Normalized defining polynomial
\( x^{32} - 97 x^{30} + 3977 x^{28} - 90598 x^{26} + 1268954 x^{24} - 11436203 x^{22} + 67589988 x^{20} + \cdots + 97 \)
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: |
\(167064287751196233763024674646775194239336663364043996417475105771225088\)
\(\medspace = 2^{32}\cdot 97^{31}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(168.16\) | 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: | $2\cdot 97^{31/32}\approx 168.15706291895154$ | ||
| 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: | \(388=2^{2}\cdot 97\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{388}(1,·)$, $\chi_{388}(131,·)$, $\chi_{388}(263,·)$, $\chi_{388}(139,·)$, $\chi_{388}(269,·)$, $\chi_{388}(143,·)$, $\chi_{388}(273,·)$, $\chi_{388}(19,·)$, $\chi_{388}(33,·)$, $\chi_{388}(175,·)$, $\chi_{388}(51,·)$, $\chi_{388}(309,·)$, $\chi_{388}(55,·)$, $\chi_{388}(313,·)$, $\chi_{388}(63,·)$, $\chi_{388}(193,·)$, $\chi_{388}(67,·)$, $\chi_{388}(161,·)$, $\chi_{388}(311,·)$, $\chi_{388}(341,·)$, $\chi_{388}(343,·)$, $\chi_{388}(89,·)$, $\chi_{388}(271,·)$, $\chi_{388}(221,·)$, $\chi_{388}(105,·)$, $\chi_{388}(127,·)$, $\chi_{388}(109,·)$, $\chi_{388}(239,·)$, $\chi_{388}(241,·)$, $\chi_{388}(361,·)$, $\chi_{388}(319,·)$, $\chi_{388}(85,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{61}a^{24}+\frac{15}{61}a^{22}-\frac{12}{61}a^{20}+\frac{18}{61}a^{18}+\frac{14}{61}a^{16}-\frac{19}{61}a^{14}-\frac{20}{61}a^{12}+\frac{11}{61}a^{10}+\frac{10}{61}a^{8}-\frac{7}{61}a^{6}+\frac{18}{61}a^{4}+\frac{18}{61}a^{2}+\frac{5}{61}$, $\frac{1}{61}a^{25}+\frac{15}{61}a^{23}-\frac{12}{61}a^{21}+\frac{18}{61}a^{19}+\frac{14}{61}a^{17}-\frac{19}{61}a^{15}-\frac{20}{61}a^{13}+\frac{11}{61}a^{11}+\frac{10}{61}a^{9}-\frac{7}{61}a^{7}+\frac{18}{61}a^{5}+\frac{18}{61}a^{3}+\frac{5}{61}a$, $\frac{1}{61}a^{26}+\frac{7}{61}a^{22}+\frac{15}{61}a^{20}-\frac{12}{61}a^{18}+\frac{15}{61}a^{16}+\frac{21}{61}a^{14}+\frac{6}{61}a^{12}+\frac{28}{61}a^{10}+\frac{26}{61}a^{8}+\frac{1}{61}a^{6}-\frac{8}{61}a^{4}-\frac{21}{61}a^{2}-\frac{14}{61}$, $\frac{1}{61}a^{27}+\frac{7}{61}a^{23}+\frac{15}{61}a^{21}-\frac{12}{61}a^{19}+\frac{15}{61}a^{17}+\frac{21}{61}a^{15}+\frac{6}{61}a^{13}+\frac{28}{61}a^{11}+\frac{26}{61}a^{9}+\frac{1}{61}a^{7}-\frac{8}{61}a^{5}-\frac{21}{61}a^{3}-\frac{14}{61}a$, $\frac{1}{3721}a^{28}-\frac{12}{3721}a^{26}+\frac{7}{3721}a^{24}-\frac{1289}{3721}a^{22}+\frac{52}{3721}a^{20}-\frac{1000}{3721}a^{18}-\frac{1806}{3721}a^{16}+\frac{486}{3721}a^{14}+\frac{17}{3721}a^{12}+\frac{849}{3721}a^{10}-\frac{1348}{3721}a^{8}-\frac{203}{3721}a^{6}+\frac{1539}{3721}a^{4}+\frac{848}{3721}a^{2}+\frac{534}{3721}$, $\frac{1}{3721}a^{29}-\frac{12}{3721}a^{27}+\frac{7}{3721}a^{25}-\frac{1289}{3721}a^{23}+\frac{52}{3721}a^{21}-\frac{1000}{3721}a^{19}-\frac{1806}{3721}a^{17}+\frac{486}{3721}a^{15}+\frac{17}{3721}a^{13}+\frac{849}{3721}a^{11}-\frac{1348}{3721}a^{9}-\frac{203}{3721}a^{7}+\frac{1539}{3721}a^{5}+\frac{848}{3721}a^{3}+\frac{534}{3721}a$, $\frac{1}{61\!\cdots\!37}a^{30}+\frac{49\!\cdots\!70}{61\!\cdots\!37}a^{28}-\frac{84\!\cdots\!92}{61\!\cdots\!37}a^{26}+\frac{29\!\cdots\!62}{63\!\cdots\!47}a^{24}+\frac{26\!\cdots\!65}{61\!\cdots\!37}a^{22}-\frac{30\!\cdots\!88}{61\!\cdots\!37}a^{20}+\frac{82\!\cdots\!27}{61\!\cdots\!37}a^{18}+\frac{21\!\cdots\!27}{61\!\cdots\!37}a^{16}+\frac{51\!\cdots\!17}{61\!\cdots\!37}a^{14}+\frac{68\!\cdots\!27}{61\!\cdots\!37}a^{12}-\frac{11\!\cdots\!09}{61\!\cdots\!37}a^{10}-\frac{12\!\cdots\!35}{61\!\cdots\!37}a^{8}+\frac{10\!\cdots\!59}{61\!\cdots\!37}a^{6}+\frac{29\!\cdots\!03}{61\!\cdots\!37}a^{4}-\frac{15\!\cdots\!86}{61\!\cdots\!37}a^{2}-\frac{13\!\cdots\!99}{61\!\cdots\!37}$, $\frac{1}{61\!\cdots\!37}a^{31}+\frac{49\!\cdots\!70}{61\!\cdots\!37}a^{29}-\frac{84\!\cdots\!92}{61\!\cdots\!37}a^{27}+\frac{29\!\cdots\!62}{63\!\cdots\!47}a^{25}+\frac{26\!\cdots\!65}{61\!\cdots\!37}a^{23}-\frac{30\!\cdots\!88}{61\!\cdots\!37}a^{21}+\frac{82\!\cdots\!27}{61\!\cdots\!37}a^{19}+\frac{21\!\cdots\!27}{61\!\cdots\!37}a^{17}+\frac{51\!\cdots\!17}{61\!\cdots\!37}a^{15}+\frac{68\!\cdots\!27}{61\!\cdots\!37}a^{13}-\frac{11\!\cdots\!09}{61\!\cdots\!37}a^{11}-\frac{12\!\cdots\!35}{61\!\cdots\!37}a^{9}+\frac{10\!\cdots\!59}{61\!\cdots\!37}a^{7}+\frac{29\!\cdots\!03}{61\!\cdots\!37}a^{5}-\frac{15\!\cdots\!86}{61\!\cdots\!37}a^{3}-\frac{13\!\cdots\!99}{61\!\cdots\!37}a$
| 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 $16$ | $2$ | $8$ | $16$ | |||
| Deg $16$ | $2$ | $8$ | $16$ | ||||
|
\(97\)
| Deg $32$ | $32$ | $1$ | $31$ |