Normalized defining polynomial
\( x^{32} + 3x^{28} + 13x^{24} - 108x^{20} - 384x^{16} - 1728x^{12} + 3328x^{8} + 12288x^{4} + 65536 \)
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: | \(480960519379403029833827263813614000556122650443776\) \(\medspace = 2^{48}\cdot 3^{16}\cdot 13^{8}\cdot 17^{16}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(38.35\) | 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: | $2^{2}3^{1/2}13^{1/2}17^{1/2}\approx 102.99514551666987$ | ||
Ramified primes: | \(2\), \(3\), \(13\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $16$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}+\frac{1}{4}a^{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{9}+\frac{1}{4}a^{5}$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{10}+\frac{1}{8}a^{6}$, $\frac{1}{16}a^{15}-\frac{1}{16}a^{11}-\frac{1}{2}a^{9}+\frac{1}{16}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{96}a^{16}-\frac{1}{8}a^{13}-\frac{3}{32}a^{12}-\frac{1}{4}a^{10}+\frac{1}{8}a^{9}-\frac{7}{96}a^{8}+\frac{1}{4}a^{6}-\frac{1}{8}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{1}{3}$, $\frac{1}{192}a^{17}-\frac{1}{16}a^{14}+\frac{5}{64}a^{13}-\frac{1}{8}a^{11}+\frac{1}{16}a^{10}+\frac{65}{192}a^{9}+\frac{1}{8}a^{7}-\frac{1}{16}a^{6}-\frac{1}{2}a^{5}-\frac{1}{8}a^{3}+\frac{1}{3}a$, $\frac{1}{192}a^{18}-\frac{3}{64}a^{14}-\frac{1}{8}a^{12}-\frac{7}{192}a^{10}-\frac{1}{2}a^{9}+\frac{1}{8}a^{8}-\frac{1}{8}a^{6}-\frac{1}{2}a^{5}-\frac{1}{8}a^{4}-\frac{1}{6}a^{2}-\frac{1}{2}a$, $\frac{1}{384}a^{19}-\frac{3}{128}a^{15}+\frac{1}{16}a^{13}+\frac{89}{384}a^{11}-\frac{1}{4}a^{10}+\frac{7}{16}a^{9}+\frac{3}{16}a^{7}+\frac{1}{4}a^{6}-\frac{7}{16}a^{5}+\frac{1}{6}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{384}a^{20}-\frac{1}{384}a^{16}-\frac{1}{16}a^{14}+\frac{17}{384}a^{12}-\frac{1}{4}a^{11}+\frac{1}{16}a^{10}+\frac{1}{24}a^{8}+\frac{1}{4}a^{7}-\frac{1}{16}a^{6}-\frac{1}{3}a^{4}-\frac{1}{4}a^{3}+\frac{1}{3}$, $\frac{1}{768}a^{21}-\frac{1}{768}a^{17}-\frac{1}{32}a^{15}+\frac{17}{768}a^{13}-\frac{1}{8}a^{12}+\frac{1}{32}a^{11}-\frac{23}{48}a^{9}+\frac{1}{8}a^{8}+\frac{15}{32}a^{7}+\frac{1}{3}a^{5}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}+\frac{1}{6}a$, $\frac{1}{768}a^{22}-\frac{1}{768}a^{18}+\frac{17}{768}a^{14}-\frac{11}{48}a^{10}+\frac{1}{12}a^{6}+\frac{5}{12}a^{2}$, $\frac{1}{1536}a^{23}-\frac{1}{1536}a^{19}+\frac{17}{1536}a^{15}-\frac{11}{96}a^{11}-\frac{11}{24}a^{7}-\frac{7}{24}a^{3}$, $\frac{1}{9216}a^{24}-\frac{1}{1024}a^{20}-\frac{13}{3072}a^{16}-\frac{1}{16}a^{14}-\frac{7}{128}a^{12}-\frac{1}{4}a^{11}+\frac{1}{16}a^{10}-\frac{7}{24}a^{8}+\frac{1}{4}a^{7}-\frac{1}{16}a^{6}+\frac{5}{16}a^{4}-\frac{1}{4}a^{3}+\frac{1}{9}$, $\frac{1}{18432}a^{25}-\frac{1}{2048}a^{21}-\frac{13}{6144}a^{17}-\frac{1}{32}a^{15}+\frac{25}{256}a^{13}-\frac{1}{8}a^{12}-\frac{7}{32}a^{11}+\frac{11}{48}a^{9}+\frac{1}{8}a^{8}-\frac{9}{32}a^{7}-\frac{7}{32}a^{5}-\frac{1}{8}a^{4}+\frac{1}{4}a^{3}-\frac{4}{9}a$, $\frac{1}{36864}a^{26}+\frac{5}{12288}a^{22}-\frac{7}{4096}a^{18}+\frac{23}{384}a^{14}-\frac{1}{8}a^{12}+\frac{1}{8}a^{10}-\frac{1}{2}a^{9}+\frac{1}{8}a^{8}-\frac{37}{192}a^{6}-\frac{1}{2}a^{5}-\frac{1}{8}a^{4}-\frac{7}{18}a^{2}-\frac{1}{2}a$, $\frac{1}{73728}a^{27}+\frac{5}{24576}a^{23}-\frac{7}{8192}a^{19}+\frac{23}{768}a^{15}-\frac{1}{16}a^{13}+\frac{1}{16}a^{11}-\frac{1}{4}a^{10}-\frac{7}{16}a^{9}-\frac{37}{384}a^{7}+\frac{1}{4}a^{6}+\frac{7}{16}a^{5}-\frac{7}{36}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{147456}a^{28}-\frac{1}{147456}a^{24}-\frac{37}{49152}a^{20}-\frac{1}{3072}a^{16}-\frac{1}{8}a^{13}+\frac{1}{768}a^{12}-\frac{1}{4}a^{10}+\frac{1}{8}a^{9}-\frac{293}{768}a^{8}+\frac{1}{4}a^{6}-\frac{1}{8}a^{5}+\frac{1}{144}a^{4}-\frac{1}{4}a^{2}-\frac{1}{9}$, $\frac{1}{147456}a^{29}-\frac{1}{147456}a^{25}+\frac{9}{16384}a^{21}-\frac{5}{3072}a^{17}-\frac{1}{32}a^{15}+\frac{3}{128}a^{13}-\frac{1}{8}a^{12}-\frac{7}{32}a^{11}+\frac{107}{768}a^{9}+\frac{1}{8}a^{8}-\frac{9}{32}a^{7}+\frac{49}{144}a^{5}-\frac{1}{8}a^{4}+\frac{1}{4}a^{3}+\frac{1}{18}a$, $\frac{1}{589824}a^{30}-\frac{1}{294912}a^{29}-\frac{1}{294912}a^{28}-\frac{1}{589824}a^{26}+\frac{1}{294912}a^{25}+\frac{1}{294912}a^{24}-\frac{37}{196608}a^{22}+\frac{37}{98304}a^{21}+\frac{37}{98304}a^{20}+\frac{31}{12288}a^{18}+\frac{1}{6144}a^{17}-\frac{31}{6144}a^{16}+\frac{121}{3072}a^{14}-\frac{1}{1536}a^{13}-\frac{121}{1536}a^{12}+\frac{1}{8}a^{11}+\frac{227}{3072}a^{10}+\frac{293}{1536}a^{9}+\frac{541}{1536}a^{8}-\frac{1}{8}a^{7}-\frac{287}{576}a^{6}+\frac{143}{288}a^{5}+\frac{143}{288}a^{4}+\frac{1}{8}a^{3}-\frac{1}{9}a^{2}-\frac{4}{9}a-\frac{5}{18}$, $\frac{1}{1179648}a^{31}-\frac{1}{589824}a^{29}-\frac{1}{294912}a^{28}-\frac{1}{1179648}a^{27}+\frac{1}{589824}a^{25}+\frac{1}{294912}a^{24}-\frac{37}{393216}a^{23}+\frac{37}{196608}a^{21}+\frac{37}{98304}a^{20}+\frac{31}{24576}a^{19}-\frac{31}{12288}a^{17}-\frac{31}{6144}a^{16}-\frac{71}{6144}a^{15}-\frac{1}{16}a^{14}-\frac{121}{3072}a^{13}+\frac{71}{1536}a^{12}-\frac{1117}{6144}a^{11}+\frac{1}{16}a^{10}-\frac{995}{3072}a^{9}+\frac{349}{1536}a^{8}-\frac{35}{1152}a^{7}-\frac{1}{16}a^{6}-\frac{145}{576}a^{5}-\frac{109}{288}a^{4}+\frac{7}{36}a^{3}-\frac{5}{36}a-\frac{5}{18}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{4}\times C_{8}$, which has order $64$ (assuming GRH)
Unit group
Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{5}{294912} a^{30} - \frac{101}{294912} a^{26} - \frac{25}{98304} a^{22} - \frac{5}{2048} a^{18} + \frac{55}{1536} a^{14} + \frac{5}{512} a^{10} + \frac{35}{288} a^{6} - \frac{55}{36} a^{2} \) (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: | $\frac{7}{294912}a^{30}-\frac{31}{294912}a^{26}+\frac{5}{98304}a^{22}+\frac{1}{12288}a^{18}+\frac{25}{1536}a^{14}-\frac{35}{1536}a^{10}-\frac{13}{576}a^{6}-\frac{35}{36}a^{2}-1$, $\frac{5}{1179648}a^{31}+\frac{43}{589824}a^{30}+\frac{29}{589824}a^{29}+\frac{11}{1179648}a^{27}+\frac{181}{589824}a^{26}+\frac{35}{589824}a^{25}+\frac{151}{393216}a^{23}-\frac{215}{196608}a^{22}+\frac{5}{65536}a^{21}-\frac{5}{12288}a^{19}-\frac{43}{4096}a^{18}-\frac{65}{12288}a^{17}-\frac{7}{6144}a^{15}-\frac{121}{3072}a^{14}-\frac{5}{1024}a^{13}-\frac{113}{6144}a^{11}+\frac{43}{1024}a^{10}-\frac{25}{3072}a^{9}+\frac{7}{576}a^{7}+\frac{301}{576}a^{6}+\frac{53}{576}a^{5}+\frac{1}{36}a^{3}+\frac{13}{9}a^{2}+\frac{5}{36}a$, $\frac{65}{589824}a^{30}-\frac{3}{32768}a^{29}+\frac{35}{294912}a^{28}+\frac{127}{589824}a^{26}+\frac{59}{294912}a^{25}+\frac{21}{32768}a^{24}+\frac{91}{196608}a^{22}+\frac{45}{32768}a^{21}+\frac{27}{32768}a^{20}-\frac{125}{12288}a^{18}+\frac{27}{2048}a^{17}-\frac{19}{2048}a^{16}-\frac{79}{3072}a^{14}-\frac{1}{512}a^{13}-\frac{27}{512}a^{12}-\frac{157}{3072}a^{10}-\frac{27}{512}a^{9}-\frac{45}{512}a^{8}+\frac{173}{576}a^{6}-\frac{21}{32}a^{5}+\frac{113}{288}a^{4}+\frac{19}{36}a^{2}-\frac{5}{9}a+\frac{3}{2}$, $\frac{1}{32768}a^{31}-\frac{1}{49152}a^{29}+\frac{1}{18432}a^{28}+\frac{5}{98304}a^{27}+\frac{3}{16384}a^{25}+\frac{5}{6144}a^{24}+\frac{43}{98304}a^{23}-\frac{25}{49152}a^{21}-\frac{5}{6144}a^{20}-\frac{49}{12288}a^{19}+\frac{7}{6144}a^{17}-\frac{1}{128}a^{16}-\frac{7}{1536}a^{15}-\frac{23}{768}a^{13}-\frac{31}{384}a^{12}-\frac{37}{1536}a^{11}+\frac{19}{768}a^{9}+\frac{1}{32}a^{8}+\frac{35}{192}a^{7}+\frac{11}{96}a^{5}+\frac{7}{18}a^{4}+\frac{1}{8}a^{3}+\frac{7}{6}a+\frac{7}{3}$, $\frac{1}{32768}a^{31}-\frac{1}{589824}a^{30}-\frac{29}{294912}a^{29}+\frac{35}{294912}a^{28}+\frac{5}{98304}a^{27}+\frac{97}{589824}a^{26}-\frac{163}{294912}a^{25}+\frac{21}{32768}a^{24}+\frac{43}{98304}a^{23}+\frac{5}{196608}a^{22}+\frac{113}{98304}a^{21}+\frac{27}{32768}a^{20}-\frac{49}{12288}a^{19}+\frac{1}{4096}a^{18}+\frac{89}{6144}a^{17}-\frac{19}{2048}a^{16}-\frac{7}{1536}a^{15}-\frac{17}{3072}a^{14}+\frac{115}{1536}a^{13}-\frac{27}{512}a^{12}-\frac{37}{1536}a^{11}-\frac{1}{1024}a^{10}+\frac{1}{1536}a^{9}-\frac{45}{512}a^{8}+\frac{35}{192}a^{7}-\frac{7}{576}a^{6}-\frac{137}{288}a^{5}+\frac{113}{288}a^{4}+\frac{1}{8}a^{3}+\frac