3 - 1/6*a^2 - 1/3*a + 4/9, 1/294912*a^32 - 1/147456*a^29 - 1/73728*a^28 - 1/24576*a^27 - 1/24576*a^26 + 5/36864*a^25 + 1/4096*a^23 - 1/2304*a^22 + 11/18432*a^21 - 1/6144*a^20 - 1/1536*a^19 - 11/9216*a^18 + 13/3072*a^17 - 5/1536*a^16 - 11/2304*a^15 + 19/4608*a^14 + 3/128*a^13 - 5/256*a^12 - 1/72*a^11 - 1/192*a^10 + 1/192*a^9 + 5/72*a^8 + 11/144*a^7 + 1/48*a^6 - 1/24*a^5 - 7/36*a^4 + 1/6*a^3 + 1/3*a^2 - 1/9*a - 2/9, 1/589824*a^33 - 1/294912*a^31 - 1/294912*a^30 - 1/73728*a^28 - 1/49152*a^27 - 1/73728*a^26 - 1/8192*a^25 + 11/73728*a^24 - 11/36864*a^23 - 1/3072*a^22 + 19/36864*a^21 - 1/1024*a^20 - 1/9216*a^19 - 5/2048*a^18 + 23/9216*a^17 + 11/9216*a^16 - 1/3072*a^15 - 5/4608*a^14 - 1/768*a^13 + 53/2304*a^12 - 1/16*a^11 + 5/144*a^10 - 17/288*a^9 - 1/8*a^8 + 1/72*a^7 - 1/24*a^6 - 1/72*a^5 - 1/4*a^4 + 7/36*a^3 - 7/18*a^2 - 1/3*a + 4/9, 1/589824*a^34 - 1/294912*a^31 - 1/147456*a^29 - 1/147456*a^28 + 1/36864*a^27 - 1/12288*a^26 + 1/8192*a^25 + 1/36864*a^24 + 1/12288*a^23 - 13/36864*a^22 + 7/18432*a^21 + 1/18432*a^20 + 13/6144*a^19 - 7/2304*a^17 - 7/3072*a^16 - 19/4608*a^15 + 47/4608*a^14 + 35/2304*a^13 - 7/256*a^12 + 1/128*a^11 - 1/576*a^10 - 7/192*a^9 + 1/9*a^8 - 17/144*a^7 + 1/36*a^6 - 1/12*a^5 - 5/36*a^3 - 1/9*a + 2/9, 1/1179648*a^35 - 1/589824*a^32 - 1/294912*a^31 - 1/294912*a^30 + 1/294912*a^29 - 1/147456*a^28 - 1/24576*a^27 - 1/49152*a^26 - 1/9216*a^25 + 5/73728*a^24 - 19/73728*a^23 + 1/4608*a^22 + 7/36864*a^21 - 11/12288*a^20 - 5/6144*a^19 + 1/576*a^18 - 41/18432*a^17 + 19/4608*a^16 + 25/9216*a^15 - 1/1152*a^14 - 1/128*a^12 + 1/576*a^11 - 17/288*a^10 - 5/288*a^9 + 5/72*a^8 - 13/144*a^7 + 1/48*a^6 - 5/24*a^5 - 1/36*a^4 - 5/36*a^3 - 1/18*a^2 + 2/9*a - 1/3

Class group and class number

$C_{3}\times C_{63}$, which has order $189$ (assuming GRH)

Unit group

Rank:		$17$	sage: UK.rank()   gp: K.fu   magma: UnitRank(K);   oscar: rank(UK)
Torsion generator:		\( -\frac{1}{589824} a^{34} - \frac{1}{589824} a^{33} + \frac{1}{147456} a^{31} + \frac{1}{294912} a^{30} + \frac{7}{147456} a^{29} - \frac{1}{147456} a^{28} - \frac{1}{147456} a^{27} + \frac{1}{73728} a^{26} - \frac{1}{73728} a^{24} + \frac{1}{18432} a^{23} + \frac{1}{36864} a^{22} - \frac{1}{36864} a^{21} - \frac{1}{18432} a^{20} - \frac{1}{18432} a^{19} + \frac{1}{2048} a^{18} - \frac{1}{9216} a^{17} - \frac{1}{4608} a^{16} - \frac{19}{9216} a^{15} - \frac{1}{1152} a^{14} + \frac{1}{576} a^{13} - \frac{1}{1152} a^{12} + \frac{1}{288} a^{10} - \frac{1}{288} a^{9} - \frac{5}{288} a^{8} - \frac{1}{72} a^{7} + \frac{1}{72} a^{6} + \frac{1}{18} a^{5} - \frac{1}{18} a^{3} - \frac{1}{9} a^{2} - \frac{5}{9} a + \frac{2}{9} \) -(1)/(589824)*a^(34) - (1)/(589824)*a^(33) + (1)/(147456)*a^(31) + (1)/(294912)*a^(30) + (7)/(147456)*a^(29) - (1)/(147456)*a^(28) - (1)/(147456)*a^(27) + (1)/(73728)*a^(26) - (1)/(73728)*a^(24) + (1)/(18432)*a^(23) + (1)/(36864)*a^(22) - (1)/(36864)*a^(21) - (1)/(18432)*a^(20) - (1)/(18432)*a^(19) + (1)/(2048)*a^(18) - (1)/(9216)*a^(17) - (1)/(4608)*a^(16) - (19)/(9216)*a^(15) - (1)/(1152)*a^(14) + (1)/(576)*a^(13) - (1)/(1152)*a^(12) + (1)/(288)*a^(10) - (1)/(288)*a^(9) - (5)/(288)*a^(8) - (1)/(72)*a^(7) + (1)/(72)*a^(6) + (1)/(18)*a^(5) - (1)/(18)*a^(3) - (1)/(9)*a^(2) - (5)/(9)*a + (2)/(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:		$\frac{1}{393216}a^{35}+\frac{1}{294912}a^{34}-\frac{11}{589824}a^{32}-\frac{1}{294912}a^{31}+\frac{7}{294912}a^{30}+\frac{1}{98304}a^{29}+\frac{1}{147456}a^{28}-\frac{1}{36864}a^{27}+\frac{1}{49152}a^{26}+\frac{1}{36864}a^{25}-\frac{7}{73728}a^{24}-\frac{5}{73728}a^{23}+\frac{7}{36864}a^{21}+\frac{1}{36864}a^{20}-\frac{5}{6144}a^{19}-\frac{7}{9216}a^{18}+\frac{1}{18432}a^{17}+\frac{7}{2304}a^{16}+\frac{1}{1024}a^{15}-\frac{7}{2304}a^{14}+\frac{7}{2304}a^{13}-\frac{1}{768}a^{12}+\frac{5}{576}a^{11}+\frac{1}{288}a^{10}+\frac{1}{144}a^{9}+\frac{1}{32}a^{8}-\frac{1}{36}a^{7}-\frac{13}{144}a^{6}+\frac{5}{18}a^{4}+\frac{7}{36}a^{3}-\frac{1}{9}a^{2}-\frac{1}{3}a-\frac{8}{9}$, $\frac{1}{393216}a^{35}-\frac{1}{196608}a^{34}-\frac{1}{196608}a^{33}-\frac{1}{196608}a^{32}+\frac{5}{294912}a^{31}+\frac{1}{36864}a^{30}+\frac{1}{294912}a^{29}-\frac{1}{36864}a^{28}+\frac{1}{49152}a^{27}-\frac{1}{49152}a^{26}+\frac{1}{12288}a^{25}-\frac{1}{4608}a^{24}-\frac{1}{73728}a^{23}+\frac{5}{36864}a^{22}-\frac{5}{18432}a^{21}+\frac{5}{12288}a^{20}-\frac{1}{3072}a^{19}+\frac{1}{6144}a^{18}+\frac{7}{18432}a^{17}-\frac{5}{4608}a^{16}-\frac{1}{4608}a^{15}-\frac{5}{2304}a^{14}+\frac{5}{768}a^{13}+\frac{1}{128}a^{12}-\frac{1}{192}a^{11}+\frac{1}{72}a^{10}+\frac{1}{288}a^{9}-\frac{1}{72}a^{8}-\frac{1}{72}a^{7}-\frac{1}{48}a^{6}+\frac{1}{12}a^{5}+\frac{1}{6}a^{4}-\frac{11}{36}a^{3}-\frac{2}{9}a^{2}-\frac{4}{9}a+\frac{2}{9}$, $\frac{1}{196608}a^{35}-\frac{1}{2048}a^{21}+\frac{1}{1536}a^{14}-\frac{7}{48}a^{7}+\frac{1}{3}$, $\frac{1}{393216}a^{35}+\frac{1}{98304}a^{34}+\frac{1}{98304}a^{33}-\frac{1}{196608}a^{32}-\frac{13}{294912}a^{31}-\frac{1}{294912}a^{30}+\frac{11}{294912}a^{29}+\frac{5}{147456}a^{28}+\frac{1}{12288}a^{27}-\frac{7}{49152}a^{26}+\frac{1}{12288}a^{25}-\frac{7}{73728}a^{24}-\frac{37}{73728}a^{23}+\frac{7}{9216}a^{22}-\frac{1}{36864}a^{21}+\frac{11}{12288}a^{20}+\frac{1}{6144}a^{19}-\frac{13}{3072}a^{18}+\frac{25}{18432}a^{17}+\frac{1}{1152}a^{16}+\frac{17}{9216}a^{15}+\frac{13}{2304}a^{14}-\frac{7}{768}a^{13}+\frac{1}{64}a^{12}-\frac{1}{192}a^{11}-\frac{5}{288}a^{10}+\frac{5}{144}a^{9}-\frac{7}{144}a^{8}+\frac{1}{9}a^{7}-\frac{7}{48}a^{6}-\frac{5}{12}a^{5}+\frac{1}{6}a^{4}+\frac{7}{36}a^{3}+\frac{7}{9}a^{2}-\frac{2}{9}a-\frac{16}{9}$, $\frac{5}{589824}a^{35}-\frac{1}{589824}a^{34}-\frac{1}{196608}a^{33}-\frac{1}{147456}a^{32}-\frac{1}{147456}a^{31}+\frac{7}{294912}a^{30}-\frac{5}{147456}a^{29}-\frac{1}{49152}a^{28}+\frac{5}{147456}a^{27}-\frac{1}{24576}a^{26}+\frac{1}{18432}a^{25}+\frac{1}{73728}a^{24}-\frac{1}{9216}a^{23}+\frac{7}{36864}a^{22}-\frac{25}{36864}a^{21}+\frac{5}{18432}a^{20}-\frac{1}{6144}a^{19}-\frac{13}{18432}a^{18}+\frac{19}{9216}a^{17}-\frac{1}{4608}a^{16}+\frac{11}{9216}a^{15}+\frac{1}{1536}a^{14}+\frac{7}{2304}a^{13}+\frac{1}{128}a^{12}-\frac{1}{288}a^{11}-\frac{1}{288}a^{10}+\frac{5}{288}a^{9}+\frac{1}{288}a^{8}-\frac{23}{144}a^{7}-\frac{7}{144}a^{6}+\frac{2}{9}a^{4}+\frac{1}{18}a^{3}-\frac{1}{9}a^{2}+\frac{1}{9}a+\frac{1}{9}$, $\frac{1}{73728}a^{33}-\frac{1}{36864}a^{31}-\frac{1}{9216}a^{26}+\frac{1}{4608}a^{24}+\frac{1}{2304}a^{19}-\frac{1}{1152}a^{17}-\frac{5}{576}a^{12}+\frac{5}{288}a^{10}-\frac{5}{72}a^{5}+\frac{5}{36}a^{3}$, $\frac{1}{393216}a^{35}+\frac{1}{294912}a^{34}+\frac{7}{589824}a^{33}-\frac{5}{196608}a^{32}-\frac{5}{147456}a^{31}-\frac{1}{73728}a^{30}+\frac{1}{98304}a^{29}+\frac{17}{147456}a^{28}-\frac{1}{147456}a^{27}-\frac{23}{147456}a^{26}+\frac{1}{8192}a^{25}-\frac{5}{36864}a^{24}+\frac{17}{73728}a^{23}+\frac{1}{6144}a^{22}-\frac{1}{2304}a^{21}+\frac{25}{36864}a^{20}-\frac{23}{18432}a^{19}-\frac{1}{2048}a^{18}+\frac{7}{18432}a^{17}-\frac{13}{9216}a^{16}+\frac{7}{768}a^{15}-\frac{13}{4608}a^{14}+\frac{13}{2304}a^{13}+\frac{1}{1152}a^{12}-\frac{1}{64}a^{11}+\frac{5}{144}a^{10}-\frac{1}{144}a^{9}-\frac{1}{96}a^{8}+\frac{7}{144}a^{7}-\frac{31}{144}a^{6}-\frac{7}{72}a^{5}+\frac{1}{3}a^{4}+\frac{5}{18}a^{3}+\frac{17}{18}a^{2}-\frac{1}{3}a-\frac{10}{9}$, $\frac{7}{1179648}a^{35}+\frac{1}{196608}a^{34}+\frac{1}{196608}a^{33}-\frac{11}{589824}a^{32}-\frac{7}{294912}a^{31}+\frac{1}{36864}a^{30}+\frac{7}{98304}a^{29}+\frac{1}{12288}a^{28}-\frac{1}{16384}a^{27}-\frac{5}{49152}a^{26}+\frac{7}{36864}a^{25}-\frac{1}{4608}a^{24}+\frac{11}{73728}a^{23}+\frac{1}{4096}a^{22}-\frac{5}{18432}a^{21}+\frac{5}{12288}a^{20}-\frac{1}{512}a^{19}-\frac{29}{18432}a^{18}+\frac{19}{18432}a^{17}+\frac{7}{4608}a^{16}+\frac{3}{512}a^{15}-\frac{5}{768}a^{14}-\frac{1}{96}a^{13}+\frac{1}{768}a^{12}-\frac{17}{1152}a^{11}+\frac{5}{144}a^{10}+\frac{1}{288}a^{9}-\frac{1}{72}a^{7}-\frac{5}{24}a^{6}-\frac{1}{18}a^{4}+\frac{13}{36}a^{3}+\frac{4}{9}a^{2}-\frac{4}{3}a-\frac{14}{9}$, $\frac{1}{196608}a^{35}+\frac{1}{147456}a^{34}+\frac{1}{589824}a^{33}-\frac{1}{49152}a^{32}-\frac{5}{294912}a^{31}-\frac{11}{294912}a^{30}+\frac{1}{18432}a^{29}+\frac{1}{36864}a^{28}-\frac{5}{147456}a^{27}-\frac{1}{73728}a^{26}+\frac{1}{24576}a^{25}-\frac{11}{73728}a^{24}+\frac{11}{36864}a^{23}+\frac{1}{18432}a^{22}+\frac{7}{36864}a^{21}+\frac{1}{4608}a^{20}-\frac{13}{9216}a^{19}-\frac{1}{6144}a^{18}+\frac{7}{9216}a^{17}-\frac{5}{9216}a^{16}+\frac{67}{9216}a^{15}-\frac{5}{1152}a^{14}+\frac{1}{1152}a^{13}+\frac{1}{1152}a^{12}-\frac{1}{96}a^{11}+\frac{1}{144}a^{10}-\frac{1}{72}a^{9}-\frac{1}{72}a^{8}-\frac{1}{36}a^{7}-\frac{13}{72}a^{6}-\frac{1}{18}a^{5}+\frac{1}{6}a^{4}+\frac{7}{18}a^{3}+\frac{19}{18}a^{2}-\frac{4}{9}a-\frac{2}{9}$, $\frac{1}{1179648}a^{35}-\frac{1}{589824}a^{34}-\frac{1}{147456}a^{33}-\frac{5}{589824}a^{32}+\frac{13}{294912}a^{30}-\frac{1}{294912}a^{29}+\frac{1}{24576}a^{28}-\frac{1}{36864}a^{27}+\frac{5}{147456}a^{26}-\frac{1}{73728}a^{25}+\frac{1}{8192}a^{24}+\frac{7}{73728}a^{23}+\frac{7}{36864}a^{22}-\frac{25}{36864}a^{21}-\frac{11}{36864}a^{20}-\frac{5}{9216}a^{19}+\frac{11}{9216}a^{18}+\frac{3}{2048}a^{17}-\frac{53}{9216}a^{16}-\frac{7}{9216}a^{15}-\frac{13}{1536}a^{14}+\frac{5}{1152}a^{13}-\frac{1}{288}a^{12}-\frac{1}{576}a^{11}-\frac{1}{72}a^{9}-\frac{17}{288}a^{8}-\frac{1}{72}a^{7}-\frac{1}{36}a^{6}+\frac{5}{36}a^{5}+\frac{1}{9}a^{4}-\frac{1}{4}a^{3}-\frac{4}{9}a^{2}-\frac{8}{9}a+\frac{4}{9}$, $\frac{7}{1179648}a^{35}+\frac{1}{147456}a^{34}+\frac{1}{147456}a^{33}+\frac{5}{589824}a^{32}-\frac{5}{98304}a^{31}+\frac{1}{98304}a^{30}+\frac{23}{294912}a^{29}+\frac{11}{147456}a^{28}+\frac{1}{36864}a^{27}-\frac{29}{147456}a^{26}+\frac{1}{4608}a^{25}-\frac{1}{24576}a^{24}-\frac{1}{8192}a^{23}+\frac{5}{18432}a^{22}-\frac{17}{36864}a^{21}+\frac{29}{36864}a^{20}-\frac{17}{18432}a^{19}-\frac{11}{9216}a^{18}+\frac{19}{6144}a^{17}-\frac{1}{1536}a^{16}+\frac{5}{9216}a^{15}+\frac{1}{2304}a^{14}-\frac{37}{2304}a^{13}+\frac{29}{2304}a^{12}-\frac{5}{576}a^{11}+\frac{1}{96}a^{10}+\frac{5}{192}a^{9}-\frac{13}{144}a^{8}+\frac{1}{18}a^{7}-\frac{29}{144}a^{6}-\frac{2}{9}a^{5}-\frac{7}{36}a^{4}+\frac{5}{12}a^{3}+\frac{5}{6}a^{2}-\frac{11}{9}a-\frac{5}{3}$, $\frac{5}{1179648}a^{35}-\frac{1}{147456}a^{34}-\frac{1}{24576}a^{33}+\frac{17}{589824}a^{32}+\frac{43}{294912}a^{31}+\frac{13}{294912}a^{30}-\frac{61}{294912}a^{29}-\frac{29}{147456}a^{28}+\frac{19}{73728}a^{27}+\frac{3}{16384}a^{26}-\frac{13}{36864}a^{25}+\frac{7}{73728}a^{24}+\frac{7}{73728}a^{23}-\frac{7}{18432}a^{22}-\frac{37}{36864}a^{21}-\frac{23}{36864}a^{20}+\frac{19}{6144}a^{19}+\frac{1}{576}a^{18}+\frac{41}{18432}a^{17}-\frac{1}{288}a^{16}-\frac{157}{9216}a^{15}+\frac{25}{4608}a^{14}+\frac{37}{2304}a^{13}+\frac{1}{192}a^{12}-\frac{1}{72}a^{11}-\frac{7}{288}a^{10}+\frac{43}{576}a^{9}-\frac{5}{72}a^{8}-\frac{19}{72}a^{7}+\frac{13}{72}a^{6}+\frac{19}{24}a^{5}-\frac{1}{36}a^{4}-\frac{79}{36}a^{3}-\frac{23}{18}a^{2}+\frac{25}{9}a+\frac{8}{3}$, $\frac{1}{131072}a^{35}-\frac{7}{589824}a^{34}+\frac{1}{36864}a^{33}-\frac{13}{589824}a^{32}-\frac{5}{147456}a^{31}+\frac{11}{294912}a^{30}-\frac{1}{32768}a^{29}+\frac{5}{73728}a^{28}+\frac{7}{73728}a^{27}-\frac{71}{147456}a^{26}+\frac{37}{73728}a^{25}-\frac{25}{73728}a^{24}-\frac{13}{73728}a^{23}+\frac{13}{12288}a^{22}-\frac{59}{36864}a^{21}+\frac{73}{36864}a^{20}-\frac{5}{4608}a^{19}-\frac{13}{4608}a^{18}+\frac{127}{18432}a^{17}-\frac{79}{9216}a^{16}+\frac{41}{3072}a^{15}-\frac{17}{4608}a^{14}-\frac{47}{2304}a^{13}+\frac{83}{2304}a^{12}-\frac{29}{576}a^{11}+\frac{19}{288}a^{10}+\frac{1}{288}a^{9}-\frac{11}{96}a^{8}+\frac{29}{144}a^{7}-\frac{31}{144}a^{6}-\frac{11}{36}a^{5}+\frac{5}{9}a^{4}-\frac{11}{36}a^{3}+\frac{23}{18}a^{2}-a-\frac{8}{9}$, $\frac{1}{49152}a^{35}+\frac{13}{589824}a^{34}-\frac{1}{98304}a^{33}-\frac{1}{16384}a^{32}-\frac{25}{294912}a^{31}+\frac{1}{12288}a^{30}+\frac{31}{147456}a^{29}+\frac{7}{147456}a^{28}-\frac{5}{36864}a^{27}-\frac{1}{4096}a^{26}+\frac{1}{8192}a^{25}+\frac{1}{36864}a^{24}-\frac{1}{4096}a^{23}+\frac{43}{36864}a^{22}-\frac{1}{18432}a^{21}-\frac{23}{18432}a^{20}-\frac{7}{2048}a^{19}-\frac{7}{1536}a^{18}+\frac{7}{1152}a^{17}+\frac{17}{3072}a^{16}+\frac{1}{4608}a^{15}-\frac{17}{4608}a^{14}-\frac{67}{2304}a^{13}+\frac{1}{128}a^{12}-\frac{1}{192}a^{11}-\frac{13}{576}a^{10}+\frac{7}{192}a^{9}+\frac{1}{288}a^{8}-\frac{7}{36}a^{7}-\frac{37}{72}a^{6}-\frac{5}{12}a^{5}+\frac{11}{12}a^{4}+\frac{13}{9}a^{3}-\frac{1}{3}a^{2}-\frac{35}{9}a-\frac{32}{9}$, $\frac{1}{1179648}a^{35}-\frac{1}{294912}a^{34}+\frac{1}{294912}a^{33}+\frac{1}{65536}a^{32}+\frac{1}{98304}a^{31}-\frac{1}{294912}a^{30}-\frac{1}{294912}a^{29}-\frac{11}{147456}a^{28}-\frac{1}{18432}a^{27}-\frac{1}{147456}a^{26}+\frac{1}{12288}a^{25}+\frac{1}{8192}a^{24}-\frac{25}{73728}a^{23}-\frac{1}{4608}a^{22}+\frac{11}{12288}a^{21}+\frac{5}{36864}a^{20}+\frac{11}{18432}a^{19}+\frac{1}{3072}a^{18}-\frac{1}{6144}a^{17}+\frac{1}{1152}a^{16}-\frac{55}{9216}a^{15}-\frac{5}{4608}a^{14}+\frac{29}{2304}a^{13}+\frac{1}{144}a^{12}-\frac{7}{192}a^{10}+\frac{1}{72}a^{9}+\frac{11}{144}a^{8}-\frac{1}{12}a^{7}+\frac{1}{144}a^{6}+\frac{1}{18}a^{5}-\frac{1}{6}a^{4}-\frac{1}{6}a^{3}-\frac{7}{18}a^{2}+\frac{1}{9}a+\frac{11}{9}$, $\frac{1}{294912}a^{35}+\frac{1}{147456}a^{34}-\frac{1}{98304}a^{33}-\frac{1}{73728}a^{32}-\frac{1}{147456}a^{31}+\frac{7}{73728}a^{29}-\frac{1}{73728}a^{28}+\frac{1}{36864}a^{27}-\frac{5}{36864}a^{25}+\frac{1}{4608}a^{24}-\frac{1}{6144}a^{23}+\frac{7}{18432}a^{22}+\frac{1}{3072}a^{21}-\frac{5}{4608}a^{20}+\frac{1}{1536}a^{19}-\frac{11}{4608}a^{18}+\frac{5}{1152}a^{17}+\frac{5}{1536}a^{16}-\frac{7}{1152}a^{15}+\frac{7}{1152}a^{14}-\frac{13}{2304}a^{13}-\frac{5}{768}a^{12}+\frac{7}{288}a^{11}-\frac{11}{576}a^{10}+\frac{3}{64}a^{9}-\frac{1}{72}a^{8}-\frac{1}{12}a^{7}-\frac{5}{144}a^{6}-\frac{1}{12}a^{5}+\frac{4}{9}a^{4}-\frac{1}{9}a^{3}-\frac{1}{6}a^{2}-\frac{4}{9}a-\frac{16}{9}$, $\frac{5}{1179648}a^{35}-\frac{1}{589824}a^{34}-\frac{1}{65536}a^{33}+\frac{7}{196608}a^{32}-\frac{7}{294912}a^{31}+\frac{1}{49152}a^{30}-\frac{23}{294912}a^{29}+\frac{5}{73728}a^{28}+\frac{17}{147456}a^{27}-\frac{3}{16384}a^{26}-\frac{1}{12288}a^{25}+\frac{13}{36864}a^{24}-\frac{5}{8192}a^{23}+\frac{35}{36864}a^{22}-\frac{17}{18432}a^{21}-\frac{17}{36864}a^{20}+\frac{1}{512}a^{19}-\frac{37}{6144}a^{18}+\frac{175}{18432}a^{17}-\frac{5}{512}a^{16}+\frac{1}{2304}a^{15}+\frac{49}{4608}a^{14}-\frac{17}{2304}a^{13}-\frac{1}{384}a^{11}-\frac{5}{288}a^{10}+\frac{3}{32}a^{9}-\frac{7}{72}a^{8}-\frac{1}{72}a^{7}+\frac{13}{72}a^{6}-\frac{1}{4}a^{5}+\frac{1}{3}a^{4}-\frac{23}{36}a^{3}+\frac{5}{6}a^{2}+\frac{2}{9}a-\frac{2}{3}$ 1/393216*a^35 + 1/294912*a^34 - 11/589824*a^32 - 1/294912*a^31 + 7/294912*a^30 + 1/98304*a^29 + 1/147456*a^28 - 1/36864*a^27 + 1/49152*a^26 + 1/36864*a^25 - 7/73728*a^24 - 5/73728*a^23 + 7/36864*a^21 + 1/36864*a^20 - 5/6144*a^19 - 7/9216*a^18 + 1/18432*a^17 + 7/2304*a^16 + 1/1024*a^15 - 7/2304*a^14 + 7/2304*a^13 - 1/768*a^12 + 5/576*a^11 + 1/288*a^10 + 1/144*a^9 + 1/32*a^8 - 1/36*a^7 - 13/144*a^6 + 5/18*a^4 + 7/36*a^3 - 1/9*a^2 - 1/3*a - 8/9, 1/393216*a^35 - 1/196608*a^34 - 1/196608*a^33 - 1/196608*a^32 + 5/294912*a^31 + 1/36864*a^30 + 1/294912*a^29 - 1/36864*a^28 + 1/49152*a^27 - 1/49152*a^26 + 1/12288*a^25 - 1/4608*a^24 - 1/73728*a^23 + 5/36864*a^22 - 5/18432*a^21 + 5/12288*a^20 - 1/3072*a^19 + 1/6144*a^18 + 7/18432*a^17 - 5/4608*a^16 - 1/4608*a^15 - 5/2304*a^14 + 5/768*a^13 + 1/128*a^12 - 1/192*a^11 + 1/72*a^10 + 1/288*a^9 - 1/72*a^8 - 1/72*a^7 - 1/48*a^6 + 1/12*a^5 + 1/6*a^4 - 11/36*a^3 - 2/9*a^2 - 4/9*a + 2/9, 1/196608*a^35 - 1/2048*a^21 + 1/1536*a^14 - 7/48*a^7 + 1/3, 1/393216*a^35 + 1/98304*a^34 + 1/98304*a^33 - 1/196608*a^32 - 13/294912*a^31 - 1/294912*a^30 + 11/294912*a^29 + 5/147456*a^28 + 1/12288*a^27 - 7/49152*a^26 + 1/12288*a^25 - 7/73728*a^24 - 37/73728*a^23 + 7/9216*a^22 - 1/36864*a^21 + 11/12288*a^20 + 1/6144*a^19 - 13/3072*a^18 + 25/18432*a^17 + 1/1152*a^16 + 17/9216*a^15 + 13/2304*a^14 - 7/768*a^13 + 1/64*a^12 - 1/192*a^11 - 5/288*a^10 + 5/144*a^9 - 7/144*a^8 + 1/9*a^7 - 7/48*a^6 - 5/12*a^5 + 1/6*a^4 + 7/36*a^3 + 7/9*a^2 - 2/9*a - 16/9, 5/589824*a^35 - 1/589824*a^34 - 1/196608*a^33 - 1/147456*a^32 - 1/147456*a^31 + 7/294912*a^30 - 5/147456*a^29 - 1/49152*a^28 + 5/147456*a^27 - 1/24576*a^26 + 1/18432*a^25 + 1/73728*a^24 - 1/9216*a^23 + 7/36864*a^22 - 25/36864*a^21 + 5/18432*a^20 - 1/6144*a^19 - 13/18432*a^18 + 19/9216*a^17 - 1/4608*a^16 + 11/9216*a^15 + 1/1536*a^14 + 7/2304*a^13 + 1/128*a^12 - 1/288*a^11 - 1/288*a^10 + 5/288*a^9 + 1/288*a^8 - 23/144*a^7 - 7/144*a^6 + 2/9*a^4 + 1/18*a^3 - 1/9*a^2 + 1/9*a + 1/9, 1/73728*a^33 - 1/36864*a^31 - 1/9216*a^26 + 1/4608*a^24 + 1/2304*a^19 - 1/1152*a^17 - 5/576*a^12 + 5/288*a^10 - 5/72*a^5 + 5/36*a^3, 1/393216*a^35 + 1/294912*a^34 + 7/589824*a^33 - 5/196608*a^32 - 5/147456*a^31 - 1/73728*a^30 + 1/98304*a^29 + 17/147456*a^28 - 1/147456*a^27 - 23/147456*a^26 + 1/8192*a^25 - 5/36864*a^24 + 17/73728*a^23 + 1/6144*a^22 - 1/2304*a^21 + 25/36864*a^20 - 23/18432*a^19 - 1/2048*a^18 + 7/18432*a^17 - 13/9216*a^16 + 7/768*a^15 - 13/4608*a^14 + 13/2304*a^13 + 1/1152*a^12 - 1/64*a^11 + 5/144*a^10 - 1/144*a^9 - 1/96*a^8 + 7/144*a^7 - 31/144*a^6 - 7/72*a^5 + 1/3*a^4 + 5/18*a^3 + 17/18*a^2 - 1/3*a - 10/9, 7/1179648*a^35 + 1/196608*a^34 + 1/196608*a^33 - 11/589824*a^32 - 7/294912*a^31 + 1/36864*a^30 + 7/98304*a^29 + 1/12288*a^28 - 1/16384*a^27 - 5/49152*a^26 + 7/36864*a^25 - 1/4608*a^24 + 11/73728*a^23 + 1/4096*a^22 - 5/18432*a^21 + 5/12288*a^20 - 1/512*a^19 - 29/18432*a^18 + 19/18432*a^17 + 7/4608*a^16 + 3/512*a^15 - 5/768*a^14 - 1/96*a^13 + 1/768*a^12 - 17/1152*a^11 + 5/144*a^10 + 1/288*a^9 - 1/72*a^7 - 5/24*a^6 - 1/18*a^4 + 13/36*a^3 + 4/9*a^2 - 4/3*a - 14/9, 1/196608*a^35 + 1/147456*a^34 + 1/589824*a^33 - 1/49152*a^32 - 5/294912*a^31 - 11/294912*a^30 + 1/18432*a^29 + 1/36864*a^28 - 5/147456*a^27 - 1/73728*a^26 + 1/24576*a^25 - 11/73728*a^24 + 11/36864*a^23 + 1/18432*a^22 + 7/36864*a^21 + 1/4608*a^20 - 13/9216*a^19 - 1/6144*a^18 + 7/9216*a^17 - 5/9216*a^16 + 67/9216*a^15 - 5/1152*a^14 + 1/1152*a^13 + 1/1152*a^12 - 1/96*a^11 + 1/144*a^10 - 1/72*a^9 - 1/72*a^8 - 1/36*a^7 - 13/72*a^6 - 1/18*a^5 + 1/6*a^4 + 7/18*a^3 + 19/18*a^2 - 4/9*a - 2/9, 1/1179648*a^35 - 1/589824*a^34 - 1/147456*a^33 - 5/589824*a^32 + 13/294912*a^30 - 1/294912*a^29 + 1/24576*a^28 - 1/36864*a^27 + 5/147456*a^26 - 1/73728*a^25 + 1/8192*a^24 + 7/73728*a^23 + 7/36864*a^22 - 25/36864*a^21 - 11/36864*a^20 - 5/9216*a^19 + 11/9216*a^18 + 3/2048*a^17 - 53/9216*a^16 - 7/9216*a^15 - 13/1536*a^14 + 5/1152*a^13 - 1/288*a^12 - 1/576*a^11 - 1/72*a^9 - 17/288*a^8 - 1/72*a^7 - 1/36*a^6 + 5/36*a^5 + 1/9*a^4 - 1/4*a^3 - 4/9*a^2 - 8/9*a + 4/9, 7/1179648*a^35 + 1/147456*a^34 + 1/147456*a^33 + 5/589824*a^32 - 5/98304*a^31 + 1/98304*a^30 + 23/294912*a^29 + 11/147456*a^28 + 1/36864*a^27 - 29/147456*a^26 + 1/4608*a^25 - 1/24576*a^24 - 1/8192*a^23 + 5/18432*a^22 - 17/36864*a^21 + 29/36864*a^20 - 17/18432*a^19 - 11/9216*a^18 + 19/6144*a^17 - 1/1536*a^16 + 5/9216*a^15 + 1/2304*a^14 - 37/2304*a^13 + 29/2304*a^12 - 5/576*a^11 + 1/96*a^10 + 5/192*a^9 - 13/144*a^8 + 1/18*a^7 - 29/144*a^6 - 2/9*a^5 - 7/36*a^4 + 5/12*a^3 + 5/6*a^2 - 11/9*a - 5/3, 5/1179648*a^35 - 1/147456*a^34 - 1/24576*a^33 + 17/589824*a^32 + 43/294912*a^31 + 13/294912*a^30 - 61/294912*a^29 - 29/147456*a^28 + 19/73728*a^27 + 3/16384*a^26 - 13/36864*a^25 + 7/73728*a^24 + 7/73728*a^23 - 7/18432*a^22 - 37/36864*a^21 - 23/36864*a^20 + 19/6144*a^19 + 1/576*a^18 + 41/18432*a^17 - 1/288*a^16 - 157/9216*a^15 + 25/4608*a^14 + 37/2304*a^13 + 1/192*a^12 - 1/72*a^11 - 7/288*a^10 + 43/576*a^9 - 5/72*a^8 - 19/72*a^7 + 13/72*a^6 + 19/24*a^5 - 1/36*a^4 - 79/36*a^3 - 23/18*a^2 + 25/9*a + 8/3, 1/131072*a^35 - 7/589824*a^34 + 1/36864*a^33 - 13/589824*a^32 - 5/147456*a^31 + 11/294912*a^30 - 1/32768*a^29 + 5/73728*a^28 + 7/73728*a^27 - 71/147456*a^26 + 37/73728*a^25 - 25/73728*a^24 - 13/73728*a^23 + 13/12288*a^22 - 59/36864*a^21 + 73/36864*a^20 - 5/4608*a^19 - 13/4608*a^18 + 127/18432*a^17 - 79/9216*a^16 + 41/3072*a^15 - 17/4608*a^14 - 47/2304*a^13 + 83/2304*a^12 - 29/576*a^11 + 19/288*a^10 + 1/288*a^9 - 11/96*a^8 + 29/144*a^7 - 31/144*a^6 - 11/36*a^5 + 5/9*a^4 - 11/36*a^3 + 23/18*a^2 - a - 8/9, 1/49152*a^35 + 13/589824*a^34 - 1/98304*a^33 - 1/16384*a^32 - 25/294912*a^31 + 1/12288*a^30 + 31/147456*a^29 + 7/147456*a^28 - 5/36864*a^27 - 1/4096*a^26 + 1/8192*a^25 + 1/36864*a^24 - 1/4096*a^23 + 43/36864*a^22 - 1/18432*a^21 - 23/18432*a^20 - 7/2048*a^19 - 7/1536*a^18 + 7/1152*a^17 + 17/3072*a^16 + 1/4608*a^15 - 17/4608*a^14 - 67/2304*a^13 + 1/128*a^12 - 1/192*a^11 - 13/576*a^10 + 7/192*a^9 + 1/288*a^8 - 7/36*a^7 - 37/72*a^6 - 5/12*a^5 + 11/12*a^4 + 13/9*a^3 - 1/3*a^2 - 35/9*a - 32/9, 1/1179648*a^35 - 1/294912*a^34 + 1/294912*a^33 + 1/65536*a^32 + 1/98304*a^31 - 1/294912*a^30 - 1/294912*a^29 - 11/147456*a^28 - 1/18432*a^27 - 1/147456*a^26 + 1/12288*a^25 + 1/8192*a^24 - 25/73728*a^23 - 1/4608*a^22 + 11/12288*a^21 + 5/36864*a^20 + 11/18432*a^19 + 1/3072*a^18 - 1/6144*a^17 + 1/1152*a^16 - 55/9216*a^15 - 5/4608*a^14 + 29/2304*a^13 + 1/144*a^12 - 7/192*a^10 + 1/72*a^9 + 11/144*a^8 - 1/12*a^7 + 1/144*a^6 + 1/18*a^5 - 1/6*a^4 - 1/6*a^3 - 7/18*a^2 + 1/9*a + 11/9, 1/294912*a^35 + 1/147456*a^34 - 1/98304*a^33 - 1/73728*a^32 - 1/147456*a^31 + 7/73728*a^29 - 1/73728*a^28 + 1/36864*a^27 - 5/36864*a^25 + 1/4608*a^24 - 1/6144*a^23 + 7/18432*a^22 + 1/3072*a^21 - 5/4608*a^20 + 1/1536*a^19 - 11/4608*a^18 + 5/1152*a^17 + 5/1536*a^16 - 7/1152*a^15 + 7/1152*a^14 - 13/2304*a^13 - 5/768*a^12 + 7/288*a^11 - 11/576*a^10 + 3/64*a^9 - 1/72*a^8 - 1/12*a^7 - 5/144*a^6 - 1/12*a^5 + 4/9*a^4 - 1/9*a^3 - 1/6*a^2 - 4/9*a - 16/9, 5/1179648*a^35 - 1/589824*a^34 - 1/65536*a^33 + 7/196608*a^32 - 7/294912*a^31 + 1/49152*a^30 - 23/294912*a^29 + 5/73728*a^28 + 17/147456*a^27 - 3/16384*a^26 - 1/12288*a^25 + 13/36864*a^24 - 5/8192*a^23 + 35/36864*a^22 - 17/18432*a^21 - 17/36864*a^20 + 1/512*a^19 - 37/6144*a^18 + 175/18432*a^17 - 5/512*a^16 + 1/2304*a^15 + 49/4608*a^14 - 17/2304*a^13 - 1/384*a^11 - 5/288*a^10 + 3/32*a^9 - 7/72*a^8 - 1/72*a^7 + 13/72*a^6 - 1/4*a^5 + 1/3*a^4 - 23/36*a^3 + 5/6*a^2 + 2/9*a - 2/3 (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:		\( 161164499152547.9 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 161164499152547.9 \cdot 189}{14\cdot\sqrt{2209286039084312039484359423586799049408706824590560826753024}}\cr\approx \mathstrut & 0.340968930793453 \end{aligned}\] (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	2.9.6.1	$x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$	$3$	$3$	$6$	$S_3\times C_3$	$[\ ]_{3}^{6}$
	2.9.6.1	$x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$	$3$	$3$	$6$	$S_3\times C_3$	$[\ ]_{3}^{6}$
	2.9.6.1	$x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$	$3$	$3$	$6$	$S_3\times C_3$	$[\ ]_{3}^{6}$
	2.9.6.1	$x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$	$3$	$3$	$6$	$S_3\times C_3$	$[\ ]_{3}^{6}$
\(7\) 7	7.18.15.5	$x^{18} + 36 x^{17} + 540 x^{16} + 4344 x^{15} + 20160 x^{14} + 55296 x^{13} + 98757 x^{12} + 161784 x^{11} + 246024 x^{10} + 264920 x^{9} + 530640 x^{8} + 156384 x^{7} - 1885725 x^{6} - 6133212 x^{5} - 3645540 x^{4} + 5968464 x^{3} + 5011344 x^{2} + 1820448 x + 2358791$	$6$	$3$	$15$	$C_6 \times C_3$	$[\ ]_{6}^{3}$
	7.18.15.5	$x^{18} + 36 x^{17} + 540 x^{16} + 4344 x^{15} + 20160 x^{14} + 55296 x^{13} + 98757 x^{12} + 161784 x^{11} + 246024 x^{10} + 264920 x^{9} + 530640 x^{8} + 156384 x^{7} - 1885725 x^{6} - 6133212 x^{5} - 3645540 x^{4} + 5968464 x^{3} + 5011344 x^{2} + 1820448 x + 2358791$	$6$	$3$	$15$	$C_6 \times C_3$	$[\ ]_{6}^{3}$
\(31\) 31	31.6.0.1	$x^{6} + 19 x^{3} + 16 x^{2} + 8 x + 3$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	31.6.0.1	$x^{6} + 19 x^{3} + 16 x^{2} + 8 x + 3$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	31.6.0.1	$x^{6} + 19 x^{3} + 16 x^{2} + 8 x + 3$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	31.6.0.1	$x^{6} + 19 x^{3} + 16 x^{2} + 8 x + 3$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	31.12.6.1	$x^{12} - 14880 x^{11} + 56210037 x^{10} + 37979282185 x^{9} + 6626678058881 x^{8} + 242960454323950 x^{7} + 6470243948822665 x^{6} + 7531774084042450 x^{5} + 6368237614584641 x^{4} + 1131440795573335 x^{3} + 51911149580277 x^{2} - 426001766880 x + 887503681$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$
\(37\) 37	37.6.0.1	$x^{6} + 35 x^{3} + 4 x^{2} + 30 x + 2$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	37.6.0.1	$x^{6} + 35 x^{3} + 4 x^{2} + 30 x + 2$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	37.12.6.1	$x^{12} + 5550 x^{11} + 12834597 x^{10} + 15830089320 x^{9} + 10983311908793 x^{8} + 4064880121459160 x^{7} + 627211360763929855 x^{6} + 150784964321118570 x^{5} + 157305214180175641 x^{4} + 21952692460017249320 x^{3} + 5259154456776668411 x^{2} + 19105722471444557170 x + 2009397505586138476$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$
	37.12.6.1	$x^{12} + 5550 x^{11} + 12834597 x^{10} + 15830089320 x^{9} + 10983311908793 x^{8} + 4064880121459160 x^{7} + 627211360763929855 x^{6} + 150784964321118570 x^{5} + 157305214180175641 x^{4} + 21952692460017249320 x^{3} + 5259154456776668411 x^{2} + 19105722471444557170 x + 2009397505586138476$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$