Normalized defining polynomial

\( x^{32} + 8x^{24} + 64x^{20} + 16x^{16} + 1024x^{12} + 2048x^{8} + 65536 \) x^32 + 8*x^24 + 64*x^20 + 16*x^16 + 1024*x^12 + 2048*x^8 + 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:		\(351509915919708142388734226325389286150171856470016\) 351509915919708142388734226325389286150171856470016 \(\medspace = 2^{76}\cdot 17^{12}\cdot 41^{8}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(37.98\)	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:		$2^{19/8}17^{3/4}41^{1/2}\approx 278.0832683343685$
Ramified primes:		\(2\), \(17\), \(41\) 2, 17, 41	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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{4}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{8}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{11}+\frac{1}{4}a^{3}$, $\frac{1}{32}a^{12}+\frac{1}{8}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{32}a^{13}+\frac{1}{8}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{32}a^{14}-\frac{1}{8}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{64}a^{15}+\frac{1}{16}a^{7}-\frac{1}{4}a^{5}$, $\frac{1}{128}a^{16}-\frac{1}{64}a^{14}-\frac{1}{64}a^{12}+\frac{1}{32}a^{8}+\frac{1}{16}a^{6}-\frac{1}{4}a^{5}-\frac{1}{16}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{128}a^{17}-\frac{1}{64}a^{13}+\frac{1}{32}a^{9}-\frac{1}{8}a^{7}+\frac{3}{16}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{256}a^{18}-\frac{1}{128}a^{14}-\frac{1}{64}a^{13}-\frac{1}{64}a^{12}-\frac{3}{64}a^{10}-\frac{1}{16}a^{8}-\frac{1}{8}a^{7}-\frac{1}{32}a^{6}+\frac{3}{16}a^{5}-\frac{3}{16}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{256}a^{19}-\frac{1}{128}a^{15}-\frac{1}{64}a^{14}-\frac{1}{64}a^{13}+\frac{1}{64}a^{11}-\frac{1}{16}a^{9}-\frac{1}{32}a^{7}-\frac{1}{16}a^{6}-\frac{3}{16}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{256}a^{20}-\frac{1}{16}a^{10}-\frac{1}{4}a^{5}+\frac{3}{16}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{256}a^{21}+\frac{3}{16}a^{5}-\frac{1}{2}a$, $\frac{1}{512}a^{22}-\frac{1}{64}a^{14}-\frac{1}{16}a^{10}-\frac{3}{32}a^{6}+\frac{1}{4}a^{2}$, $\frac{1}{1024}a^{23}-\frac{1}{128}a^{15}-\frac{1}{32}a^{11}-\frac{3}{64}a^{7}-\frac{1}{4}a^{5}-\frac{3}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{4096}a^{24}+\frac{1}{512}a^{16}-\frac{1}{64}a^{12}-\frac{1}{16}a^{10}-\frac{15}{256}a^{8}+\frac{1}{8}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{8192}a^{25}+\frac{1}{1024}a^{17}-\frac{1}{128}a^{13}-\frac{1}{32}a^{11}-\frac{15}{512}a^{9}-\frac{1}{8}a^{7}+\frac{1}{16}a^{5}+\frac{1}{8}a^{3}-\frac{3}{8}a$, $\frac{1}{8192}a^{26}+\frac{1}{1024}a^{18}-\frac{1}{128}a^{14}-\frac{15}{512}a^{10}+\frac{1}{16}a^{6}-\frac{1}{2}a^{3}-\frac{3}{8}a^{2}$, $\frac{1}{16384}a^{27}+\frac{1}{2048}a^{19}-\frac{1}{256}a^{15}-\frac{1}{64}a^{14}-\frac{1}{64}a^{13}-\frac{15}{1024}a^{11}-\frac{1}{16}a^{9}+\frac{1}{32}a^{7}-\frac{1}{16}a^{6}+\frac{1}{16}a^{5}-\frac{1}{4}a^{4}-\frac{3}{16}a^{3}$, $\frac{1}{589824}a^{28}-\frac{1}{16384}a^{26}+\frac{5}{73728}a^{24}-\frac{1}{1024}a^{22}+\frac{19}{24576}a^{20}-\frac{1}{2048}a^{18}-\frac{1}{4608}a^{16}+\frac