{13}{36}a^{2}-\frac{55}{18}a+\frac{3}{2}$, $\frac{5}{589824}a^{31}+\frac{7}{196608}a^{30}+\frac{5}{24576}a^{29}-\frac{71}{294912}a^{28}-\frac{245}{589824}a^{27}+\frac{283}{589824}a^{26}+\frac{61}{73728}a^{25}-\frac{185}{294912}a^{24}+\frac{23}{196608}a^{23}-\frac{25}{196608}a^{22}-\frac{5}{8192}a^{21}+\frac{67}{98304}a^{20}-\frac{1}{6144}a^{19}-\frac{5}{3072}a^{18}-\frac{49}{2048}a^{17}+\frac{45}{2048}a^{16}+\frac{139}{3072}a^{15}-\frac{167}{3072}a^{14}-\frac{23}{256}a^{13}+\frac{125}{1536}a^{12}-\frac{49}{3072}a^{11}-\frac{49}{3072}a^{10}+\frac{1}{128}a^{9}-\frac{7}{512}a^{8}+\frac{11}{144}a^{7}+\frac{1}{16}a^{6}+\frac{65}{96}a^{5}-\frac{227}{288}a^{4}-\frac{14}{9}a^{3}+\frac{67}{36}a^{2}+\frac{61}{18}a-\frac{35}{18}$, $\frac{1}{49152}a^{31}-\frac{5}{98304}a^{30}+\frac{1}{24576}a^{29}-\frac{5}{147456}a^{28}-\frac{35}{147456}a^{27}-\frac{89}{294912}a^{26}+\frac{5}{73728}a^{25}-\frac{91}{147456}a^{24}-\frac{5}{16384}a^{23}+\frac{35}{98304}a^{22}-\frac{13}{8192}a^{21}-\frac{13}{16384}a^{20}-\frac{17}{3072}a^{19}+\frac{47}{12288}a^{18}-\frac{31}{3072}a^{17}+\frac{19}{3072}a^{16}+\frac{3}{256}a^{15}+\frac{37}{1536}a^{14}-\frac{11}{256}a^{13}+\frac{13}{256}a^{12}-\frac{1}{768}a^{11}+\frac{11}{1536}a^{10}+\frac{17}{384}a^{9}+\frac{65}{768}a^{8}+\frac{23}{96}a^{7}-\frac{11}{64}a^{6}+\frac{7}{24}a^{5}-\frac{29}{144}a^{4}-\frac{7}{18}a^{3}-\frac{25}{36}a^{2}+\frac{35}{18}a-\frac{22}{9}$, $\frac{1}{12288}a^{31}-\frac{3}{65536}a^{30}-\frac{77}{294912}a^{29}-\frac{17}{294912}a^{28}+\frac{47}{73728}a^{27}+\frac{59}{589824}a^{26}-\frac{11}{32768}a^{25}-\frac{143}{294912}a^{24}-\frac{5}{24576}a^{23}+\frac{45}{65536}a^{22}+\frac{49}{98304}a^{21}+\frac{85}{98304}a^{20}-\frac{67}{8192}a^{19}+\frac{27}{4096}a^{18}+\frac{43}{1536}a^{17}+\frac{17}{2048}a^{16}-\frac{97}{1536}a^{15}-\frac{1}{1024}a^{14}+\frac{77}{1536}a^{13}+\frac{107}{1536}a^{12}-\frac{5}{128}a^{11}-\frac{27}{1024}a^{10}+\frac{65}{1536}a^{9}-\frac{17}{512}a^{8}+\frac{33}{128}a^{7}-\frac{21}{64}a^{6}-\frac{83}{72}a^{5}-\frac{119}{288}a^{4}+\frac{85}{36}a^{3}+\frac{2}{9}a^{2}-\frac{11}{6}a-\frac{65}{18}$, $\frac{7}{393216}a^{31}-\frac{29}{589824}a^{30}+\frac{5}{65536}a^{29}-\frac{23}{147456}a^{28}+\frac{539}{1179648}a^{27}-\frac{35}{589824}a^{26}+\frac{275}{589824}a^{25}-\frac{89}{147456}a^{24}-\frac{25}{393216}a^{23}-\frac{5}{65536}a^{22}-\frac{65}{196608}a^{21}+\frac{35}{49152}a^{20}-\frac{7}{2048}a^{19}+\frac{65}{12288}a^{18}-\frac{113}{12288}a^{17}+\frac{29}{1536}a^{16}-\frac{299}{6144}a^{15}+\frac{5}{1024}a^{14}-\frac{139}{3072}a^{13}+\frac{61}{768}a^{12}+\frac{21}{2048}a^{11}+\frac{25}{3072}a^{10}+\frac{23}{3072}a^{9}-\frac{13}{768}a^{8}+\frac{3}{32}a^{7}-\frac{53}{576}a^{6}+\frac{55}{192}a^{5}-\frac{43}{72}a^{4}+\frac{119}{72}a^{3}-\frac{5}{36}a^{2}+\frac{65}{36}a-\frac{26}{9}$, $\frac{17}{196608}a^{31}-\frac{1}{65536}a^{30}-\frac{5}{24576}a^{29}-\frac{19}{98304}a^{28}+\frac{21}{65536}a^{27}+\frac{35}{196608}a^{26}-\frac{7}{24576}a^{25}-\frac{263}{294912}a^{24}+\frac{27}{65536}a^{23}+\frac{15}{65536}a^{22}-\frac{3}{8192}a^{21}-\frac{1}{32768}a^{20}-\frac{21}{2048}a^{19}+\frac{9}{4096}a^{18}+\frac{51}{2048}a^{17}+\frac{157}{6144}a^{16}-\frac{43}{1024}a^{15}-\frac{31}{1024}a^{14}+\frac{3}{128}a^{13}+\frac{61}{512}a^{12}-\frac{29}{1024}a^{11}-\frac{9}{1024}a^{10}+\frac{5}{128}a^{9}+\frac{101}{1536}a^{8}+\frac{31}{96}a^{7}-\frac{7}{64}a^{6}-\frac{83}{96}a^{5}-\frac{79}{96}a^{4}+\frac{3}{2}a^{3}+\frac{7}{6}a^{2}-\frac{2}{3}a-\frac{83}{18}$, $\frac{35}{589824}a^{31}-\frac{79}{589824}a^{30}+\frac{29}{147456}a^{29}-\frac{73}{294912}a^{28}+\frac{61}{589824}a^{27}-\frac{65}{589824}a^{26}-\frac{53}{147456}a^{25}+\frac{1}{32768}a^{24}-\frac{175}{196608}a^{23}-\frac{101}{196608}a^{22}-\frac{41}{49152}a^{21}-\frac{179}{98304}a^{20}-\frac{35}{4096}a^{19}+\frac{31}{3072}a^{18}-\frac{107}{6144}a^{17}+\frac{31}{2048}a^{16}-\frac{59}{3072}a^{15}+\frac{29}{3072}a^{14}+\frac{23}{768}a^{13}-\frac{25}{1536}a^{12}+\frac{35}{1024}a^{11}+\frac{227}{3072}a^{10}+\frac{19}{768}a^{9}+\frac{71}{512}a^{8}+\frac{245}{576}a^{7}-\frac{5}{18}a^{6}+\frac{211}{288}a^{5}-\frac{109}{288}a^{4}+\frac{10}{9}a^{3}+\frac{4}{9}a^{2}-\frac{14}{9}a+\frac{5}{2}$, $\frac{113}{1179648}a^{31}+\frac{5}{589824}a^{30}-\frac{25}{196608}a^{29}-\frac{5}{147456}a^{28}+\frac{607}{1179648}a^{27}+\frac{11}{589824}a^{26}-\frac{71}{196608}a^{25}-\frac{91}{147456}a^{24}-\frac{23}{131072}a^{23}+\frac{151}{196608}a^{22}+\frac{55}{196608}a^{21}-\frac{13}{16384}a^{20}-\frac{145}{12288}a^{19}-\frac{5}{6144}a^{18}+\frac{181}{12288}a^{17}+\frac{19}{3072}a^{16}-\frac{125}{2048}a^{15}-\frac{7}{3072}a^{14}+\frac{137}{3072}a^{13}+\frac{13}{256}a^{12}-\frac{13}{6144}a^{11}-\frac{113}{3072}a^{10}-\frac{1}{3072}a^{9}+\frac{65}{768}a^{8}+\frac{235}{576}a^{7}+\frac{7}{288}a^{6}-\frac{25}{64}a^{5}-\frac{29}{144}a^{4}+\frac{65}{36}a^{3}+\frac{1}{18}a^{2}-\frac{19}{12}a-\frac{13}{9}$, $\frac{17}{589824}a^{31}+\frac{1}{32768}a^{29}+\frac{1}{24576}a^{28}+\frac{13}{196608}a^{27}+\frac{23}{294912}a^{25}-\frac{1}{24576}a^{24}+\frac{35}{196608}a^{23}+\frac{83}{98304}a^{21}+\frac{27}{8192}a^{20}-\frac{37}{24576}a^{19}-\frac{1}{2048}a^{17}+\frac{1}{1536}a^{16}-\frac{29}{3072}a^{15}+\frac{13}{1536}a^{13}+\frac{3}{64}a^{12}-\frac{61}{3072}a^{11}-\frac{31}{512}a^{9}-\frac{91}{384}a^{8}-\frac{23}{1152}a^{7}-\frac{1}{96}a^{5}-\frac{5}{24}a^{4}+\frac{1}{