Normalized defining polynomial
\( x^{32} - 194 x^{30} + 15908 x^{28} - 724784 x^{26} + 20303264 x^{24} - 365958496 x^{22} + 4325759232 x^{20} + \cdots + 6356992 \)
Invariants
| Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[32, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(10948725162062396375893585077651059129669167570225987349215648531823007367168\)
\(\medspace = 2^{48}\cdot 97^{31}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(237.81\) | 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: | $2^{3/2}97^{31/32}\approx 237.8099989888071$ | ||
| Ramified primes: |
\(2\), \(97\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{97}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $32$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(776=2^{3}\cdot 97\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{776}(1,·)$, $\chi_{776}(515,·)$, $\chi_{776}(139,·)$, $\chi_{776}(273,·)$, $\chi_{776}(131,·)$, $\chi_{776}(729,·)$, $\chi_{776}(33,·)$, $\chi_{776}(241,·)$, $\chi_{776}(51,·)$, $\chi_{776}(313,·)$, $\chi_{776}(699,·)$, $\chi_{776}(193,·)$, $\chi_{776}(67,·)$, $\chi_{776}(161,·)$, $\chi_{776}(707,·)$, $\chi_{776}(531,·)$, $\chi_{776}(659,·)$, $\chi_{776}(563,·)$, $\chi_{776}(697,·)$, $\chi_{776}(473,·)$, $\chi_{776}(731,·)$, $\chi_{776}(627,·)$, $\chi_{776}(609,·)$, $\chi_{776}(443,·)$, $\chi_{776}(651,·)$, $\chi_{776}(657,·)$, $\chi_{776}(105,·)$, $\chi_{776}(451,·)$, $\chi_{776}(497,·)$, $\chi_{776}(19,·)$, $\chi_{776}(89,·)$, $\chi_{776}(361,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{2048}a^{22}$, $\frac{1}{2048}a^{23}$, $\frac{1}{249856}a^{24}+\frac{15}{124928}a^{22}-\frac{3}{15616}a^{20}+\frac{9}{15616}a^{18}+\frac{7}{7808}a^{16}-\frac{19}{7808}a^{14}-\frac{5}{976}a^{12}+\frac{11}{1952}a^{10}+\frac{5}{488}a^{8}-\frac{7}{488}a^{6}+\frac{9}{122}a^{4}+\frac{9}{61}a^{2}+\frac{5}{61}$, $\frac{1}{249856}a^{25}+\frac{15}{124928}a^{23}-\frac{3}{15616}a^{21}+\frac{9}{15616}a^{19}+\frac{7}{7808}a^{17}-\frac{19}{7808}a^{15}-\frac{5}{976}a^{13}+\frac{11}{1952}a^{11}+\frac{5}{488}a^{9}-\frac{7}{488}a^{7}+\frac{9}{122}a^{5}+\frac{9}{61}a^{3}+\frac{5}{61}a$, $\frac{1}{499712}a^{26}+\frac{7}{124928}a^{22}+\frac{15}{62464}a^{20}-\frac{3}{7808}a^{18}+\frac{15}{15616}a^{16}+\frac{21}{7808}a^{14}+\frac{3}{1952}a^{12}+\frac{7}{488}a^{10}+\frac{13}{488}a^{8}+\frac{1}{488}a^{6}-\frac{2}{61}a^{4}-\frac{21}{122}a^{2}-\frac{14}{61}$, $\frac{1}{499712}a^{27}+\frac{7}{124928}a^{23}+\frac{15}{62464}a^{21}-\frac{3}{7808}a^{19}+\frac{15}{15616}a^{17}+\frac{21}{7808}a^{15}+\frac{3}{1952}a^{13}+\frac{7}{488}a^{11}+\frac{13}{488}a^{9}+\frac{1}{488}a^{7}-\frac{2}{61}a^{5}-\frac{21}{122}a^{3}-\frac{14}{61}a$, $\frac{1}{60964864}a^{28}-\frac{3}{7620608}a^{26}+\frac{7}{15241216}a^{24}-\frac{1289}{7620608}a^{22}+\frac{13}{952576}a^{20}-\frac{125}{238144}a^{18}-\frac{903}{476288}a^{16}+\frac{243}{238144}a^{14}+\frac{17}{238144}a^{12}+\frac{849}{119072}a^{10}-\frac{337}{14884}a^{8}-\frac{203}{29768}a^{6}+\frac{1539}{14884}a^{4}+\frac{424}{3721}a^{2}+\frac{534}{3721}$, $\frac{1}{60964864}a^{29}-\frac{3}{7620608}a^{27}+\frac{7}{15241216}a^{25}-\frac{1289}{7620608}a^{23}+\frac{13}{952576}a^{21}-\frac{125}{238144}a^{19}-\frac{903}{476288}a^{17}+\frac{243}{238144}a^{15}+\frac{17}{238144}a^{13}+\frac{849}{119072}a^{11}-\frac{337}{14884}a^{9}-\frac{203}{29768}a^{7}+\frac{1539}{14884}a^{5}+\frac{424}{3721}a^{3}+\frac{534}{3721}a$, $\frac{1}{20\!\cdots\!16}a^{30}+\frac{24\!\cdots\!85}{50\!\cdots\!04}a^{28}-\frac{21\!\cdots\!23}{12\!\cdots\!76}a^{26}+\frac{14\!\cdots\!31}{12\!\cdots\!56}a^{24}+\frac{26\!\cdots\!65}{12\!\cdots\!76}a^{22}-\frac{75\!\cdots\!47}{15\!\cdots\!72}a^{20}+\frac{82\!\cdots\!27}{31\!\cdots\!44}a^{18}+\frac{21\!\cdots\!27}{15\!\cdots\!72}a^{16}+\frac{51\!\cdots\!17}{78\!\cdots\!36}a^{14}+\frac{68\!\cdots\!27}{39\!\cdots\!68}a^{12}-\frac{11\!\cdots\!09}{19\!\cdots\!84}a^{10}-\frac{12\!\cdots\!35}{98\!\cdots\!92}a^{8}+\frac{10\!\cdots\!59}{49\!\cdots\!96}a^{6}+\frac{29\!\cdots\!03}{24\!\cdots\!48}a^{4}-\frac{75\!\cdots\!93}{61\!\cdots\!37}a^{2}-\frac{13\!\cdots\!99}{61\!\cdots\!37}$, $\frac{1}{20\!\cdots\!16}a^{31}+\frac{24\!\cdots\!85}{50\!\cdots\!04}a^{29}-\frac{21\!\cdots\!23}{12\!\cdots\!76}a^{27}+\frac{14\!\cdots\!31}{12\!\cdots\!56}a^{25}+\frac{26\!\cdots\!65}{12\!\cdots\!76}a^{23}-\frac{75\!\cdots\!47}{15\!\cdots\!72}a^{21}+\frac{82\!\cdots\!27}{31\!\cdots\!44}a^{19}+\frac{21\!\cdots\!27}{15\!\cdots\!72}a^{17}+\frac{51\!\cdots\!17}{78\!\cdots\!36}a^{15}+\frac{68\!\cdots\!27}{39\!\cdots\!68}a^{13}-\frac{11\!\cdots\!09}{19\!\cdots\!84}a^{11}-\frac{12\!\cdots\!35}{98\!\cdots\!92}a^{9}+\frac{10\!\cdots\!59}{49\!\cdots\!96}a^{7}+\frac{29\!\cdots\!03}{24\!\cdots\!48}a^{5}-\frac{75\!\cdots\!93}{61\!\cdots\!37}a^{3}-\frac{13\!\cdots\!99}{61\!\cdots\!37}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $31$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: | not computed | 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: | not computed | 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 $
Galois group
| A cyclic group of order 32 |
| The 32 conjugacy class representatives for $C_{32}$ |
| Character table for $C_{32}$ |
Intermediate fields
| \(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1, 16.16.633251189136789386043275954593.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $16^{2}$ | $32$ | $32$ | $16^{2}$ | $32$ | $32$ | $32$ | $32$ | $32$ | $16^{2}$ | $32$ | $32$ | ${\href{/padicField/43.8.0.1}{8} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{4}$ | $16^{2}$ | $32$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $2$ | $8$ | $24$ | |||
| Deg $16$ | $2$ | $8$ | $24$ | ||||
|
\(97\)
| Deg $32$ | $32$ | $1$ | $31$ |