Normalized defining polynomial
\( x^{32} + 96 x^{30} + 4176 x^{28} + 108864 x^{26} + 1895400 x^{24} + 23250240 x^{22} + 206569440 x^{20} + 1345652352 x^{18} + 6433900308 x^{16} + \cdots + 86093442 \)
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: | \(135104323545903136978453058557785670637514001130337144105502507008\) \(\medspace = 2^{191}\cdot 3^{16}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(108.48\) | 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: | $2^{191/32}3^{1/2}\approx 108.47593794581393$ | ||
Ramified primes: | \(2\), \(3\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
$\card{ \Gal(K/\Q) }$: | $32$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(384=2^{7}\cdot 3\) | ||
Dirichlet character group: | $\lbrace$$\chi_{384}(1,·)$, $\chi_{384}(5,·)$, $\chi_{384}(265,·)$, $\chi_{384}(269,·)$, $\chi_{384}(145,·)$, $\chi_{384}(149,·)$, $\chi_{384}(25,·)$, $\chi_{384}(29,·)$, $\chi_{384}(289,·)$, $\chi_{384}(293,·)$, $\chi_{384}(169,·)$, $\chi_{384}(173,·)$, $\chi_{384}(49,·)$, $\chi_{384}(53,·)$, $\chi_{384}(313,·)$, $\chi_{384}(317,·)$, $\chi_{384}(193,·)$, $\chi_{384}(197,·)$, $\chi_{384}(73,·)$, $\chi_{384}(77,·)$, $\chi_{384}(337,·)$, $\chi_{384}(341,·)$, $\chi_{384}(217,·)$, $\chi_{384}(221,·)$, $\chi_{384}(97,·)$, $\chi_{384}(101,·)$, $\chi_{384}(361,·)$, $\chi_{384}(365,·)$, $\chi_{384}(241,·)$, $\chi_{384}(245,·)$, $\chi_{384}(121,·)$, $\chi_{384}(125,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{32768}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{9}a^{4}$, $\frac{1}{9}a^{5}$, $\frac{1}{27}a^{6}$, $\frac{1}{27}a^{7}$, $\frac{1}{81}a^{8}$, $\frac{1}{81}a^{9}$, $\frac{1}{243}a^{10}$, $\frac{1}{243}a^{11}$, $\frac{1}{729}a^{12}$, $\frac{1}{729}a^{13}$, $\frac{1}{2187}a^{14}$, $\frac{1}{2187}a^{15}$, $\frac{1}{6561}a^{16}$, $\frac{1}{6561}a^{17}$, $\frac{1}{19683}a^{18}$, $\frac{1}{19683}a^{19}$, $\frac{1}{59049}a^{20}$, $\frac{1}{59049}a^{21}$, $\frac{1}{177147}a^{22}$, $\frac{1}{177147}a^{23}$, $\frac{1}{531441}a^{24}$, $\frac{1}{531441}a^{25}$, $\frac{1}{1594323}a^{26}$, $\frac{1}{1594323}a^{27}$, $\frac{1}{4782969}a^{28}$, $\frac{1}{4782969}a^{29}$, $\frac{1}{14348907}a^{30}$, $\frac{1}{14348907}a^{31}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $15$ | 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 $C_{32}$ is not computed |
Intermediate fields
\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\), \(\Q(\zeta_{64})^+\) |
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 | R | $32$ | $16^{2}$ | $32$ | $32$ | ${\href{/padicField/17.8.0.1}{8} }^{4}$ | $32$ | $16^{2}$ | $32$ | ${\href{/padicField/31.4.0.1}{4} }^{8}$ | $32$ | $16^{2}$ | $32$ | ${\href{/padicField/47.8.0.1}{8} }^{4}$ | $32$ | $32$ |
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 $32$ | $32$ | $1$ | $191$ | |||
\(3\) | Deg $32$ | $2$ | $16$ | $16$ |