6}a^{3}-\frac{5}{9}a-3$, $\frac{1}{294912}a^{31}-\frac{13}{147456}a^{30}+\frac{1}{12288}a^{29}+\frac{3}{32768}a^{27}-\frac{23}{147456}a^{26}-\frac{1}{36864}a^{25}+\frac{13}{32768}a^{23}-\frac{83}{49152}a^{22}+\frac{11}{12288}a^{21}-\frac{7}{24576}a^{19}+\frac{27}{4096}a^{18}-\frac{37}{6144}a^{17}-\frac{1}{256}a^{15}-\frac{1}{768}a^{14}+\frac{7}{768}a^{13}-\frac{29}{1536}a^{11}+\frac{27}{256}a^{10}-\frac{13}{192}a^{9}+\frac{7}{1152}a^{7}-\frac{61}{576}a^{6}+\frac{3}{32}a^{5}+\frac{5}{24}a^{3}+\frac{13}{36}a^{2}-\frac{7}{9}a$, $\frac{7}{1179648}a^{31}+\frac{77}{589824}a^{30}-\frac{23}{196608}a^{29}+\frac{41}{147456}a^{28}-\frac{47}{131072}a^{27}+\frac{227}{589824}a^{26}+\frac{37}{589824}a^{25}+\frac{29}{49152}a^{24}-\frac{35}{393216}a^{23}-\frac{49}{196608}a^{22}+\frac{89}{196608}a^{21}+\frac{19}{49152}a^{20}+\frac{43}{24576}a^{19}-\frac{17}{1024}a^{18}+\frac{53}{4096}a^{17}-\frac{73}{3072}a^{16}+\frac{275}{6144}a^{15}-\frac{143}{3072}a^{14}-\frac{65}{3072}a^{13}-\frac{25}{768}a^{12}+\frac{101}{6144}a^{11}-\frac{3}{1024}a^{10}-\frac{53}{1024}a^{9}-\frac{29}{768}a^{8}-\frac{59}{1152}a^{7}+\frac{23}{36}a^{6}-\frac{35}{64}a^{5}+\frac{77}{144}a^{4}-\frac{19}{12}a^{3}+\frac{59}{36}a^{2}+\frac{43}{36}a-\frac{2}{3}$ (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: | \( 950005676337.5021 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 950005676337.5021 \cdot 64}{12\cdot\sqrt{480960519379403029833827263813614000556122650443776}}\cr\approx \mathstrut & 1.36316399913236 \end{aligned}\] (assuming GRH)
Galois group
$C_2^4:D_4$ (as 32T1369):
A solvable group of order 128 |
The 56 conjugacy class representatives for $C_2^4:D_4$ |
Character table for $C_2^4:D_4$ |
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: | 32.0.480960519379403029833827263813614000556122650443776.1 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{8}$ | ${\href{/padicField/7.4.0.1}{4} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{8}$ | R | R | ${\href{/padicField/19.2.0.1}{2} }^{16}$ | ${\href{/padicField/23.2.0.1}{2} }^{16}$ | ${\href{/padicField/29.4.0.1}{4} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{8}$ | ${\href{/padicField/43.2.0.1}{2} }^{16}$ | ${\href{/padicField/47.2.0.1}{2} }^{16}$ | ${\href{/padicField/53.2.0.1}{2} }^{16}$ | ${\href{/padicField/59.2.0.1}{2} }^{16}$ |
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 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
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 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(13\) | 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
\(17\) | 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |