Normalized defining polynomial

\( x^{32} - 4x^{28} + 22x^{24} - 36x^{20} + 209x^{16} - 576x^{12} + 5632x^{8} - 16384x^{4} + 65536 \) x^32 - 4*x^28 + 22*x^24 - 36*x^20 + 209*x^16 - 576*x^12 + 5632*x^8 - 16384*x^4 + 65536

Invariants

Degree:		$32$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[0, 16]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(90220386589172305242198553166127486468521067544576\) 90220386589172305242198553166127486468521067544576 \(\medspace = 2^{76}\cdot 89^{4}\cdot 257^{8}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(36.40\)	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:		$2^{19/8}89^{1/2}257^{1/2}\approx 784.5268373963652$
Ramified primes:		\(2\), \(89\), \(257\) 2, 89, 257	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q\)
$\card{ \Aut(K/\Q) }$:		$8$	sage: K.automorphisms()   magma: Automorphisms(K);   oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:		unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{8}a^{7}+\frac{1}{8}a^{6}+\frac{1}{8}a^{5}+\frac{1}{8}a^{4}+\frac{1}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{11}-\frac{1}{4}a^{6}-\frac{1}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{12}-\frac{1}{4}a^{6}-\frac{1}{8}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{16}a^{13}-\frac{1}{16}a^{11}-\frac{1}{8}a^{8}-\frac{1}{8}a^{6}+\frac{3}{16}a^{5}+\frac{1}{8}a^{4}-\frac{7}{16}a^{3}-\frac{3}{8}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{16}a^{14}-\frac{1}{16}a^{12}-\frac{1}{8}a^{9}-\frac{1}{8}a^{7}+\frac{3}{16}a^{6}+\frac{1}{8}a^{5}+\frac{1}{16}a^{4}+\frac{1}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{16}a^{15}-\frac{1}{16}a^{11}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}+\frac{1}{16}a^{7}-\frac{1}{8}a^{6}-\frac{1}{8}a^{5}+\frac{1}{8}a^{4}+\frac{7}{16}a^{3}-\frac{3}{8}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{32}a^{16}+\frac{1}{32}a^{12}+\frac{3}{32}a^{8}-\frac{1}{4}a^{6}-\frac{5}{32}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{32}a^{17}-\frac{1}{32}a^{13}-\frac{1}{16}a^{11}+\frac{3}{32}a^{9}-\frac{1}{8}a^{8}-\frac{1}{8}a^{6}+\frac{5}{32}a^{5}+\frac{1}{8}a^{4}-\frac{7}{16}a^{3}-\frac{3}{8}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{32}a^{18}-\frac{1}{32}a^{14}-\frac{1}{16}a^{12}-\frac{1}{32}a^{10}-\frac{1}{8}a^{8}-\frac{7}{32}a^{6}+\frac{3}{16}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{64}a^{19}-\frac{1}{64}a^{17}+\frac{1}{64}a^{15}-\frac{1}{64}a^{13}+\frac{3}{64}a^{11}+\frac{5}{64}a^{9}+\frac{3}{64}a^{7}+\frac{13}{64}a^{5}-\frac{1}{8}a^{3}-\frac{1}{4}a$, $\frac{1}{64}a^{20}-\frac{1}{64}a^{18}-\frac{1}{64}a^{16}-\frac{1}{64}a^{14}+\frac{1}{64}a^{12}-\frac{3}{64}a^{10}-\frac{1}{8}a^{9}+\frac{5}{64}a^{8}-\frac{1}{8}a^{7}-\frac{11}{64}a^{6}+\frac{1}{8}a^{5}-\frac{3}{32}a^{4}-\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{128}a^{21}-\frac{1}{64}a^{18}-\frac{1}{64}a^{16}-\frac{1}{64}a^{14}+\frac{1}{64}a^{13}-\frac{1}{64}a^{12}-\frac{1}{16}a^{11}-\frac{3}{64}a^{10}+\frac{1}{16}a^{9}+\frac{5}{64}a^{8}-\frac{1}{8}a^{7}+\frac{13}{64}a^{6}-\frac{27}{128}a^{5}+\frac{13}{64}a^{4}-\frac{5}{16}a^{3}+\frac{3}{8}a^{2}-\frac{3}{8}a+\frac{1}{4}$, $\frac{1}{256}a^{22}-\frac{1}{64}a^{16}-\frac{3}{128}a^{14}+\frac{3}{64}a^{12}-\frac{1}{32}a^{10}-\frac{3}{64}a^{8}+\frac{29}{256}a^{6}-\frac{15}{64}a^{4}-\frac{1}{2}a^{3}-\frac{1}{16}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{1024}a^{23}-\frac{1}{512}a^{22}-\frac{1}{256}a^{21}-\frac{1}{128}a^{20}-\frac{1}{256}a^{19}-\frac{1}{128}a^{18}-\frac{1}{64}a^{16}-\frac{13}{512}a^{15}-\frac{7}{256}a^{14}+\frac{3}{128}a^{13}-\frac{1}{32}a^{12}-\frac{5}{256}a^{11}-\frac{1}{128}a^{10}+\frac{1}{32}a^{9}+\frac{1}{64}a^{8}+\frac{65}{1024}a^{7}+\frac{7}{512}a^{6}+\frac{35}{256}a^{5}+\frac{21}{128}a^{4}-\frac{25}{64}a^{3}+\frac{9}{32}a^{2}-\frac{3}{16}a+\frac{3}{8}$, $\frac{1}{10240}a^{24}-\frac{1}{2560}a^{20}-\frac{1}{64}a^{18}+\frac{59}{5120}a^{16}-\frac{1}{64}a^{14}+\frac{3}{512}a^{12}-\frac{3}{64}a^{10}-\frac{1}{8}a^{9}-\frac{783}{10240}a^{8}-\frac{1}{8}a^{7}+\frac{5}{64}a^{6}+\frac{1}{8}a^{5}+\frac{47}{320}a^{4}+\frac{1}{8}a^{3}+\frac{9}{40}$, $\frac{1}{20480}a^{25}-\frac{1}{5120}a^{21}-\frac{1}{64}a^{18}+\frac{139}{10240}a^{17}-\frac{1}{64}a^{16}-\frac{1}{64}a^{14}+\frac{11}{1024}a^{13}-\frac{1}{64}a^{12}-\frac{1}{16}a^{11}-\frac{3}{64}a^{10}+\frac{2257}{20480}a^{9}+\frac{5}{64}a^{8}-\frac{1}{8}a^{7}-\frac{3}{64}a^{6}+\frac{11}{320}a^{5}+\frac{13}{64}a^{4}-\frac{5}{16}a^{3}-\frac{3}{8}a^{2}-\frac{1}{80}a+\frac{1}{4}$, $\frac{1}{81920}a^{26}-\frac{1}{20480}a^{24}+\frac{39}{20480}a^{22}-\frac{39}{5120}a^{20}-\frac{181}{40960}a^{18}-\frac{139}{10240}a^{16}-\frac{5}{4096}a^{14}-\frac{27}{1024}a^{12}+\frac{1617}{81920}a^{10}-\frac{1}{8}a^{9}-\frac{977}{20480}a^{8}-\frac{1}{8}a^{7}+\frac{307}{2560}a^{6}+\frac{1}{8}a^{5}+\frac{33}{640}a^{4}+\frac{1}{8}a^{3}-\frac{71}{320}a^{2}+\frac{31}{80}$, $\frac{1}{163840}a^{27}-\frac{1}{40960}a^{25}-\frac{1}{40960}a^{23}-\frac{1}{512}a^{22}+\frac{1}{10240}a^{21}-\frac{1}{128}a^{20}+\frac{139}{81920}a^{19}-\frac{1}{128}a^{18}-\frac{139}{20480}a^{17}-\frac{1}{64}a^{16}+\frac{203}{8192}a^{15}-\frac{7}{256}a^{14}+\frac{53}{2048}a^{13}-\frac{1}{32}a^{12}-\frac{5423}{163840}a^{11}-\frac{1}{128}a^{10}-\frac{4817}{40960}a^{9}+\frac{1}{64}a^{8}-\frac{169}{2560}a^{7}+\frac{7}{512}a^{6}-\frac{71}{640}a^{5}+\frac{21}{128}a^{4}-\frac{61}{640}a^{3}+\frac{9}{32}a^{2}+\frac{61}{160}a+\frac{3}{8}$, $\frac{1}{3276800}a^{28}+\frac{7}{163840}a^{24}-\frac{5269}{1638400}a^{20}-\frac{1}{64}a^{18}-\frac{3057}{819200}a^{16}-\frac{1}{64}a^{14}+\frac{92817}{3276800}a^{12}-\frac{3}{64}a^{10}-\frac{1}{8}a^{9}+\frac{6517}{204800}a^{8}-\frac{1}{8}a^{7}-\frac{11}{64}a^{6}+\frac{1}{8}a^{5}-\frac{473}{2560}a^{4}+\frac{1}{8}a^{3}+\frac{1}{4}a^{2}-\frac{169}{800}$, $\frac{1}{6553600}a^{29}+\frac{7}{327680}a^{25}-\frac{5269}{3276800}a^{21}-\frac{1}{64}a^{18}+\frac{9743}{1638400}a^{17}-\frac{1}{64}a^{16}-\frac{1}{64}a^{14}+\frac{144017}{6553600}a^{13}-\frac{1}{64}a^{12}-\frac{3}{64}a^{10}-\frac{35083}{409600}a^{9}-\frac{3}{64}a^{8}-\frac{1}{8}a^{7}+\frac{5}{64}a^{6}-\frac{673}{5120}a^{5}-\frac{11}{64}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}+\frac{31}{1600}a-\frac{1}{4}$, $\frac{1}{26214400}a^{30}-\frac{1}{6553600}a^{28}+\frac{7}{1310720}a^{26}-\frac{7}{327680}a^{24}+\frac{20331}{13107200}a^{22}-\frac{20331}{3276800}a^{20}+\frac{9743}{6553600}a^{18}-\frac{9743}{1638400}a^{16}+\frac{656017}{26214400}a^{14}-\frac{246417}{6553600}a^{12}-\frac{9483}{1638400}a^{10}-\frac{1}{8}a^{9}+\frac{9483}{409600}a^{8}-\frac{1}{8}a^{7}+\frac{4327}{20480}a^{6}+\frac{1}{8}a^{5}+\frac{473}{5120}a^{4}+\frac{1}{8}a^{3}+\frac{631}{6400}a^{2}-\frac{631}{1600}$, $\frac{1}{52428800}a^{31}-\frac{1}{13107200}a^{29}+\frac{7}{2621440}a^{27}-\frac{7}{655360}a^{25}-\frac{5269}{26214400}a^{23}-\frac{1}{512}a^{22}+\frac{5269}{6553600}a^{21}-\frac{1}{128}a^{20}+\frac{60943}{13107200}a^{19}-\frac{1}{128}a^{18}+\frac{41457}{3276800}a^{17}-\frac{1}{64}a^{16}+\frac{348817}{52428800}a^{15}-\frac{7}{256}a^{14}+\frac{60783}{13107200}a^{13}-\frac{1}{32}a^{12}+\frac{54517}{3276800}a^{11}-\frac{1}{128}a^{10}+\frac{22283}{819200}a^{9}+\frac{1}{64}a^{8}+\frac{3007}{40960}a^{7}+\frac{7}{512}a^{6}+\frac{513}{10240}a^{5}+\frac{21}{128}a^{4}+\frac{5631}{12800}a^{3}+\frac{9}{32}a^{2}+\frac{769}{3200}a+\frac{3}{8}$ 1, a, a^2, a^3, 1/2*a^4 - 1/2*a^3 - 1/2*a^2 - 1/2*a, 1/2*a^5 - 1/2*a, 1/2*a^6 - 1/2*a^2, 1/4*a^7 - 1/4*a^6 - 1/4*a^3 + 1/4*a^2, 1/4*a^8 - 1/4*a^6 - 1/4*a^4 + 1/4*a^2, 1/4*a^9 - 1/4*a^6 - 1/4*a^5 + 1/4*a^2, 1/8*a^10 - 1/8*a^9 - 1/8*a^8 - 1/8*a^7 + 1/8*a^6 + 1/8*a^5 + 1/8*a^4 + 1/8*a^3 - 1/4*a^2, 1/8*a^11 - 1/4*a^6 - 1/8*a^3 + 1/4*a^2, 1/8*a^12 - 1/4*a^6 - 1/8*a^4 + 1/4*a^2, 1/16*a^13 - 1/16*a^11 - 1/8*a^8 - 1/8*a^6 + 3/16*a^5 + 1/8*a^4 - 7/16*a^3 - 3/8*a^2 + 1/4*a - 1/2, 1/16*a^14 - 1/16*a^12 - 1/8*a^9 - 1/8*a^7 + 3/16*a^6 + 1/8*a^5 + 1/16*a^4 + 1/8*a^3 - 1/4*a^2, 1/16*a^15 - 1/16*a^11 - 1/8*a^9 - 1/8*a^8 + 1/16*a^7 - 1/8*a^6 - 1/8*a^5 + 1/8*a^4 + 7/16*a^3 - 3/8*a^2 - 1/4*a - 1/2, 1/32*a^16 + 1/32*a^12 + 3/32*a^8 - 1/4*a^6 - 5/32*a^4 - 1/2*a^3 - 1/4*a^2 - 1/2*a - 1/2, 1/32*a^17 - 1/32*a^13 - 1/16*a^11 + 3/32*a^9 - 1/8*a^8 - 1/8*a^6 + 5/32*a^5 + 1/8*a^4 - 7/16*a^3 - 3/8*a^2 + 1/4*a - 1/2, 1/32*a^18 - 1/32*a^14 - 1/16*a^12 - 1/32*a^10 - 1/8*a^8 - 7/32*a^6 + 3/16*a^4 + 1/4*a^2, 1/64*a^19 - 1/64*a^17 + 1/64*a^15 - 1/64*a^13 + 3/64*a^11 + 5/64*a^9 + 3/64*a^7 + 13/64*a^5 - 1/8*a^3 - 1/4*a, 1/64*a^20 - 1/64*a^18 - 1/64*a^16 - 1/64*a^14 + 1/64*a^12 - 3/64*a^10 - 1/8*a^9 + 5/64*a^8 - 1/8*a^7 - 11/64*a^6 + 1/8*a^5 - 3/32*a^4 - 3/8*a^3 - 1/4*a^2 - 1/2*a - 1/2, 1/128*a^21 - 1/64*a^18 - 1/64*a^16 - 1/64*a^14 + 1/64*a^13 - 1/64*a^12 - 1/16*a^11 - 3/64*a^10 + 1/16*a^9 + 5/64*a^8 - 1/8*a^7 + 13/64*a^6 - 27/128*a^5 + 13/64*a^4 - 5/16*a^3 + 3/8*a^2 - 3/8*a + 1/4, 1/256*a^22 - 1/64*a^16 - 3/128*a^14 + 3/64*a^12 - 1/32*a^10 - 3/64*a^8 + 29/256*a^6 - 15/64*a^4 - 1/2*a^3 - 1/16*a^2 - 1/2*a - 1/4, 1/1024*a^23 - 1/512*a^22 - 1/256*a^21 - 1/128*a^20 - 1/256*a^19 - 1/128*a^18 - 1/64*a^16 - 13/512*a^15 - 7/256*a^14 + 3/128*a^13 - 1/32*a^12 - 5/256*a^11 - 1/128*a^10 + 1/32*a^9 + 1/64*a^8 + 65/1024*a^7 + 7/512*a^6 + 35/256*a^5 + 21/128*a^4 - 25/64*a^3 + 9/32*a^2 - 3/16*a + 3/8, 1/10240*a^24 - 1/2560*a^20 - 1/64*a^18 + 59/5120*a^16 - 1/64*a^14 + 3/512*a^12 - 3/64*a^10 - 1/8*a^9 - 783/10240*a^8 - 1/8*a^7 + 5/64*a^6 + 1/8*a^5 + 47/320*a^4 + 1/8*a^3 + 9/40, 1/20480*a^25 - 1/5120*a^21 - 1/64*a^18 + 139/10240*a^17 - 1/64*a^16 - 1/64*a^14 + 11/1024*a^13 - 1/64*a^12 - 1/16*a^11 - 3/64*a^10 + 2257/20480*a^9 + 5/64*a^8 - 1/8*a^7 - 3/64*a^6 + 11/320*a^5 + 13/64*a^4 - 5/16*a^3 - 3/8*a^2 - 1/80*a + 1/4, 1/81920*a^26 - 1/20480*a^24 + 39/20480*a^22 - 39/5120*a^20 - 181/40960*a^18 - 139/10240*a^16 - 5/4096*a^14 - 27/1024*a^12 + 1617/81920*a^10 - 1/8*a^9 - 977/20480*a^8 - 1/8*a^7 + 307/2560*a^6 + 1/8*a^5 + 33/640*a^4 + 1/8*a^3 - 71/320*a^2 + 31/80, 1/163840*a^27 - 1/40960*a^25 - 1/40960*a^23 - 1/512*a^22 + 1/10240*a^21 - 1/128*a^20 + 139/81920*a^19 - 1/128*a^18 - 139/20480*a^17 - 1/64*a^16 + 203/8192*a^15 - 7/256*a^14 + 53/2048*a^13 - 1/32*a^12 - 5423/163840*a^11 - 1/128*a^10 - 4817/40960*a^9 + 1/64*a^8 - 169/2560*a^7 + 7/512*a^6 - 71/640*a^5 + 21/128*a^4 - 61/640*a^3 + 9/32*a^2 + 61/160*a + 3/8, 1/3276800*a^28 + 7/163840*a^24 - 5269/1638400*a^20 - 1/64*a^18 - 3057/819200*a^16 - 1/64*a^14 + 92817/3276800*a^12 - 3/64*a^10 - 1/8*a^9 + 6517/204800*a^8 - 1/8*a^7 - 11/64*a^6 + 1/8*a^5 - 473/2560*a^4 + 1/8*a^3 + 1/4*a^2 - 169/800, 1/6553600*a^29 + 7/327680*a^25 - 5269/3276800*a^21 - 1/64*a^18 + 9743/1638400*a^17 - 1/64*a^16 - 1/64*a^14 + 144017/6553600*a^13 - 1/64*a^12 - 3/64*a^10 - 35083/409600*a^9 - 3/64*a^8 - 1/8*a^7 + 5/64*a^6 - 673/5120*a^5 - 11/64*a^4 - 3/8*a^3 - 1/2*a^2 + 31/1600*a - 1/4, 1/26214400*a^30 - 1/6553600*a^28 + 7/1310720*a^26 - 7/327680*a^24 + 20331/13107200*a^22 - 20331/3276800*a^20 + 9743/6553600*a^18 - 9743/1638400*a^16 + 656017/26214400*a^14 - 246417/6553600*a^12 - 9483/1638400*a^10 - 1/8*a^9 + 9483/409600*a^8 - 1/8*a^7 + 4327/20480*a^6 + 1/8*a^5 + 473/5120*a^4 + 1/8*a^3 + 631/6400*a^2 - 631/1600, 1/52428800*a^31 - 1/13107200*a^29 + 7/2621440*a^27 - 7/655360*a^25 - 5269/26214400*a^23 - 1/512*a^22 + 5269/6553600*a^21 - 1/128*a^20 + 60943/13107200*a^19 - 1/128*a^18 + 41457/3276800*a^17 - 1/64*a^16 + 348817/52428800*a^15 - 7/256*a^14 + 60783/13107200*a^13 - 1/32*a^12 + 54517/3276800*a^11 - 1/128*a^10 + 22283/819200*a^9 + 1/64*a^8 + 3007/40960*a^7 + 7/512*a^6 + 513/10240*a^5 + 21/128*a^4 + 5631/12800*a^3 + 9/32*a^2 + 769/3200*a + 3/8

Class group and class number

$C_{24}$, which has order $24$ (assuming GRH)

Unit group

Rank:		$15$	sage: UK.rank()   gp: K.fu   magma: UnitRank(K);   oscar: rank(UK)
Torsion generator:		\( \frac{441}{26214400} a^{31} + \frac{31}{1310720} a^{27} + \frac{211}{13107200} a^{23} - \frac{2377}{6553600} a^{19} + \frac{93897}{26214400} a^{15} - \frac{16543}{1638400} a^{11} + \frac{871}{20480} a^{7} - \frac{909}{6400} a^{3} \) (441)/(26214400)*a^(31) + (31)/(1310720)*a^(27) + (211)/(13107200)*a^(23) - (2377)/(6553600)*a^(19) + (93897)/(26214400)*a^(15) - (16543)/(1638400)*a^(11) + (871)/(20480)*a^(7) - (909)/(6400)*a^(3)  (order $8$)	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:		$\frac{713}{26214400}a^{31}+\frac{277}{13107200}a^{30}-\frac{81}{1310720}a^{27}-\frac{9}{131072}a^{26}+\frac{4483}{13107200}a^{23}+\frac{2247}{6553600}a^{22}-\frac{921}{6553600}a^{19}-\frac{7749}{3276800}a^{18}-\frac{16679}{26214400}a^{15}+\frac{33509}{13107200}a^{14}-\frac{1959}{1638400}a^{11}-\frac{19351}{819200}a^{10}+\frac{639}{20480}a^{7}+\frac{31}{2048}a^{6}-\frac{1357}{6400}a^{3}-\frac{533}{3200}a^{2}$, $\frac{1047}{52428800}a^{31}-\frac{57}{26214400}a^{30}+\frac{421}{13107200}a^{29}-\frac{15}{262144}a^{28}-\frac{271}{2621440}a^{27}+\frac{29}{262144}a^{26}-\frac{77}{655360}a^{25}+\frac{19}{65536}a^{24}+\frac{9757}{26214400}a^{23}-\frac{3027}{13107200}a^{22}+\frac{1911}{6553600}a^{21}-\frac{69}{131072}a^{20}-\frac{4679}{13107200}a^{19}-\frac{3991}{6553600}a^{18}+\frac{843}{3276800}a^{17}-\frac{33}{65536}a^{16}-\frac{4601}{52428800}a^{15}+\frac{21431}{26214400}a^{14}+\frac{16757}{13107200}a^{13}-\frac{2175}{262144}a^{12}-\frac{48601}{3276800}a^{11}+\frac{27291}{1638400}a^{10}-\frac{31003}{819200}a^{9}+\frac{941}{16384}a^{8}+\frac{5049}{40960}a^{7}-\frac{339}{4096}a^{6}+\frac{1883}{10240}a^{5}-\frac{205}{1024}a^{4}-\frac{4043}{12800}a^{3}+\frac{1353}{6400}a^{2}+\frac{31}{3200}a+\frac{31}{64}$, $\frac{17}{1638400}a^{31}+\frac{1}{16384}a^{28}-\frac{7}{81920}a^{27}-\frac{1}{4096}a^{24}+\frac{267}{819200}a^{23}+\frac{11}{8192}a^{20}+\frac{91}{409600}a^{19}-\frac{9}{4096}a^{16}-\frac{6911}{1638400}a^{15}+\frac{209}{16384}a^{12}+\frac{1823}{204800}a^{11}-\frac{9}{256}a^{8}-\frac{29}{2560}a^{7}+\frac{3}{32}a^{4}-\frac{7}{100}a^{3}+\frac{1}{4}$, $\frac{57}{26214400}a^{30}+\frac{467}{6553600}a^{29}+\frac{15}{262144}a^{28}-\frac{29}{262144}a^{26}-\frac{15}{65536}a^{25}-\frac{19}{65536}a^{24}+\frac{3027}{13107200}a^{22}+\frac{2737}{3276800}a^{21}+\frac{69}{131072}a^{20}+\frac{3991}{6553600}a^{18}+\frac{4221}{1638400}a^{17}+\frac{33}{65536}a^{16}-\frac{21431}{26214400}a^{14}+\frac{11139}{6553600}a^{13}+\frac{2175}{262144}a^{12}-\frac{27291}{1638400}a^{10}-\frac{13221}{409600}a^{9}-\frac{941}{16384}a^{8}+\frac{339}{4096}a^{6}+\frac{305}{1024}a^{5}+\frac{205}{1024}a^{4}-\frac{1353}{6400}a^{2}+\frac{57}{1600}a-\frac{31}{64}$, $\frac{599}{52428800}a^{31}-\frac{139}{13107200}a^{30}-\frac{137}{13107200}a^{29}-\frac{7}{409600}a^{28}+\frac{81}{2621440}a^{27}+\frac{43}{655360}a^{26}-\frac{79}{655360}a^{25}-\frac{3}{10240}a^{24}+\frac{3549}{26214400}a^{23}+\frac{551}{6553600}a^{22}-\frac{4547}{6553600}a^{21}-\frac{37}{204800}a^{20}+\frac{17977}{13107200}a^{19}+\frac{6763}{3276800}a^{18}-\frac{9191}{3276800}a^{17}-\frac{431}{102400}a^{16}+\frac{282183}{52428800}a^{15}+\frac{119237}{13107200}a^{14}-\frac{191129}{13107200}a^{13}-\frac{6119}{409600}a^{12}+\frac{49903}{3276800}a^{11}+\frac{24527}{819200}a^{10}-\frac{32529}{819200}a^{9}-\frac{6021}{102400}a^{8}+\frac{2881}{40960}a^{7}+\frac{763}{10240}a^{6}-\frac{479}{10240}a^{5}-\frac{1}{20}a^{4}+\frac{3909}{12800}a^{3}+\frac{1221}{3200}a^{2}-\frac{1147}{3200}a-\frac{383}{400}$, $\frac{593}{52428800}a^{31}+\frac{69}{3276800}a^{30}-\frac{481}{13107200}a^{29}-\frac{251}{3276800}a^{28}+\frac{119}{2621440}a^{27}+\frac{9}{163840}a^{26}-\frac{39}{655360}a^{25}+\frac{11}{163840}a^{24}+\frac{6363}{26214400}a^{23}+\frac{999}{1638400}a^{22}-\frac{5131}{6553600}a^{21}-\frac{2601}{1638400}a^{20}+\frac{19519}{13107200}a^{19}+\frac{607}{819200}a^{18}-\frac{9263}{3276800}a^{17}-\frac{773}{819200}a^{16}+\frac{282081}{52428800}a^{15}+\frac{17973}{3276800}a^{14}-\frac{152177}{13107200}a^{13}-\frac{55467}{3276800}a^{12}+\frac{12501}{3276800}a^{11}+\frac{4001}{409600}a^{10}-\frac{14357}{819200}a^{9}+\frac{11403}{204800}a^{8}+\frac{1359}{40960}a^{7}+\frac{249}{2560}a^{6}-\frac{1679}{10240}a^{5}-\frac{589}{2560}a^{4}+\frac{1823}{12800}a^{3}-\frac{237}{1600}a^{2}+\frac{1329}{3200}a+\frac{1009}{800}$, $\frac{593}{52428800}a^{31}-\frac{873}{26214400}a^{30}+\frac{481}{13107200}a^{29}-\frac{113}{6553600}a^{28}+\frac{119}{2621440}a^{27}+\frac{129}{1310720}a^{26}+\frac{39}{655360}a^{25}+\frac{41}{327680}a^{24}+\frac{6363}{26214400}a^{23}-\frac{1763}{13107200}a^{22}+\frac{5131}{6553600}a^{21}-\frac{1083}{3276800}a^{20}+\frac{19519}{13107200}a^{19}+\frac{14361}{6553600}a^{18}+\frac{9263}{3276800}a^{17}+\frac{4321}{1638400}a^{16}+\frac{282081}{52428800}a^{15}+\frac{33159}{26214400}a^{14}+\frac{152177}{13107200}a^{13}+\frac{84479}{6553600}a^{12}+\frac{12501}{3276800}a^{11}+\frac{91259}{1638400}a^{10}+\frac{14357}{819200}a^{9}+\frac{26859}{409600}a^{8}+\frac{1359}{40960}a^{7}-\frac{711}{20480}a^{6}+\frac{1679}{10240}a^{5}+\frac{1}{5120}a^{4}+\frac{1823}{12800}a^{3}+\frac{4737}{6400}a^{2}-\frac{1329}{3200}a+\frac{1457}{1600}$, $\frac{201}{26214400}a^{31}+\frac{437}{26214400}a^{30}+\frac{41}{819200}a^{29}-\frac{273}{6553600}a^{28}-\frac{153}{1310720}a^{27}-\frac{157}{1310720}a^{26}+\frac{1}{20480}a^{25}+\frac{153}{327680}a^{24}+\frac{1731}{13107200}a^{23}+\frac{2087}{13107200}a^{22}+\frac{171}{409600}a^{21}-\frac{923}{3276800}a^{20}-\frac{5257}{6553600}a^{19}-\frac{7749}{6553600}a^{18}+\frac{113}{204800}a^{17}+\frac{4321}{1638400}a^{16}+\frac{67417}{26214400}a^{15}+\frac{145029}{26214400}a^{14}+\frac{4297}{819200}a^{13}-\frac{27041}{6553600}a^{12}-\frac{35233}{1638400}a^{11}-\frac{7611}{1638400}a^{10}-\frac{6637}{204800}a^{9}+\frac{15119}{409600}a^{8}+\frac{167}{20480}a^{7}+\frac{3043}{20480}a^{6}+\frac{91}{320}a^{5}-\frac{327}{5120}a^{4}-\frac{3019}{6400}a^{3}-\frac{3033}{6400}a^{2}+\frac{9}{800}a+\frac{1557}{1600}$, $\frac{51}{6553600}a^{31}-\frac{31}{3276800}a^{30}-\frac{39}{1638400}a^{29}-\frac{13}{1638400}a^{28}-\frac{43}{327680}a^{27}+\frac{29}{163840}a^{26}-\frac{9}{81920}a^{25}+\frac{33}{81920}a^{24}+\frac{81}{3276800}a^{23}-\frac{101}{1638400}a^{22}-\frac{269}{819200}a^{21}-\frac{463}{819200}a^{20}-\frac{1507}{1638400}a^{19}+\frac{1507}{819200}a^{18}-\frac{217}{409600}a^{17}+\frac{901}{409600}a^{16}+\frac{16867}{6553600}a^{15}-\frac{2927}{3276800}a^{14}-\frac{13463}{1638400}a^{13}-\frac{21021}{1638400}a^{12}-\frac{4133}{409600}a^{11}+\frac{11201}{409600}a^{10}+\frac{1047}{102400}a^{9}+\frac{2157}{51200}a^{8}+\frac{29}{640}a^{7}-\frac{191}{2560}a^{6}-\frac{107}{640}a^{5}-\frac{167}{1280}a^{4}-\frac{247}{800}a^{3}+\frac{1363}{1600}a^{2}-\frac{157}{200}a+\frac{171}{200}$, $\frac{889}{52428800}a^{31}-\frac{87}{2621440}a^{29}-\frac{29}{524288}a^{27}-\frac{37}{655360}a^{25}-\frac{621}{26214400}a^{23}-\frac{861}{1310720}a^{21}+\frac{14807}{13107200}a^{19}-\frac{1177}{655360}a^{17}+\frac{171913}{52428800}a^{15}-\frac{31559}{2621440}a^{13}-\frac{28107}{3276800}a^{11}+\frac{137}{32768}a^{9}+\frac{419}{8192}a^{7}-\frac{2037}{10240}a^{5}-\frac{681}{12800}a^{3}-\frac{49}{640}a$, $\frac{87}{10485760}a^{31}-\frac{533}{26214400}a^{30}-\frac{281}{13107200}a^{29}-\frac{9}{262144}a^{28}+\frac{69}{2621440}a^{27}-\frac{7}{262144}a^{26}-\frac{111}{655360}a^{25}-\frac{11}{65536}a^{24}+\frac{121}{1048576}a^{23}-\frac{1863}{13107200}a^{22}-\frac{1651}{6553600}a^{21}-\frac{67}{131072}a^{20}-\frac{167}{2621440}a^{19}-\frac{4379}{6553600}a^{18}-\frac{4343}{3276800}a^{17}-\frac{103}{65536}a^{16}+\frac{18759}{10485760}a^{15}-\frac{82661}{26214400}a^{14}-\frac{46377}{13107200}a^{13}-\frac{537}{262144}a^{12}-\frac{10789}{655360}a^{11}-\frac{19021}{1638400}a^{10}-\frac{22237}{819200}a^{9}+\frac{111}{16384}a^{8}-\frac{651}{40960}a^{7}-\frac{543}{4096}a^{6}-\frac{671}{10240}a^{5}-\frac{51}{1024}a^{4}-\frac{599}{2560}a^{3}-\frac{1543}{6400}a^{2}-\frac{1631}{3200}a+\frac{29}{64}$, $\frac{1513}{52428800}a^{31}+\frac{79}{13107200}a^{30}+\frac{37}{13107200}a^{29}-\frac{249}{3276800}a^{28}-\frac{241}{2621440}a^{27}+\frac{57}{655360}a^{26}-\frac{77}{655360}a^{25}+\frac{49}{163840}a^{24}+\frac{13283}{26214400}a^{23}+\frac{389}{6553600}a^{22}-\frac{2313}{6553600}a^{21}-\frac{1299}{1638400}a^{20}-\frac{8121}{13107200}a^{19}+\frac{3457}{3276800}a^{18}-\frac{4149}{3276800}a^{17}+\frac{873}{819200}a^{16}+\frac{150521}{52428800}a^{15}+\frac{30143}{13107200}a^{14}+\frac{61429}{13107200}a^{13}-\frac{20233}{3276800}a^{12}-\frac{30759}{3276800}a^{11}-\frac{17297}{819200}a^{10}+\frac{6789}{819200}a^{9}+\frac{8947}{204800}a^{8}+\frac{4159}{40960}a^{7}-\frac{403}{10240}a^{6}+\frac{3}{10240}a^{5}-\frac{291}{2560}a^{4}-\frac{4557}{12800}a^{3}+\frac{369}{3200}a^{2}+\frac{567}{3200}a+\frac{741}{800}$, $\frac{1513}{52428800}a^{31}-\frac{79}{13107200}a^{30}+\frac{37}{13107200}a^{29}+\frac{249}{3276800}a^{28}-\frac{241}{2621440}a^{27}-\frac{57}{655360}a^{26}-\frac{77}{655360}a^{25}-\frac{49}{163840}a^{24}+\frac{13283}{26214400}a^{23}-\frac{389}{6553600}a^{22}-\frac{2313}{6553600}a^{21}+\frac{1299}{1638400}a^{20}-\frac{8121}{13107200}a^{19}-\frac{3457}{3276800}a^{18}-\frac{4149}{3276800}a^{17}-\frac{873}{819200}a^{16}+\frac{150521}{52428800}a^{15}-\frac{30143}{13107200}a^{14}+\frac{61429}{13107200}a^{13}+\frac{20233}{3276800}a^{12}-\frac{30759}{3276800}a^{11}+\frac{17297}{819200}a^{10}+\frac{6789}{819200}a^{9}-\frac{8947}{204800}a^{8}+\frac{4159}{40960}a^{7}+\frac{403}{10240}a^{6}+\frac{3}{10240}a^{5}+\frac{291}{2560}a^{4}-\frac{4557}{12800}a^{3}-\frac{369}{3200}a^{2}+\frac{567}{3200}a-\frac{741}{800}$, $\frac{3}{26214400}a^{31}+\frac{1333}{26214400}a^{30}+\frac{189}{3276800}a^{29}-\frac{9}{262144}a^{28}-\frac{19}{1310720}a^{27}-\frac{25}{262144}a^{26}-\frac{49}{163840}a^{25}-\frac{11}{65536}a^{24}-\frac{1407}{13107200}a^{23}+\frac{10663}{13107200}a^{22}-\frac{161}{1638400}a^{21}-\frac{67}{131072}a^{20}-\frac{771}{6553600}a^{19}-\frac{2821}{6553600}a^{18}-\frac{1653}{819200}a^{17}-\frac{103}{65536}a^{16}+\frac{51}{26214400}a^{15}+\frac{249861}{26214400}a^{14}+\frac{11213}{3276800}a^{13}-\frac{537}{262144}a^{12}+\frac{18701}{1638400}a^{11}-\frac{9779}{1638400}a^{10}-\frac{7521}{102400}a^{9}+\frac{111}{16384}a^{8}+\frac{761}{20480}a^{7}+\frac{1247}{4096}a^{6}+\frac{221}{2560}a^{5}-\frac{51}{1024}a^{4}+\frac{1043}{6400}a^{3}-\frac{4857}{6400}a^{2}-\frac{269}{200}a-\frac{3}{64}$, $\frac{577}{26214400}a^{31}+\frac{191}{26214400}a^{30}+\frac{1}{819200}a^{29}+\frac{631}{6553600}a^{28}-\frac{5}{262144}a^{27}-\frac{87}{1310720}a^{26}+\frac{1}{8192}a^{25}-\frac{79}{327680}a^{24}+\frac{2347}{13107200}a^{23}-\frac{2219}{13107200}a^{22}-\frac{389}{409600}a^{21}+\frac{3901}{3276800}a^{20}-\frac{1649}{6553600}a^{19}-\frac{16047}{6553600}a^{18}+\frac{563}{204800}a^{17}+\frac{793}{1638400}a^{16}+\frac{38609}{26214400}a^{15}+\frac{44847}{26214400}a^{14}+\frac{817}{819200}a^{13}+\frac{129127}{6553600}a^{12}-\frac{9251}{1638400}a^{11}-\frac{17513}{1638400}a^{10}+\frac{1749}{102400}a^{9}-\frac{5713}{409600}a^{8}+\frac{151}{4096}a^{7}-\frac{167}{20480}a^{6}-\frac{7}{256}a^{5}+\frac{1681}{5120}a^{4}-\frac{1133}{6400}a^{3}-\frac{899}{6400}a^{2}+\frac{12}{25}a-\frac{619}{1600}$ 713/26214400*a^31 + 277/13107200*a^30 - 81/1310720*a^27 - 9/131072*a^26 + 4483/13107200*a^23 + 2247/6553600*a^22 - 921/6553600*a^19 - 7749/3276800*a^18 - 16679/26214400*a^15 + 33509/13107200*a^14 - 1959/1638400*a^11 - 19351/819200*a^10 + 639/20480*a^7 + 31/2048*a^6 - 1357/6400*a^3 - 533/3200*a^2, 1047/52428800*a^31 - 57/26214400*a^30 + 421/13107200*a^29 - 15/262144*a^28 - 271/2621440*a^27 + 29/262144*a^26 - 77/655360*a^25 + 19/65536*a^24 + 9757/26214400*a^23 - 3027/13107200*a^22 + 1911/6553600*a^21 - 69/131072*a^20 - 4679/13107200*a^19 - 3991/6553600*a^18 + 843/3276800*a^17 - 33/65536*a^16 - 4601/52428800*a^15 + 21431/26214400*a^14 + 16757/13107200*a^13 - 2175/262144*a^12 - 48601/3276800*a^11 + 27291/1638400*a^10 - 31003/819200*a^9 + 941/16384*a^8 + 5049/40960*a^7 - 339/4096*a^6 + 1883/10240*a^5 - 205/1024*a^4 - 4043/12800*a^3 + 1353/6400*a^2 + 31/3200*a + 31/64, 17/1638400*a^31 + 1/16384*a^28 - 7/81920*a^27 - 1/4096*a^24 + 267/819200*a^23 + 11/8192*a^20 + 91/409600*a^19 - 9/4096*a^16 - 6911/1638400*a^15 + 209/16384*a^12 + 1823/204800*a^11 - 9/256*a^8 - 29/2560*a^7 + 3/32*a^4 - 7/100*a^3 + 1/4, 57/26214400*a^30 + 467/6553600*a^29 + 15/262144*a^28 - 29/262144*a^26 - 15/65536*a^25 - 19/65536*a^24 + 3027/13107200*a^22 + 2737/3276800*a^21 + 69/131072*a^20 + 3991/6553600*a^18 + 4221/1638400*a^17 + 33/65536*a^16 - 21431/26214400*a^14 + 11139/6553600*a^13 + 2175/262144*a^12 - 27291/1638400*a^10 - 13221/409600*a^9 - 941/16384*a^8 + 339/4096*a^6 + 305/1024*a^5 + 205/1024*a^4 - 1353/6400*a^2 + 57/1600*a - 31/64, 599/52428800*a^31 - 139/13107200*a^30 - 137/13107200*a^29 - 7/409600*a^28 + 81/2621440*a^27 + 43/655360*a^26 - 79/655360*a^25 - 3/10240*a^24 + 3549/26214400*a^23 + 551/6553600*a^22 - 4547/6553600*a^21 - 37/204800*a^20 + 17977/13107200*a^19 + 6763/3276800*a^18 - 9191/3276800*a^17 - 431/102400*a^16 + 282183/52428800*a^15 + 119237/13107200*a^14 - 191129/13107200*a^13 - 6119/409600*a^12 + 49903/3276800*a^11 + 24527/819200*a^10 - 32529/819200*a^9 - 6021/102400*a^8 + 2881/40960*a^7 + 763/10240*a^6 - 479/10240*a^5 - 1/20*a^4 + 3909/12800*a^3 + 1221/3200*a^2 - 1147/3200*a - 383/400, 593/52428800*a^31 + 69/3276800*a^30 - 481/13107200*a^29 - 251/3276800*a^28 + 119/2621440*a^27 + 9/163840*a^26 - 39/655360*a^25 + 11/163840*a^24 + 6363/26214400*a^23 + 999/1638400*a^22 - 5131/6553600*a^21 - 2601/1638400*a^20 + 19519/13107200*a^19 + 607/819200*a^18 - 9263/3276800*a^17 - 773/819200*a^16 + 282081/52428800*a^15 + 17973/3276800*a^14 - 152177/13107200*a^13 - 55467/3276800*a^12 + 12501/3276800*a^11 + 4001/409600*a^10 - 14357/819200*a^9 + 11403/204800*a^8 + 1359/40960*a^7 + 249/2560*a^6 - 1679/10240*a^5 - 589/2560*a^4 + 1823/12800*a^3 - 237/1600*a^2 + 1329/3200*a + 1009/800, 593/52428800*a^31 - 873/26214400*a^30 + 481/13107200*a^29 - 113/6553600*a^28 + 119/2621440*a^27 + 129/1310720*a^26 + 39/655360*a^25 + 41/327680*a^24 + 6363/26214400*a^23 - 1763/13107200*a^22 + 5131/6553600*a^21 - 1083/3276800*a^20 + 19519/13107200*a^19 + 14361/6553600*a^18 + 9263/3276800*a^17 + 4321/1638400*a^16 + 282081/52428800*a^15 + 33159/26214400*a^14 + 152177/13107200*a^13 + 84479/6553600*a^12 + 12501/3276800*a^11 + 91259/1638400*a^10 + 14357/819200*a^9 + 26859/409600*a^8 + 1359/40960*a^7 - 711/20480*a^6 + 1679/10240*a^5 + 1/5120*a^4 + 1823/12800*a^3 + 4737/6400*a^2 - 1329/3200*a + 1457/1600, 201/26214400*a^31 + 437/26214400*a^30 + 41/819200*a^29 - 273/6553600*a^28 - 153/1310720*a^27 - 157/1310720*a^26 + 1/20480*a^25 + 153/327680*a^24 + 1731/13107200*a^23 + 2087/13107200*a^22 + 171/409600*a^21 - 923/3276800*a^20 - 5257/6553600*a^19 - 7749/6553600*a^18 + 113/204800*a^17 + 4321/1638400*a^16 + 67417/26214400*a^15 + 145029/26214400*a^14 + 4297/819200*a^13 - 27041/6553600*a^12 - 35233/1638400*a^11 - 7611/1638400*a^10 - 6637/204800*a^9 + 15119/409600*a^8 + 167/20480*a^7 + 3043/20480*a^6 + 91/320*a^5 - 327/5120*a^4 - 3019/6400*a^3 - 3033/6400*a^2 + 9/800*a + 1557/1600, 51/6553600*a^31 - 31/3276800*a^30 - 39/1638400*a^29 - 13/1638400*a^28 - 43/327680*a^27 + 29/163840*a^26 - 9/81920*a^25 + 33/81920*a^24 + 81/3276800*a^23 - 101/1638400*a^22 - 269/819200*a^21 - 463/819200*a^20 - 1507/1638400*a^19 + 1507/819200*a^18 - 217/409600*a^17 + 901/409600*a^16 + 16867/6553600*a^15 - 2927/3276800*a^14 - 13463/1638400*a^13 - 21021/1638400*a^12 - 4133/409600*a^11 + 11201/409600*a^10 + 1047/102400*a^9 + 2157/51200*a^8 + 29/640*a^7 - 191/2560*a^6 - 107/640*a^5 - 167/1280*a^4 - 247/800*a^3 + 1363/1600*a^2 - 157/200*a + 171/200, 889/52428800*a^31 - 87/2621440*a^29 - 29/524288*a^27 - 37/655360*a^25 - 621/26214400*a^23 - 861/1310720*a^21 + 14807/13107200*a^19 - 1177/655360*a^17 + 171913/52428800*a^15 - 31559/2621440*a^13 - 28107/3276800*a^11 + 137/32768*a^9 + 419/8192*a^7 - 2037/10240*a^5 - 681/12800*a^3 - 49/640*a, 87/10485760*a^31 - 533/26214400*a^30 - 281/13107200*a^29 - 9/262144*a^28 + 69/2621440*a^27 - 7/262144*a^26 - 111/655360*a^25 - 11/65536*a^24 + 121/1048576*a^23 - 1863/13107200*a^22 - 1651/6553600*a^21 - 67/131072*a^20 - 167/2621440*a^19 - 4379/6553600*a^18 - 4343/3276800*a^17 - 103/65536*a^16 + 18759/10485760*a^15 - 82661/26214400*a^14 - 46377/13107200*a^13 - 537/262144*a^12 - 10789/655360*a^11 - 19021/1638400*a^10 - 22237/819200*a^9 + 111/16384*a^8 - 651/40960*a^7 - 543/4096*a^6 - 671/10240*a^5 - 51/1024*a^4 - 599/2560*a^3 - 1543/6400*a^2 - 1631/3200*a + 29/64, 1513/52428800*a^31 + 79/13107200*a^30 + 37/13107200*a^29 - 249/3276800*a^28 - 241/2621440*a^27 + 57/655360*a^26 - 77/655360*a^25 + 49/163840*a^24 + 13283/26214400*a^23 + 389/6553600*a^22 - 2313/6553600*a^21 - 1299/1638400*a^20 - 8121/13107200*a^19 + 3457/3276800*a^18 - 4149/3276800*a^17 + 873/819200*a^16 + 150521/52428800*a^15 + 30143/13107200*a^14 + 61429/13107200*a^13 - 20233/3276800*a^12 - 30759/3276800*a^11 - 17297/819200*a^10 + 6789/819200*a^9 + 8947/204800*a^8 + 4159/40960*a^7 - 403/10240*a^6 + 3/10240*a^5 - 291/2560*a^4 - 4557/12800*a^3 + 369/3200*a^2 + 567/3200*a + 741/800, 1513/52428800*a^31 - 79/13107200*a^30 + 37/13107200*a^29 + 249/3276800*a^28 - 241/2621440*a^27 - 57/655360*a^26 - 77/655360*a^25 - 49/163840*a^24 + 13283/26214400*a^23 - 389/6553600*a^22 - 2313/6553600*a^21 + 1299/1638400*a^20 - 8121/13107200*a^19 - 3457/3276800*a^18 - 4149/3276800*a^17 - 873/819200*a^16 + 150521/52428800*a^15 - 30143/13107200*a^14 + 61429/13107200*a^13 + 20233/3276800*a^12 - 30759/3276800*a^11 + 17297/819200*a^10 + 6789/819200*a^9 - 8947/204800*a^8 + 4159/40960*a^7 + 403/10240*a^6 + 3/10240*a^5 + 291/2560*a^4 - 4557/12800*a^3 - 369/3200*a^2 + 567/3200*a - 741/800, 3/26214400*a^31 + 1333/26214400*a^30 + 189/3276800*a^29 - 9/262144*a^28 - 19/1310720*a^27 - 25/262144*a^26 - 49/163840*a^25 - 11/65536*a^24 - 1407/13107200*a^23 + 10663/13107200*a^22 - 161/1638400*a^21 - 67/131072*a^20 - 771/6553600*a^19 - 2821/6553600*a^18 - 1653/819200*a^17 - 103/65536*a^16 + 51/26214400*a^15 + 249861/26214400*a^14 + 11213/3276800*a^13 - 537/262144*a^12 + 18701/1638400*a^11 - 9779/1638400*a^10 - 7521/102400*a^9 + 111/16384*a^8 + 761/20480*a^7 + 1247/4096*a^6 + 221/2560*a^5 - 51/1024*a^4 + 1043/6400*a^3 - 4857/6400*a^2 - 269/200*a - 3/64, 577/26214400*a^31 + 191/26214400*a^30 + 1/819200*a^29 + 631/6553600*a^28 - 5/262144*a^27 - 87/1310720*a^26 + 1/8192*a^25 - 79/327680*a^24 + 2347/13107200*a^23 - 2219/13107200*a^22 - 389/409600*a^21 + 3901/3276800*a^20 - 1649/6553600*a^19 - 16047/6553600*a^18 + 563/204800*a^17 + 793/1638400*a^16 + 38609/26214400*a^15 + 44847/26214400*a^14 + 817/819200*a^13 + 129127/6553600*a^12 - 9251/1638400*a^11 - 17513/1638400*a^10 + 1749/102400*a^9 - 5713/409600*a^8 + 151/4096*a^7 - 167/20480*a^6 - 7/256*a^5 + 1681/5120*a^4 - 1133/6400*a^3 - 899/6400*a^2 + 12/25*a - 619/1600 (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:		\( 588107386863.9401 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 588107386863.9401 \cdot 24}{8\cdot\sqrt{90220386589172305242198553166127486468521067544576}}\cr\approx \mathstrut & 1.09598204904774 \end{aligned}\] (assuming GRH)

Galois group

$C_2^6:S_4$ (as 32T96908):

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		Deg $16$	$16$	$1$	$38$
		Deg $16$	$16$	$1$	$38$
\(89\) 89	89.2.1.2	$x^{2} + 267$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	89.2.1.2	$x^{2} + 267$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	89.2.1.2	$x^{2} + 267$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	89.2.1.2	$x^{2} + 267$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	89.6.0.1	$x^{6} + x^{4} + 82 x^{3} + 80 x^{2} + 15 x + 3$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	89.6.0.1	$x^{6} + x^{4} + 82 x^{3} + 80 x^{2} + 15 x + 3$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	89.6.0.1	$x^{6} + x^{4} + 82 x^{3} + 80 x^{2} + 15 x + 3$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	89.6.0.1	$x^{6} + x^{4} + 82 x^{3} + 80 x^{2} + 15 x + 3$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
\(257\) 257		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $4$	$2$	$2$	$2$
		Deg $4$	$2$	$2$	$2$
		Deg $4$	$2$	$2$	$2$
		Deg $4$	$2$	$2$	$2$