{3}{256}a^{14}-\frac{1}{64}a^{13}-\frac{319}{36864}a^{12}+\frac{47}{1024}a^{10}-\frac{49}{1536}a^{8}-\frac{1}{8}a^{7}+\frac{1}{64}a^{6}-\frac{1}{16}a^{5}-\frac{67}{576}a^{4}-\frac{1}{2}a^{3}+\frac{1}{16}a^{2}-\frac{1}{2}a-\frac{11}{72}$, $\frac{1}{1179648}a^{29}-\frac{1}{32768}a^{27}-\frac{1}{16384}a^{26}+\frac{5}{147456}a^{25}-\frac{1}{2048}a^{23}-\frac{77}{49152}a^{21}+\frac{7}{4096}a^{19}-\frac{1}{2048}a^{18}-\frac{1}{9216}a^{17}+\frac{1}{512}a^{15}-\frac{3}{256}a^{14}+\frac{257}{73728}a^{13}-\frac{1}{64}a^{12}+\frac{63}{2048}a^{11}+\frac{15}{1024}a^{10}-\frac{145}{3072}a^{9}-\frac{1}{16}a^{8}-\frac{1}{128}a^{7}-\frac{3}{32}a^{6}+\frac{77}{1152}a^{5}-\frac{3}{16}a^{4}+\frac{13}{32}a^{3}-\frac{5}{16}a^{2}-\frac{11}{144}a$, $\frac{1}{2359296}a^{30}-\frac{1}{32768}a^{27}-\frac{13}{294912}a^{26}-\frac{1}{8192}a^{24}+\frac{19}{98304}a^{22}-\frac{1}{512}a^{20}+\frac{7}{4096}a^{19}-\frac{5}{9216}a^{18}-\frac{1}{1024}a^{16}-\frac{1}{512}a^{15}+\frac{1985}{147456}a^{14}-\frac{1}{64}a^{13}+\frac{1}{128}a^{12}-\frac{33}{2048}a^{11}+\frac{329}{6144}a^{10}-\frac{1}{16}a^{9}+\frac{15}{512}a^{8}-\frac{1}{32}a^{7}+\frac{185}{2304}a^{6}-\frac{1}{16}a^{5}-\frac{5}{32}a^{4}-\frac{5}{32}a^{3}+\frac{79}{288}a^{2}-\frac{1}{4}a-\frac{3}{8}$, $\frac{1}{2359296}a^{31}+\frac{5}{294912}a^{27}+\frac{19}{98304}a^{23}-\frac{1}{512}a^{21}-\frac{1}{18432}a^{19}-\frac{895}{147456}a^{15}-\frac{1}{64}a^{13}+\frac{47}{6144}a^{11}-\frac{1}{16}a^{9}-\frac{175}{2304}a^{7}+\frac{7}{32}a^{5}-\frac{83}{288}a^{3}+\frac{1}{4}a$ 1, a, a^2, a^3, 1/2*a^4, 1/2*a^5, 1/4*a^6 - 1/2*a^2, 1/4*a^7 - 1/2*a^3, 1/8*a^8 - 1/4*a^4 - 1/2*a^3 - 1/2*a^2, 1/8*a^9 - 1/4*a^5 - 1/2*a^3, 1/8*a^10 - 1/2*a^2, 1/16*a^11 + 1/4*a^3, 1/32*a^12 + 1/8*a^4 - 1/2*a^2, 1/32*a^13 + 1/8*a^5 - 1/2*a^3, 1/32*a^14 - 1/8*a^6 - 1/2*a^2, 1/64*a^15 + 1/16*a^7 - 1/4*a^5, 1/128*a^16 - 1/64*a^14 - 1/64*a^12 + 1/32*a^8 + 1/16*a^6 - 1/4*a^5 - 1/16*a^4 + 1/4*a^2 - 1/2*a - 1/2, 1/128*a^17 - 1/64*a^13 + 1/32*a^9 - 1/8*a^7 + 3/16*a^5 - 1/4*a^3 - 1/2*a, 1/256*a^18 - 1/128*a^14 - 1/64*a^13 - 1/64*a^12 - 3/64*a^10 - 1/16*a^8 - 1/8*a^7 - 1/32*a^6 + 3/16*a^5 - 3/16*a^4 - 1/2*a^3 - 1/2, 1/256*a^19 - 1/128*a^15 - 1/64*a^14 - 1/64*a^13 + 1/64*a^11 - 1/16*a^9 - 1/32*a^7 - 1/16*a^6 - 3/16*a^5 - 1/4*a^4 - 1/4*a^3 - 1/2*a, 1/256*a^20 - 1/16*a^10 - 1/4*a^5 + 3/16*a^4 - 1/4*a^2 - 1/2*a - 1/2, 1/256*a^21 + 3/16*a^5 - 1/2*a, 1/512*a^22 - 1/64*a^14 - 1/16*a^10 - 3/32*a^6 + 1/4*a^2, 1/1024*a^23 - 1/128*a^15 - 1/32*a^11 - 3/64*a^7 - 1/4*a^5 - 3/8*a^3 - 1/2*a, 1/4096*a^24 + 1/512*a^16 - 1/64*a^12 - 1/16*a^10 - 15/256*a^8 + 1/8*a^4 - 1/4*a^2 + 1/4, 1/8192*a^25 + 1/1024*a^17 - 1/128*a^13 - 1/32*a^11 - 15/512*a^9 - 1/8*a^7 + 1/16*a^5 + 1/8*a^3 - 3/8*a, 1/8192*a^26 + 1/1024*a^18 - 1/128*a^14 - 15/512*a^10 + 1/16*a^6 - 1/2*a^3 - 3/8*a^2, 1/16384*a^27 + 1/2048*a^19 - 1/256*a^15 - 1/64*a^14 - 1/64*a^13 - 15/1024*a^11 - 1/16*a^9 + 1/32*a^7 - 1/16*a^6 + 1/16*a^5 - 1/4*a^4 - 3/16*a^3, 1/589824*a^28 - 1/16384*a^26 + 5/73728*a^24 - 1/1024*a^22 + 19/24576*a^20 - 1/2048*a^18 - 1/4608*a^16 + 3/256*a^14 - 1/64*a^13 - 319/36864*a^12 + 47/1024*a^10 - 49/1536*a^8 - 1/8*a^7 + 1/64*a^6 - 1/16*a^5 - 67/576*a^4 - 1/2*a^3 + 1/16*a^2 - 1/2*a - 11/72, 1/1179648*a^29 - 1/32768*a^27 - 1/16384*a^26 + 5/147456*a^25 - 1/2048*a^23 - 77/49152*a^21 + 7/4096*a^19 - 1/2048*a^18 - 1/9216*a^17 + 1/512*a^15 - 3/256*a^14 + 257/73728*a^13 - 1/64*a^12 + 63/2048*a^11 + 15/1024*a^10 - 145/3072*a^9 - 1/16*a^8 - 1/128*a^7 - 3/32*a^6 + 77/1152*a^5 - 3/16*a^4 + 13/32*a^3 - 5/16*a^2 - 11/144*a, 1/2359296*a^30 - 1/32768*a^27 - 13/294912*a^26 - 1/8192*a^24 + 19/98304*a^22 - 1/512*a^20 + 7/4096*a^19 - 5/9216*a^18 - 1/1024*a^16 - 1/512*a^15 + 1985/147456*a^14 - 1/64*a^13 + 1/128*a^12 - 33/2048*a^11 + 329/6144*a^10 - 1/16*a^9 + 15/512*a^8 - 1/32*a^7 + 185/2304*a^6 - 1/16*a^5 - 5/32*a^4 - 5/32*a^3 + 79/288*a^2 - 1/4*a - 3/8, 1/2359296*a^31 + 5/294912*a^27 + 19/98304*a^23 - 1/512*a^21 - 1/18432*a^19 - 895/147456*a^15 - 1/64*a^13 + 47/6144*a^11 - 1/16*a^9 - 175/2304*a^7 + 7/32*a^5 - 83/288*a^3 + 1/4*a

Class group and class number

$C_{2}\times C_{24}$, which has order $48$ (assuming GRH)

Unit group

Rank:		$15$	sage: UK.rank()   gp: K.fu   magma: UnitRank(K);   oscar: rank(UK)
Torsion generator:		\( \frac{1}{147456} a^{31} - \frac{5}{147456} a^{27} + \frac{1}{6144} a^{23} + \frac{11}{18432} a^{19} + \frac{5}{9216} a^{15} - \frac{23}{3072} a^{11} - \frac{7}{576} a^{7} - \frac{7}{144} a^{3} \) (1)/(147456)*a^(31) - (5)/(147456)*a^(27) + (1)/(6144)*a^(23) + (11)/(18432)*a^(19) + (5)/(9216)*a^(15) - (23)/(3072)*a^(11) - (7)/(576)*a^(7) - (7)/(144)*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:		$\frac{1}{147456}a^{31}-\frac{5}{294912}a^{30}-\frac{5}{147456}a^{27}-\frac{5}{73728}a^{26}+\frac{1}{6144}a^{23}+\frac{1}{12288}a^{22}+\frac{11}{18432}a^{19}-\frac{7}{9216}a^{18}+\frac{5}{9216}a^{15}+\frac{11}{18432}a^{14}-\frac{23}{3072}a^{11}+\frac{25}{1536}a^{10}-\frac{7}{576}a^{7}-\frac{25}{288}a^{6}-\frac{7}{144}a^{3}+\frac{11}{72}a^{2}+1$, $\frac{5}{294912}a^{31}-\frac{13}{1179648}a^{30}+\frac{11}{589824}a^{29}-\frac{1}{65536}a^{28}+\frac{5}{73728}a^{27}-\frac{5}{36864}a^{26}+\frac{5}{36864}a^{25}+\frac{1}{8192}a^{24}-\frac{1}{12288}a^{23}-\frac{7}{49152}a^{22}+\frac{17}{24576}a^{21}+\frac{7}{8192}a^{20}+\frac{7}{9216}a^{19}-\frac{1}{18432}a^{18}+\frac{5}{9216}a^{17}-\frac{11}{18432}a^{15}-\frac{173}{73728}a^{14}+\frac{235}{36864}a^{13}-\frac{1}{4096}a^{12}+\frac{23}{1536}a^{11}-\frac{5}{768}a^{10}+\frac{11}{768}a^{9}+\frac{9}{512}a^{8}+\frac{25}{288}a^{7}-\frac{155}{1152}a^{6}+\frac{91}{576}a^{5}+\frac{3}{64}a^{4}-\frac{1}{36}a^{3}-\frac{5}{72}a^{2}+\frac{25}{36}a+\frac{9}{8}$, $\frac{41}{2359296}a^{31}+\frac{5}{786432}a^{30}+\frac{7}{589824}a^{29}+\frac{23}{589824}a^{28}-\frac{5}{73728}a^{27}+\frac{1}{98304}a^{26}-\frac{1}{73728}a^{25}-\frac{5}{18432}a^{24}+\frac{11}{98304}a^{23}-\frac{1}{32768}a^{22}-\frac{11}{24576}a^{21}+\frac{5}{24576}a^{20}+\frac{53}{36864}a^{19}+\frac{7}{6144}a^{18}+\frac{11}{4608}a^{17}+\frac{35}{9216}a^{16}-\frac{119}{147456}a^{15}+\frac{133}{49152}a^{14}+\frac{71}{36864}a^{13}-\frac{713}{36864}a^{12}+\frac{7}{1536}a^{11}+\frac{3}{2048}a^{10}-\frac{19}{1536}a^{9}-\frac{5}{384}a^{8}+\frac{97}{2304}a^{7}+\frac{73}{768}a^{6}+\frac{17}{576}a^{5}+\frac{97}{576}a^{4}+\frac{1}{36}a^{3}+\frac{17}{96}a^{2}+\frac{31}{72}a-\frac{23}{36}$, $\frac{35}{2359296}a^{31}+\frac{7}{786432}a^{30}+\frac{7}{393216}a^{29}+\frac{11}{147456}a^{28}+\frac{13}{294912}a^{27}+\frac{11}{98304}a^{26}+\frac{5}{49152}a^{25}+\frac{13}{73728}a^{24}+\frac{41}{98304}a^{23}+\frac{5}{32768}a^{22}+\frac{5}{16384}a^{21}+\frac{5}{6144}a^{20}+\frac{7}{4608}a^{19}+\frac{5}{6144}a^{18}+\frac{1}{1536}a^{17}+\frac{65}{9216}a^{16}-\frac{221}{147456}a^{15}+\frac{263}{49152}a^{14}+\frac{71}{24576}a^{13}+\frac{19}{9216}a^{12}+\frac{55}{6144}a^{11}+\frac{33}{2048}a^{10}-\frac{1}{1024}a^{9}+\frac{79}{1536}a^{8}-\frac{23}{2304}a^{7}+\frac{47}{768}a^{6}-\frac{1}{384}a^{5}+\frac{47}{288}a^{4}-\frac{61}{288}a^{3}+\frac{7}{96}a^{2}+\frac{13}{48}a-\frac{61}{72}$, $\frac{7}{294912}a^{31}-\frac{7}{1179648}a^{30}-\frac{5}{294912}a^{29}+\frac{1}{65536}a^{28}+\frac{5}{147456}a^{27}+\frac{5}{73728}a^{26}-\frac{5}{73728}a^{25}-\frac{1}{8192}a^{24}+\frac{1}{12288}a^{23}+\frac{11}{49152}a^{22}+\frac{1}{12288}a^{21}-\frac{7}{8192}a^{20}+\frac{25}{18432}a^{19}-\frac{13}{18432}a^{18}-\frac{7}{9216}a^{17}-\frac{1}{18432}a^{15}+\frac{217}{73728}a^{14}+\frac{11}{18432}a^{13}+\frac{1}{4096}a^{12}+\frac{23}{3072}a^{11}+\frac{35}{1536}a^{10}+\frac{25}{1536}a^{9}-\frac{9}{512}a^{8}+\frac{43}{576}a^{7}+\frac{55}{1152}a^{6}-\frac{25}{288}a^{5}-\frac{3}{64}a^{4}-\frac{11}{144}a^{3}+\frac{2}{9}a^{2}-\frac{25}{72}a-\frac{1}{8}$, $\frac{41}{2359296}a^{31}+\frac{1}{262144}a^{30}-\frac{7}{589824}a^{29}-\frac{1}{32768}a^{28}-\frac{5}{73728}a^{27}+\frac{3}{32768}a^{26}+\frac{1}{73728}a^{25}-\frac{1}{8192}a^{24}+\frac{11}{98304}a^{23}-\frac{7}{32768}a^{22}+\frac{11}{24576}a^{21}-\frac{1}{4096}a^{20}+\frac{53}{36864}a^{19}-\frac{1}{1024}a^{18}-\frac{11}{4608}a^{17}-\frac{3}{1024}a^{16}-\frac{119}{147456}a^{15}+\frac{193}{16384}a^{14}-\frac{71}{36864}a^{13}-\frac{17}{2048}a^{12}+\frac{7}{1536}a^{11}-\frac{21}{2048}a^{10}+\frac{19}{1536}a^{9}-\frac{1}{512}a^{8}+\frac{97}{2304}a^{7}-\frac{15}{256}a^{6}-\frac{17}{576}a^{5}-\frac{1}{16}a^{4}+\frac{1}{36}a^{3}+\frac{15}{32}a^{2}-\frac{31}{72}a-\frac{3}{8}$, $\frac{25}{589824}a^{30}-\frac{1}{65536}a^{28}-\frac{11}{147456}a^{26}-\frac{1}{8192}a^{24}+\frac{19}{24576}a^{22}+\frac{7}{8192}a^{20}-\frac{1}{18432}a^{18}-\frac{1}{512}a^{16}+\frac{233}{36864}a^{14}+\frac{63}{4096}a^{12}+\frac{151}{3072}a^{10}-\frac{25}{512}a^{8}-\frac{23}{288}a^{6}+\frac{11}{64}a^{4}+\frac{71}{144}a^{2}-\frac{1}{8}$, $\frac{17}{589824}a^{30}+\frac{17}{196608}a^{28}+\frac{17}{147456}a^{26}+\frac{1}{24576}a^{24}+\frac{11}{24576}a^{22}+\frac{3}{8192}a^{20}-\frac{5}{18432}a^{18}+\frac{1}{1536}a^{16}-\frac{95}{36864}a^{14}-\frac{47}{12288}a^{12}+\frac{107}{3072}a^{10}+\frac{19}{512}a^{8}+\frac{1}{144}a^{6}+\frac{37}{192}a^{4}+\frac{31}{144}a^{2}-\frac{7}{24}$, $\frac{31}{1179648}a^{31}-\frac{1}{65536}a^{30}-\frac{17}{1179648}a^{29}-\frac{1}{65536}a^{28}-\frac{5}{294912}a^{27}+\frac{1}{8192}a^{26}+\frac{5}{147456}a^{25}+\frac{1}{8192}a^{24}-\frac{11}{49152}a^{23}-\frac{1}{8192}a^{22}-\frac{35}{49152}a^{21}+\frac{7}{8192}a^{20}+\frac{65}{36864}a^{19}-\frac{5}{4608}a^{17}-\frac{241}{73728}a^{15}+\frac{31}{4096}a^{14}-\frac{337}{73728}a^{13}-\frac{1}{4096}a^{12}+\frac{121}{6144}a^{11}-\frac{7}{512}a^{10}-\frac{1}{3072}a^{9}+\frac{9}{512}a^{8}+\frac{19}{576}a^{7}-\frac{1}{32}a^{6}-\frac{13}{1152}a^{5}+\frac{3}{64}a^{4}-\frac{43}{288}a^{3}+\frac{1}{2}a^{2}-\frac{155}{144}a+\frac{5}{8}$, $\frac{1}{786432}a^{31}-\frac{5}{262144}a^{30}-\frac{7}{393216}a^{29}-\frac{1}{65536}a^{28}+\frac{5}{98304}a^{27}+\frac{1}{32768}a^{26}-\frac{11}{49152}a^{25}-\frac{1}{4096}a^{24}+\frac{3}{32768}a^{23}+\frac{3}{32768}a^{22}-\frac{5}{16384}a^{21}-\frac{9}{8192}a^{20}-\frac{1}{6144}a^{19}+\frac{1}{1024}a^{18}-\frac{5}{3072}a^{17}-\frac{3}{1024}a^{16}+\frac{65}{49152}a^{15}-\frac{69}{16384}a^{14}-\frac{263}{24576}a^{13}-\frac{33}{4096}a^{12}+\frac{15}{2048}a^{11}-\frac{7}{2048}a^{10}-\frac{33}{1024}a^{9}-\frac{5}{256}a^{8}+\frac{83}{768}a^{7}+\frac{7}{256}a^{6}-\frac{95}{384}a^{5}-\frac{7}{64}a^{4}+\frac{19}{96}a^{3}+\frac{1}{32}a^{2}-\frac{19}{48}a-1$, $\frac{19}{2359296}a^{31}+\frac{7}{786432}a^{30}+\frac{19}{1179648}a^{29}+\frac{1}{294912}a^{28}+\frac{5}{294912}a^{27}+\frac{11}{98304}a^{26}+\frac{23}{147456}a^{25}+\frac{19}{73728}a^{24}+\frac{25}{98304}a^{23}+\frac{5}{32768}a^{22}-\frac{23}{49152}a^{21}-\frac{5}{12288}a^{20}+\frac{1}{2304}a^{19}+\frac{5}{6144}a^{18}+\frac{17}{9216}a^{17}+\frac{5}{9216}a^{16}+\frac{275}{147456}a^{15}+\frac{263}{49152}a^{14}+\frac{275}{73728}a^{13}+\frac{113}{18432}a^{12}-\frac{1}{6144}a^{11}+\frac{33}{2048}a^{10}+\frac{5}{3072}a^{9}+\frac{49}{1536}a^{8}+\frac{221}{2304}a^{7}+\frac{143}{768}a^{6}+\frac{203}{1152}a^{5}+\frac{1}{9}a^{4}+\frac{43}{288}a^{3}+\frac{31}{96}a^{2}+\frac{7}{144}a+\frac{5}{72}$, $\frac{17}{2359296}a^{31}-\frac{1}{131072}a^{30}+\frac{19}{1179648}a^{29}+\frac{1}{16384}a^{28}+\frac{1}{73728}a^{27}+\frac{1}{8192}a^{26}-\frac{13}{147456}a^{25}+\frac{1}{4096}a^{24}-\frac{13}{98304}a^{23}+\frac{7}{16384}a^{22}-\frac{23}{49152}a^{21}+\frac{1}{2048}a^{20}-\frac{43}{36864}a^{19}+\frac{1}{2048}a^{18}-\frac{1}{9216}a^{17}+\frac{3}{512}a^{16}+\frac{49}{147456}a^{15}+\frac{95}{8192}a^{14}+\frac{275}{73728}a^{13}+\frac{17}{1024}a^{12}+\frac{1}{1536}a^{11}-\frac{3}{512}a^{10}-\frac{7}{3072}a^{9}+\frac{1}{256}a^{8}-\frac{77}{2304}a^{7}-\frac{1}{128}a^{6}-\frac{13}{1152}a^{5}+\frac{1}{8}a^{4}-\frac{17}{144}a^{3}+\frac{5}{8}a^{2}+\frac{7}{144}a+\frac{5}{4}$, $\frac{5}{2359296}a^{31}-\frac{17}{2359296}a^{30}-\frac{1}{32768}a^{29}-\frac{1}{589824}a^{28}-\frac{5}{73728}a^{27}-\frac{13}{294912}a^{26}+\frac{1}{8192}a^{25}+\frac{1}{18432}a^{24}-\frac{1}{98304}a^{23}-\frac{35}{98304}a^{22}-\frac{1}{4096}a^{21}+\frac{29}{24576}a^{20}+\frac{17}{36864}a^{19}-\frac{19}{18432}a^{18}-\frac{1}{1024}a^{17}+\frac{11}{9216}a^{16}-\frac{155}{147456}a^{15}-\frac{913}{147456}a^{14}+\frac{15}{2048}a^{13}+\frac{607}{36864}a^{12}-\frac{17}{1536}a^{11}-\frac{103}{6144}a^{10}-\frac{31}{512}a^{9}+\frac{25}{384}a^{8}+\frac{25}{2304}a^{7}-\frac{301}{2304}a^{6}+\frac{1}{16}a^{5}-\frac{23}{576}a^{4}+\frac{1}{36}a^{3}+\frac{43}{288}a^{2}-\frac{3}{8}a+\frac{37}{36}$, $\frac{65}{2359296}a^{31}-\frac{29}{1179648}a^{29}-\frac{1}{65536}a^{28}-\frac{13}{147456}a^{27}+\frac{1}{8192}a^{26}+\frac{17}{147456}a^{25}-\frac{1}{8192}a^{24}+\frac{35}{98304}a^{23}-\frac{1}{1024}a^{22}+\frac{25}{49152}a^{21}+\frac{7}{8192}a^{20}+\frac{23}{36864}a^{19}+\frac{1}{1024}a^{18}-\frac{17}{4608}a^{17}-\frac{1}{512}a^{16}+\frac{289}{147456}a^{15}+\frac{1187}{73728}a^{13}-\frac{65}{4096}a^{12}+\frac{29}{3072}a^{11}+\frac{1}{512}a^{10}-\frac{61}{3072}a^{9}+\frac{39}{512}a^{8}-\frac{161}{2304}a^{7}-\frac{1}{64}a^{6}+\frac{107}{1152}a^{5}-\frac{13}{64}a^{4}+\frac{13}{36}a^{3}-\frac{1}{4}a^{2}+\frac{13}{144}a+\frac{3}{8}$, $\frac{3}{262144}a^{31}+\frac{17}{2359296}a^{30}+\frac{7}{589824}a^{29}-\frac{1}{589824}a^{28}+\frac{13}{294912}a^{26}-\frac{1}{73728}a^{25}+\frac{1}{18432}a^{24}+\frac{11}{32768}a^{23}+\frac{35}{98304}a^{22}-\frac{11}{24576}a^{21}+\frac{29}{24576}a^{20}+\frac{3}{4096}a^{19}+\frac{19}{18432}a^{18}+\frac{11}{4608}a^{17}+\frac{11}{9216}a^{16}+\frac{35}{16384}a^{15}+\frac{913}{147456}a^{14}+\frac{71}{36864}a^{13}+\frac{607}{36864}a^{12}+\frac{7}{256}a^{11}+\frac{103}{6144}a^{10}-\frac{19}{1536}a^{9}+\frac{25}{384}a^{8}-\frac{9}{256}a^{7}+\frac{301}{2304}a^{6}+\frac{17}{576}a^{5}-\frac{23}{576}a^{4}+\frac{1}{2}a^{3}-\frac{43}{288}a^{2}-\frac{5}{72}a+\frac{37}{36}$ 1/147456*a^31 - 5/294912*a^30 - 5/147456*a^27 - 5/73728*a^26 + 1/6144*a^23 + 1/12288*a^22 + 11/18432*a^19 - 7/9216*a^18 + 5/9216*a^15 + 11/18432*a^14 - 23/3072*a^11 + 25/1536*a^10 - 7/576*a^7 - 25/288*a^6 - 7/144*a^3 + 11/72*a^2 + 1, 5/294912*a^31 - 13/1179648*a^30 + 11/589824*a^29 - 1/65536*a^28 + 5/73728*a^27 - 5/36864*a^26 + 5/36864*a^25 + 1/8192*a^24 - 1/12288*a^23 - 7/49152*a^22 + 17/24576*a^21 + 7/8192*a^20 + 7/9216*a^19 - 1/18432*a^18 + 5/9216*a^17 - 11/18432*a^15 - 173/73728*a^14 + 235/36864*a^13 - 1/4096*a^12 + 23/1536*a^11 - 5/768*a^10 + 11/768*a^9 + 9/512*a^8 + 25/288*a^7 - 155/1152*a^6 + 91/576*a^5 + 3/64*a^4 - 1/36*a^3 - 5/72*a^2 + 25/36*a + 9/8, 41/2359296*a^31 + 5/786432*a^30 + 7/589824*a^29 + 23/589824*a^28 - 5/73728*a^27 + 1/98304*a^26 - 1/73728*a^25 - 5/18432*a^24 + 11/98304*a^23 - 1/32768*a^22 - 11/24576*a^21 + 5/24576*a^20 + 53/36864*a^19 + 7/6144*a^18 + 11/4608*a^17 + 35/9216*a^16 - 119/147456*a^15 + 133/49152*a^14 + 71/36864*a^13 - 713/36864*a^12 + 7/1536*a^11 + 3/2048*a^10 - 19/1536*a^9 - 5/384*a^8 + 97/2304*a^7 + 73/768*a^6 + 17/576*a^5 + 97/576*a^4 + 1/36*a^3 + 17/96*a^2 + 31/72*a - 23/36, 35/2359296*a^31 + 7/786432*a^30 + 7/393216*a^29 + 11/147456*a^28 + 13/294912*a^27 + 11/98304*a^26 + 5/49152*a^25 + 13/73728*a^24 + 41/98304*a^23 + 5/32768*a^22 + 5/16384*a^21 + 5/6144*a^20 + 7/4608*a^19 + 5/6144*a^18 + 1/1536*a^17 + 65/9216*a^16 - 221/147456*a^15 + 263/49152*a^14 + 71/24576*a^13 + 19/9216*a^12 + 55/6144*a^11 + 33/2048*a^10 - 1/1024*a^9 + 79/1536*a^8 - 23/2304*a^7 + 47/768*a^6 - 1/384*a^5 + 47/288*a^4 - 61/288*a^3 + 7/96*a^2 + 13/48*a - 61/72, 7/294912*a^31 - 7/1179648*a^30 - 5/294912*a^29 + 1/65536*a^28 + 5/147456*a^27 + 5/73728*a^26 - 5/73728*a^25 - 1/8192*a^24 + 1/12288*a^23 + 11/49152*a^22 + 1/12288*a^21 - 7/8192*a^20 + 25/18432*a^19 - 13/18432*a^18 - 7/9216*a^17 - 1/18432*a^15 + 217/73728*a^14 + 11/18432*a^13 + 1/4096*a^12 + 23/3072*a^11 + 35/1536*a^10 + 25/1536*a^9 - 9/512*a^8 + 43/576*a^7 + 55/1152*a^6 - 25/288*a^5 - 3/64*a^4 - 11/144*a^3 + 2/9*a^2 - 25/72*a - 1/8, 41/2359296*a^31 + 1/262144*a^30 - 7/589824*a^29 - 1/32768*a^28 - 5/73728*a^27 + 3/32768*a^26 + 1/73728*a^25 - 1/8192*a^24 + 11/98304*a^23 - 7/32768*a^22 + 11/24576*a^21 - 1/4096*a^20 + 53/36864*a^19 - 1/1024*a^18 - 11/4608*a^17 - 3/1024*a^16 - 119/147456*a^15 + 193/16384*a^14 - 71/36864*a^13 - 17/2048*a^12 + 7/1536*a^11 - 21/2048*a^10 + 19/1536*a^9 - 1/512*a^8 + 97/2304*a^7 - 15/256*a^6 - 17/576*a^5 - 1/16*a^4 + 1/36*a^3 + 15/32*a^2 - 31/72*a - 3/8, 25/589824*a^30 - 1/65536*a^28 - 11/147456*a^26 - 1/8192*a^24 + 19/24576*a^22 + 7/8192*a^20 - 1/18432*a^18 - 1/512*a^16 + 233/36864*a^14 + 63/4096*a^12 + 151/3072*a^10 - 25/512*a^8 - 23/288*a^6 + 11/64*a^4 + 71/144*a^2 - 1/8, 17/589824*a^30 + 17/196608*a^28 + 17/147456*a^26 + 1/24576*a^24 + 11/24576*a^22 + 3/8192*a^20 - 5/18432*a^18 + 1/1536*a^16 - 95/36864*a^14 - 47/12288*a^12 + 107/3072*a^10 + 19/512*a^8 + 1/144*a^6 + 37/192*a^4 + 31/144*a^2 - 7/24, 31/1179648*a^31 - 1/65536*a^30 - 17/1179648*a^29 - 1/65536*a^28 - 5/294912*a^27 + 1/8192*a^26 + 5/147456*a^25 + 1/8192*a^24 - 11/49152*a^23 - 1/8192*a^22 - 35/49152*a^21 + 7/8192*a^20 + 65/36864*a^19 - 5/4608*a^17 - 241/73728*a^15 + 31/4096*a^14 - 337/73728*a^13 - 1/4096*a^12 + 121/6144*a^11 - 7/512*a^10 - 1/3072*a^9 + 9/512*a^8 + 19/576*a^7 - 1/32*a^6 - 13/1152*a^5 + 3/64*a^4 - 43/288*a^3 + 1/2*a^2 - 155/144*a + 5/8, 1/786432*a^31 - 5/262144*a^30 - 7/393216*a^29 - 1/65536*a^28 + 5/98304*a^27 + 1/32768*a^26 - 11/49152*a^25 - 1/4096*a^24 + 3/32768*a^23 + 3/32768*a^22 - 5/16384*a^21 - 9/8192*a^20 - 1/6144*a^19 + 1/1024*a^18 - 5/3072*a^17 - 3/1024*a^16 + 65/49152*a^15 - 69/16384*a^14 - 263/24576*a^13 - 33/4096*a^12 + 15/2048*a^11 - 7/2048*a^10 - 33/1024*a^9 - 5/256*a^8 + 83/768*a^7 + 7/256*a^6 - 95/384*a^5 - 7/64*a^4 + 19/96*a^3 + 1/32*a^2 - 19/48*a - 1, 19/2359296*a^31 + 7/786432*a^30 + 19/1179648*a^29 + 1/294912*a^28 + 5/294912*a^27 + 11/98304*a^26 + 23/147456*a^25 + 19/73728*a^24 + 25/98304*a^23 + 5/32768*a^22 - 23/49152*a^21 - 5/12288*a^20 + 1/2304*a^19 + 5/6144*a^18 + 17/9216*a^17 + 5/9216*a^16 + 275/147456*a^15 + 263/49152*a^14 + 275/73728*a^13 + 113/18432*a^12 - 1/6144*a^11 + 33/2048*a^10 + 5/3072*a^9 + 49/1536*a^8 + 221/2304*a^7 + 143/768*a^6 + 203/1152*a^5 + 1/9*a^4 + 43/288*a^3 + 31/96*a^2 + 7/144*a + 5/72, 17/2359296*a^31 - 1/131072*a^30 + 19/1179648*a^29 + 1/16384*a^28 + 1/73728*a^27 + 1/8192*a^26 - 13/147456*a^25 + 1/4096*a^24 - 13/98304*a^23 + 7/16384*a^22 - 23/49152*a^21 + 1/2048*a^20 - 43/36864*a^19 + 1/2048*a^18 - 1/9216*a^17 + 3/512*a^16 + 49/147456*a^15 + 95/8192*a^14 + 275/73728*a^13 + 17/1024*a^12 + 1/1536*a^11 - 3/512*a^10 - 7/3072*a^9 + 1/256*a^8 - 77/2304*a^7 - 1/128*a^6 - 13/1152*a^5 + 1/8*a^4 - 17/144*a^3 + 5/8*a^2 + 7/144*a + 5/4, 5/2359296*a^31 - 17/2359296*a^30 - 1/32768*a^29 - 1/589824*a^28 - 5/73728*a^27 - 13/294912*a^26 + 1/8192*a^25 + 1/18432*a^24 - 1/98304*a^23 - 35/98304*a^22 - 1/4096*a^21 + 29/24576*a^20 + 17/36864*a^19 - 19/18432*a^18 - 1/1024*a^17 + 11/9216*a^16 - 155/147456*a^15 - 913/147456*a^14 + 15/2048*a^13 + 607/36864*a^12 - 17/1536*a^11 - 103/6144*a^10 - 31/512*a^9 + 25/384*a^8 + 25/2304*a^7 - 301/2304*a^6 + 1/16*a^5 - 23/576*a^4 + 1/36*a^3 + 43/288*a^2 - 3/8*a + 37/36, 65/2359296*a^31 - 29/1179648*a^29 - 1/65536*a^28 - 13/147456*a^27 + 1/8192*a^26 + 17/147456*a^25 - 1/8192*a^24 + 35/98304*a^23 - 1/1024*a^22 + 25/49152*a^21 + 7/8192*a^20 + 23/36864*a^19 + 1/1024*a^18 - 17/4608*a^17 - 1/512*a^16 + 289/147456*a^15 + 1187/73728*a^13 - 65/4096*a^12 + 29/3072*a^11 + 1/512*a^10 - 61/3072*a^9 + 39/512*a^8 - 161/2304*a^7 - 1/64*a^6 + 107/1152*a^5 - 13/64*a^4 + 13/36*a^3 - 1/4*a^2 + 13/144*a + 3/8, 3/262144*a^31 + 17/2359296*a^30 + 7/589824*a^29 - 1/589824*a^28 + 13/294912*a^26 - 1/73728*a^25 + 1/18432*a^24 + 11/32768*a^23 + 35/98304*a^22 - 11/24576*a^21 + 29/24576*a^20 + 3/4096*a^19 + 19/18432*a^18 + 11/4608*a^17 + 11/9216*a^16 + 35/16384*a^15 + 913/147456*a^14 + 71/36864*a^13 + 607/36864*a^12 + 7/256*a^11 + 103/6144*a^10 - 19/1536*a^9 + 25/384*a^8 - 9/256*a^7 + 301/2304*a^6 + 17/576*a^5 - 23/576*a^4 + 1/2*a^3 - 43/288*a^2 - 5/72*a + 37/36 (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:		\( 955272330542.1476 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 955272330542.1476 \cdot 48}{8\cdot\sqrt{351509915919708142388734226325389286150171856470016}}\cr\approx \mathstrut & 1.80379655157594 \end{aligned}\] (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 $16$	$16$	$1$	$38$
\(17\) 17	$\Q_{17}$	$x + 14$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{17}$	$x + 14$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{17}$	$x + 14$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{17}$	$x + 14$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{17}$	$x + 14$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{17}$	$x + 14$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{17}$	$x + 14$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{17}$	$x + 14$	$1$	$1$	$0$	Trivial	$[\ ]$
	17.2.0.1	$x^{2} + 16 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	17.2.0.1	$x^{2} + 16 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	17.2.0.1	$x^{2} + 16 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	17.2.0.1	$x^{2} + 16 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	17.4.3.4	$x^{4} + 102$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
	17.4.3.4	$x^{4} + 102$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
	17.4.3.4	$x^{4} + 102$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
	17.4.3.4	$x^{4} + 102$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
\(41\) 41	41.4.0.1	$x^{4} + 23 x + 6$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	41.4.2.1	$x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	41.4.2.1	$x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	41.4.0.1	$x^{4} + 23 x + 6$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	41.4.0.1	$x^{4} + 23 x + 6$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	41.4.2.1	$x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	41.4.2.1	$x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	41.4.0.1	$x^{4} + 23 x + 6$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$