Normalized defining polynomial
\( x^{32} - x^{30} - 3 x^{28} + 20 x^{26} - 12 x^{24} - 64 x^{22} + 160 x^{20} + 64 x^{18} - 704 x^{16} + 256 x^{14} + 2560 x^{12} - 4096 x^{10} - 3072 x^{8} + 20480 x^{6} - 12288 x^{4} + \cdots + 65536 \)
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: | \(334285232879825411498534254159851729760457815228416\) \(\medspace = 2^{44}\cdot 3^{16}\cdot 5077^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(37.92\) | 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: | $2^{19/12}3^{1/2}5077^{1/2}\approx 369.8239522562059$ | ||
Ramified primes: | \(2\), \(3\), \(5077\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}+\frac{1}{4}a^{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{5}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{8}+\frac{1}{8}a^{6}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{9}+\frac{1}{8}a^{7}$, $\frac{1}{16}a^{12}-\frac{1}{16}a^{10}+\frac{1}{16}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{16}a^{13}-\frac{1}{16}a^{11}+\frac{1}{16}a^{9}-\frac{1}{2}a^{5}$, $\frac{1}{32}a^{14}-\frac{1}{32}a^{12}+\frac{1}{32}a^{10}-\frac{1}{4}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{32}a^{15}-\frac{1}{32}a^{13}+\frac{1}{32}a^{11}-\frac{1}{4}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{64}a^{16}-\frac{1}{64}a^{14}+\frac{1}{64}a^{12}-\frac{1}{8}a^{8}-\frac{1}{4}a^{4}$, $\frac{1}{128}a^{17}+\frac{1}{128}a^{15}-\frac{1}{128}a^{13}-\frac{3}{64}a^{11}-\frac{1}{8}a^{9}-\frac{1}{16}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{256}a^{18}+\frac{1}{256}a^{16}-\frac{1}{256}a^{14}-\frac{1}{32}a^{13}-\frac{3}{128}a^{12}-\frac{1}{32}a^{11}-\frac{1}{16}a^{10}+\frac{1}{32}a^{9}-\frac{1}{32}a^{8}-\frac{1}{16}a^{7}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}+\frac{3}{8}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{256}a^{19}-\frac{1}{256}a^{17}-\frac{3}{256}a^{15}-\frac{1}{64}a^{13}-\frac{1}{64}a^{11}+\frac{3}{32}a^{9}-\frac{1}{16}a^{7}-\frac{3}{8}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{512}a^{20}-\frac{1}{512}a^{18}-\frac{3}{512}a^{16}-\frac{1}{64}a^{15}-\frac{1}{128}a^{14}+\frac{1}{64}a^{13}-\frac{1}{128}a^{12}+\frac{3}{64}a^{11}+\frac{3}{64}a^{10}-\frac{1}{16}a^{9}-\frac{1}{32}a^{8}-\frac{1}{16}a^{7}-\frac{3}{16}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{512}a^{21}-\frac{1}{512}a^{19}+\frac{1}{512}a^{17}-\frac{1}{64}a^{13}+\frac{3}{32}a^{9}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{1024}a^{22}-\frac{1}{1024}a^{20}+\frac{1}{1024}a^{18}-\frac{1}{64}a^{15}-\frac{1}{128}a^{14}+\frac{1}{64}a^{13}-\frac{1}{64}a^{11}+\frac{3}{64}a^{10}+\frac{1}{8}a^{7}-\frac{1}{8}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{1024}a^{23}-\frac{1}{1024}a^{21}+\frac{1}{1024}a^{19}-\frac{1}{128}a^{15}+\frac{3}{64}a^{11}-\frac{1}{8}a^{7}-\frac{1}{4}a^{5}+\frac{1}{4}a^{3}$, $\frac{1}{2048}a^{24}-\frac{1}{2048}a^{22}+\frac{1}{2048}a^{20}-\frac{1}{256}a^{16}-\frac{1}{64}a^{15}+\frac{1}{64}a^{13}+\frac{3}{128}a^{12}+\frac{3}{64}a^{11}+\frac{1}{16}a^{9}-\frac{1}{16}a^{8}-\frac{3}{16}a^{7}-\frac{1}{8}a^{6}-\frac{1}{8}a^{5}+\frac{1}{8}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2048}a^{25}-\frac{1}{2048}a^{23}+\frac{1}{2048}a^{21}-\frac{1}{256}a^{17}+\frac{3}{128}a^{13}-\frac{1}{16}a^{9}-\frac{1}{8}a^{7}+\frac{1}{8}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4096}a^{26}-\frac{1}{4096}a^{24}+\frac{1}{4096}a^{22}-\frac{1}{512}a^{18}-\frac{1}{64}a^{15}+\frac{3}{256}a^{14}-\frac{1}{64}a^{13}+\frac{1}{64}a^{11}-\frac{1}{32}a^{10}-\frac{1}{32}a^{9}-\frac{1}{16}a^{8}-\frac{1}{8}a^{7}+\frac{1}{16}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a$, $\frac{1}{4096}a^{27}-\frac{1}{4096}a^{25}+\frac{1}{4096}a^{23}-\frac{1}{512}a^{19}+\frac{3}{256}a^{15}-\frac{1}{32}a^{11}-\frac{1}{16}a^{9}+\frac{1}{16}a^{7}-\frac{1}{4}a^{5}$, $\frac{1}{24576}a^{28}-\frac{1}{8192}a^{26}-\frac{1}{24576}a^{24}+\frac{5}{12288}a^{22}-\frac{1}{6144}a^{20}-\frac{1}{1024}a^{18}+\frac{1}{192}a^{16}+\frac{1}{128}a^{14}-\frac{1}{384}a^{12}-\frac{1}{32}a^{10}-\frac{1}{8}a^{9}+\frac{11}{96}a^{8}-\frac{1}{8}a^{7}+\frac{1}{6}a^{6}+\frac{1}{8}a^{5}-\frac{1}{6}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{3}$, $\frac{1}{24576}a^{29}-\frac{1}{8192}a^{27}-\frac{1}{24576}a^{25}+\frac{5}{12288}a^{23}-\frac{1}{6144}a^{21}-\frac{1}{1024}a^{19}-\frac{1}{384}a^{17}+\frac{1}{192}a^{13}+\frac{1}{64}a^{11}-\frac{1}{96}a^{9}-\frac{1}{48}a^{7}+\frac{1}{3}a^{5}-\frac{1}{2}a^{3}+\frac{1}{6}a$, $\frac{1}{147456}a^{30}+\frac{1}{147456}a^{28}-\frac{13}{147456}a^{26}+\frac{5}{24576}a^{24}+\frac{5}{12288}a^{22}-\frac{1}{2304}a^{20}+\frac{5}{9216}a^{18}+\frac{1}{288}a^{16}-\frac{19}{2304}a^{14}-\frac{35}{1152}a^{12}-\frac{11}{288}a^{10}-\frac{1}{24}a^{8}-\frac{1}{4}a^{7}+\frac{1}{6}a^{6}+\frac{1}{4}a^{5}-\frac{13}{36}a^{4}-\frac{1}{4}a^{3}+\frac{1}{36}a^{2}+\frac{4}{9}$, $\frac{1}{294912}a^{31}-\frac{5}{294912}a^{29}-\frac{31}{294912}a^{27}+\frac{1}{8192}a^{23}-\frac{7}{18432}a^{21}-\frac{1}{4608}a^{19}-\frac{1}{1152}a^{17}+\frac{17}{4608}a^{15}+\frac{1}{576}a^{13}-\frac{5}{144}a^{11}+\frac{3}{32}a^{9}+\frac{5}{32}a^{7}+\frac{5}{18}a^{5}-\frac{17}{72}a^{3}+\frac{7}{18}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{4}\times C_{4}$, which has order $16$ (assuming GRH)
Unit group
Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{7}{147456} a^{31} - \frac{35}{147456} a^{29} + \frac{35}{147456} a^{27} + \frac{5}{4096} a^{25} - \frac{7}{2048} a^{23} + \frac{7}{4608} a^{21} + \frac{49}{4608} a^{19} - \frac{7}{576} a^{17} - \frac{35}{1152} a^{15} + \frac{91}{1152} a^{13} + \frac{35}{576} a^{11} - \frac{7}{16} a^{9} + \frac{7}{16} a^{7} + \frac{7}{18} a^{5} - \frac{14}{9} a^{3} + \frac{35}{18} a \) (order $12$) | 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: | $\frac{37}{294912}a^{31}-\frac{1}{49152}a^{30}-\frac{137}{294912}a^{29}-\frac{7}{49152}a^{28}+\frac{149}{294912}a^{27}+\frac{43}{49152}a^{26}+\frac{7}{3072}a^{25}-\frac{5}{4096}a^{24}-\frac{173}{24576}a^{23}-\frac{7}{4096}a^{22}+\frac{49}{9216}a^{21}+\frac{7}{768}a^{20}+\frac{169}{9216}a^{19}-\frac{7}{1536}a^{18}-\frac{151}{4608}a^{17}-\frac{5}{192}a^{16}-\frac{217}{4608}a^{15}+\frac{19}{768}a^{14}+\frac{47}{288}a^{13}+\frac{19}{192}a^{12}+\frac{43}{576}a^{11}-\frac{25}{96}a^{10}-\frac{85}{96}a^{9}-\frac{1}{8}a^{8}+\frac{127}{96}a^{7}+\frac{17}{16}a^{6}+\frac{11}{18}a^{5}-\frac{17}{12}a^{4}-\frac{269}{72}a^{3}-\frac{1}{12}a^{2}+\frac{50}{9}a+\frac{5}{3}$, $\frac{127}{294912}a^{31}-\frac{7}{147456}a^{30}-\frac{335}{294912}a^{29}-\frac{31}{147456}a^{28}+\frac{347}{294912}a^{27}+\frac{127}{147456}a^{26}+\frac{143}{24576}a^{25}-\frac{85}{24576}a^{24}-\frac{401}{24576}a^{23}+\frac{11}{3072}a^{22}+\frac{85}{9216}a^{21}+\frac{59}{18432}a^{20}+\frac{367}{9216}a^{19}-\frac{179}{9216}a^{18}-\frac{277}{4608}a^{17}+\frac{29}{1152}a^{16}-\frac{631}{4608}a^{15}+\frac{43}{2304}a^{14}+\frac{101}{288}a^{13}+\frac{5}{1152}a^{12}+\frac{71}{288}a^{11}-\frac{19}{72}a^{10}-\frac{49}{24}a^{9}+\frac{59}{96}a^{8}+\frac{313}{96}a^{7}-\frac{7}{48}a^{6}+\frac{49}{36}a^{5}-\frac{55}{18}a^{4}-\frac{521}{72}a^{3}+\frac{50}{9}a^{2}+\frac{104}{9}a-\frac{70}{9}$, $\frac{11}{98304}a^{31}-\frac{7}{98304}a^{29}-\frac{53}{98304}a^{27}+\frac{3}{2048}a^{25}-\frac{7}{8192}a^{23}-\frac{19}{3072}a^{21}+\frac{5}{768}a^{19}+\frac{25}{1536}a^{17}-\frac{53}{1536}a^{15}-\frac{7}{192}a^{13}+\frac{47}{192}a^{11}-\frac{1}{8}a^{9}-\frac{13}{32}a^{7}+\frac{25}{24}a^{5}-\frac{13}{24}a^{3}-\frac{7}{6}a$, $\frac{49}{98304}a^{31}+\frac{11}{49152}a^{30}-\frac{149}{98304}a^{29}-\frac{31}{49152}a^{28}+\frac{137}{98304}a^{27}+\frac{7}{49152}a^{26}+\frac{33}{4096}a^{25}+\frac{21}{4096}a^{24}-\frac{191}{8192}a^{23}-\frac{27}{2048}a^{22}+\frac{5}{384}a^{21}+\frac{35}{6144}a^{20}+\frac{193}{3072}a^{19}+\frac{121}{3072}a^{18}-\frac{151}{1536}a^{17}-\frac{91}{1536}a^{16}-\frac{277}{1536}a^{15}-\frac{71}{768}a^{14}+\frac{197}{384}a^{13}+\frac{107}{384}a^{12}+\frac{67}{192}a^{11}+\frac{23}{96}a^{10}-3a^{9}-\frac{27}{16}a^{8}+\frac{139}{32}a^{7}+\frac{33}{16}a^{6}+\frac{31}{12}a^{5}+\frac{31}{12}a^{4}-\frac{305}{24}a^{3}-\frac{28}{3}a^{2}+\frac{103}{6}a+\frac{32}{3}$, $\frac{1}{294912}a^{31}-\frac{29}{147456}a^{30}+\frac{7}{294912}a^{29}+\frac{73}{147456}a^{28}+\frac{5}{294912}a^{27}-\frac{73}{147456}a^{26}-\frac{7}{24576}a^{25}-\frac{11}{4096}a^{24}+\frac{19}{24576}a^{23}+\frac{1}{128}a^{22}-\frac{5}{9216}a^{21}-\frac{71}{18432}a^{20}-\frac{29}{9216}a^{19}-\frac{181}{9216}a^{18}+\frac{5}{1152}a^{17}+\frac{71}{2304}a^{16}+\frac{35}{4608}a^{15}+\frac{73}{1152}a^{14}-\frac{19}{1152}a^{13}-\frac{215}{1152}a^{12}-\frac{11}{576}a^{11}-\frac{73}{576}a^{10}+\frac{17}{96}a^{9}+a^{8}-\frac{5}{96}a^{7}-\frac{13}{8}a^{6}-\frac{19}{72}a^{5}-\frac{71}{72}a^{4}+\frac{55}{72}a^{3}+\frac{71}{18}a^{2}-\frac{4}{9}a-\frac{50}{9}$, $\frac{23}{294912}a^{31}-\frac{7}{36864}a^{30}-\frac{79}{294912}a^{29}+\frac{5}{36864}a^{28}+\frac{43}{294912}a^{27}+\frac{19}{36864}a^{26}+\frac{11}{8192}a^{25}-\frac{19}{6144}a^{24}-\frac{27}{8192}a^{23}+\frac{5}{3072}a^{22}-\frac{53}{18432}a^{21}+\frac{55}{9216}a^{20}+\frac{125}{9216}a^{19}-\frac{77}{9216}a^{18}+\frac{1}{288}a^{17}-\frac{49}{4608}a^{16}-\frac{293}{4608}a^{15}+\frac{25}{576}a^{14}+\frac{1}{18}a^{13}+\frac{47}{1152}a^{12}+\frac{29}{144}a^{11}-\frac{35}{144}a^{10}-\frac{7}{16}a^{9}+\frac{29}{96}a^{8}+\frac{1}{32}a^{7}-\frac{7}{48}a^{6}+\frac{73}{72}a^{5}-\frac{35}{36}a^{4}-\frac{31}{72}a^{3}+\frac{11}{9}a^{2}-\frac{14}{9}a-\frac{46}{9}$, $\frac{25}{294912}a^{31}+\frac{1}{49152}a^{30}-\frac{53}{294912}a^{29}-\frac{1}{16384}a^{28}+\frac{89}{294912}a^{27}-\frac{1}{49152}a^{26}+\frac{1}{2048}a^{25}-\frac{7}{24576}a^{24}-\frac{9}{8192}a^{23}+\frac{17}{12288}a^{22}+\frac{1}{576}a^{21}-\frac{7}{4608}a^{19}-\frac{19}{3072}a^{18}-\frac{1}{4608}a^{17}+\frac{5}{512}a^{16}-\frac{25}{4608}a^{15}+\frac{5}{768}a^{14}+\frac{7}{576}a^{13}-\frac{1}{64}a^{12}-\frac{7}{288}a^{11}-\frac{13}{192}a^{10}-\frac{1}{16}a^{9}+\frac{17}{96}a^{8}+\frac{15}{32}a^{7}+\frac{5}{48}a^{6}-\frac{49}{72}a^{5}-a^{4}+\frac{43}{72}a^{3}+\frac{5}{6}a^{2}+\frac{31}{18}a$, $\frac{127}{294912}a^{31}+\frac{7}{147456}a^{30}-\frac{335}{294912}a^{29}+\frac{31}{147456}a^{28}+\frac{347}{294912}a^{27}-\frac{127}{147456}a^{26}+\frac{143}{24576}a^{25}+\frac{85}{24576}a^{24}-\frac{401}{24576}a^{23}-\frac{11}{3072}a^{22}+\frac{85}{9216}a^{21}-\frac{59}{18432}a^{20}+\frac{367}{9216}a^{19}+\frac{179}{9216}a^{18}-\frac{277}{4608}a^{17}-\frac{29}{1152}a^{16}-\frac{631}{4608}a^{15}-\frac{43}{2304}a^{14}+\frac{101}{288}a^{13}-\frac{5}{1152}a^{12}+\frac{71}{288}a^{11}+\frac{19}{72}a^{10}-\frac{49}{24}a^{9}-\frac{59}{96}a^{8}+\frac{313}{96}a^{7}+\frac{7}{48}a^{6}+\frac{49}{36}a^{5}+\frac{55}{18}a^{4}-\frac{521}{72}a^{3}-\frac{50}{9}a^{2}+\frac{104}{9}a+\frac{70}{9}$, $\frac{1}{9216}a^{31}+\frac{1}{49152}a^{30}-\frac{1}{4608}a^{29}-\frac{5}{49152}a^{28}+\frac{1}{18432}a^{27}+\frac{5}{49152}a^{26}+\frac{13}{6144}a^{25}-\frac{1}{4096}a^{24}-\frac{19}{6144}a^{23}-\frac{1}{512}a^{22}-\frac{25}{9216}a^{21}+\frac{7}{6144}a^{20}+\frac{29}{2304}a^{19}+\frac{17}{3072}a^{18}-\frac{1}{576}a^{17}-\frac{7}{768}a^{16}-\frac{133}{2304}a^{15}-\frac{5}{384}a^{14}+\frac{5}{144}a^{13}+\frac{31}{384}a^{12}+\frac{29}{144}a^{11}+\frac{5}{192}a^{10}-\frac{19}{48}a^{9}-\frac{1}{4}a^{8}+\frac{5}{24}a^{7}+\frac{1}{4}a^{6}+\frac{19}{18}a^{5}+\frac{7}{24}a^{4}-\frac{29}{36}a^{3}-\frac{7}{6}a^{2}+\frac{16}{9}a-\frac{5}{3}$, $\frac{7}{147456}a^{31}-\frac{5}{49152}a^{30}-\frac{23}{147456}a^{29}+\frac{13}{49152}a^{28}-\frac{1}{147456}a^{27}-\frac{13}{49152}a^{26}+\frac{7}{6144}a^{25}-\frac{1}{2048}a^{24}-\frac{1}{384}a^{23}+\frac{3}{2048}a^{22}+\frac{11}{9216}a^{21}+\frac{1}{1536}a^{20}+\frac{5}{576}a^{19}-\frac{7}{3072}a^{18}-\frac{11}{1152}a^{17}-\frac{1}{192}a^{16}-\frac{13}{576}a^{15}+\frac{13}{384}a^{14}+\frac{29}{576}a^{13}-\frac{1}{48}a^{12}+\frac{11}{144}a^{11}-\frac{13}{192}a^{10}-\frac{35}{96}a^{9}+\frac{3}{16}a^{8}+\frac{1}{3}a^{7}-\frac{1}{4}a^{6}+\frac{5}{9}a^{5}+\frac{1}{6}a^{4}-\frac{37}{18}a^{3}-\frac{2}{3}a^{2}+\frac{16}{9}a+\frac{1}{3}$, $\frac{5}{16384}a^{31}-\frac{23}{73728}a^{30}-\frac{9}{16384}a^{29}+\frac{47}{36864}a^{28}+\frac{9}{16384}a^{27}-\frac{35}{36864}a^{26}+\frac{17}{4096}a^{25}-\frac{107}{24576}a^{24}-\frac{5}{512}a^{23}+\frac{43}{3072}a^{22}+\frac{11}{2048}a^{21}-\frac{25}{4608}a^{20}+\frac{3}{128}a^{19}-\frac{365}{9216}a^{18}-\frac{11}{256}a^{17}+\frac{191}{4608}a^{16}-\frac{9}{128}a^{15}+\frac{79}{576}a^{14}+\frac{25}{128}a^{13}-\frac{185}{576}a^{12}+\frac{9}{64}a^{11}-\frac{167}{576}a^{10}-\frac{5}{4}a^{9}+\frac{169}{96}a^{8}+\frac{35}{16}a^{7}-\frac{55}{24}a^{6}+\frac{11}{8}a^{5}-\frac{23}{36}a^{4}-\frac{11}{2}a^{3}+\frac{233}{36}a^{2}+11a-\frac{49}{9}$, $\frac{43}{294912}a^{31}+\frac{41}{147456}a^{30}-\frac{143}{294912}a^{29}-\frac{79}{147456}a^{28}-\frac{37}{294912}a^{27}-\frac{137}{147456}a^{26}+\frac{9}{2048}a^{25}+\frac{61}{8192}a^{24}-\frac{79}{8192}a^{23}-\frac{25}{2048}a^{22}-\frac{13}{18432}a^{21}-\frac{97}{18432}a^{20}+\frac{319}{9216}a^{19}+\frac{7}{144}a^{18}-\frac{145}{4608}a^{17}-\frac{83}{2304}a^{16}-\frac{457}{4608}a^{15}-\frac{311}{2304}a^{14}+\frac{97}{576}a^{13}+\frac{173}{1152}a^{12}+\frac{193}{576}a^{11}+\frac{349}{576}a^{10}-\frac{41}{32}a^{9}-\frac{55}{32}a^{8}+\frac{27}{32}a^{7}+\frac{15}{16}a^{6}+\frac{97}{36}a^{5}+\frac{163}{36}a^{4}-\frac{461}{72}a^{3}-\frac{373}{36}a^{2}+\frac{47}{9}a+\frac{98}{9}$, $\frac{13}{73728}a^{31}-\frac{7}{147456}a^{30}-\frac{23}{73728}a^{29}+\frac{35}{147456}a^{28}+\frac{29}{73728}a^{27}-\frac{71}{147456}a^{26}+\frac{5}{3072}a^{25}-\frac{1}{512}a^{24}-\frac{47}{12288}a^{23}+\frac{29}{4096}a^{22}+\frac{31}{9216}a^{21}-\frac{17}{2304}a^{20}+\frac{5}{1152}a^{19}-\frac{179}{9216}a^{18}-\frac{61}{4608}a^{17}+\frac{227}{4608}a^{16}-\frac{31}{1152}a^{15}+\frac{61}{2304}a^{14}+\frac{5}{72}a^{13}-\frac{181}{1152}a^{12}+\frac{31}{576}a^{11}+\frac{7}{144}a^{10}-\frac{19}{48}a^{9}+\frac{29}{32}a^{8}+\frac{55}{48}a^{7}-\frac{11}{8}a^{6}-\frac{1}{72}a^{5}-\frac{8}{9}a^{4}-\frac{37}{36}a^{3}+\frac{173}{36}a^{2}+\frac{91}{18}a-\frac{58}{9}$, $\frac{7}{49152}a^{31}+\frac{11}{147456}a^{30}-\frac{17}{49152}a^{29}-\frac{19}{147456}a^{28}+\frac{5}{49152}a^{27}-\frac{17}{147456}a^{26}+\frac{15}{8192}a^{25}+\frac{3}{4096}a^{24}-\frac{17}{4096}a^{23}-\frac{3}{4096}a^{22}-\frac{5}{6144}a^{21}-\frac{41}{9216}a^{20}+\frac{1}{96}a^{19}+\frac{37}{9216}a^{18}-\frac{5}{1536}a^{17}+\frac{37}{2304}a^{16}-\frac{37}{768}a^{15}-\frac{101}{2304}a^{14}+\frac{11}{192}a^{13}+\frac{1}{144}a^{12}+\frac{19}{96}a^{11}+\frac{109}{576}a^{10}-\frac{17}{32}a^{9}-\frac{3}{16}a^{8}+\frac{1}{2}a^{7}-\frac{1}{16}a^{6}+\frac{19}{24}a^{5}+\frac{11}{18}a^{4}-\frac{23}{12}a^{3}-\frac{4}{9}a^{2}+\frac{4}{3}a-\frac{13}{9}$, $\frac{5}{32768}a^{31}+\frac{1}{16384}a^{30}-\frac{23}{98304}a^{29}-\frac{31}{49152}a^{28}+\frac{9}{32768}a^{27}+\frac{17}{16384}a^{26}+\frac{65}{24576}a^{25}+\frac{1}{3072}a^{24}-\frac{125}{24576}a^{23}-\frac{97}{12288}a^{22}+\frac{5}{1536}a^{21}+\frac{17}{1536}a^{20}+\frac{7}{512}a^{19}+\frac{13}{1024}a^{18}-\frac{5}{192}a^{17}-\frac{1}{24}a^{16}-\frac{21}{512}a^{15}-\frac{7}{256}a^{14}+\frac{35}{384}a^{13}+\frac{23}{96}a^{12}+\frac{1}{16}a^{11}-\frac{7}{64}a^{10}-\frac{35}{48}a^{9}-\frac{11}{12}a^{8}+\frac{121}{96}a^{7}+\frac{83}{48}a^{6}+\frac{5}{6}a^{5}-\frac{11}{12}a^{4}-\frac{23}{8}a^{3}-4a^{2}+\frac{23}{3}a+\frac{8}{3}$ (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: | \( 1000974895371.5886 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 1000974895371.5886 \cdot 16}{12\cdot\sqrt{334285232879825411498534254159851729760457815228416}}\cr\approx \mathstrut & 0.430706295531231 \end{aligned}\] (assuming GRH)
Galois group
$C_2^5:S_4$ (as 32T34907):
A solvable group of order 768 |
The 52 conjugacy class representatives for $C_2^5:S_4$ are not computed |
Character table for $C_2^5:S_4$ is not computed |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 32 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.4.0.1}{4} }^{8}$ | ${\href{/padicField/7.4.0.1}{4} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{8}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{16}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.6.0.1}{6} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
Deg $24$ | $12$ | $2$ | $36$ | ||||
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
\(5077\) | $\Q_{5077}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{5077}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5077}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5077}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5077}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5077}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5077}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5077}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5077}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5077}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5077}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5077}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5077}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5077}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5077}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5077}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |