Normalized defining polynomial
\( x^{16} + 1 \)
Invariants
| Degree: | $16$ |
| |
| Signature: | $(0, 8)$ |
| |
| Discriminant: |
\(18446744073709551616\)
\(\medspace = 2^{64}\)
|
| |
| Root discriminant: | \(16.00\) |
| |
| Galois root discriminant: | $2^{4}\approx 16.0$ | ||
| Ramified primes: |
\(2\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_2\times C_8$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(32=2^{5}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{32}(1,·)$, $\chi_{32}(3,·)$, $\chi_{32}(5,·)$, $\chi_{32}(7,·)$, $\chi_{32}(9,·)$, $\chi_{32}(11,·)$, $\chi_{32}(13,·)$, $\chi_{32}(15,·)$, $\chi_{32}(17,·)$, $\chi_{32}(19,·)$, $\chi_{32}(21,·)$, $\chi_{32}(23,·)$, $\chi_{32}(25,·)$, $\chi_{32}(27,·)$, $\chi_{32}(29,·)$, $\chi_{32}(31,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
| |
| Relative class number: | $1$ |
Unit group
| Rank: | $7$ |
| |
| Torsion generator: |
\( a \)
(order $32$)
|
| |
| Fundamental units: |
$a^{12}-a^{4}-1$, $a^{4}+a^{2}+1$, $a^{12}-a^{6}+a^{2}$, $a^{2}+a+1$, $a^{11}+a^{6}+a$, $a^{13}-a^{10}+a^{7}$, $a^{14}-a^{10}+a^{8}-a^{4}-a$
|
| |
| Regulator: | \( 15753.9498624 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 15753.9498624 \cdot 1}{32\cdot\sqrt{18446744073709551616}}\cr\approx \mathstrut & 0.278431627704 \end{aligned}\]
Galois group
$C_2\times C_8$ (as 16T5):
| An abelian group of order 16 |
| The 16 conjugacy class representatives for $C_8\times C_2$ |
| Character table for $C_8\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{8})\), \(\Q(\zeta_{16})^+\), \(\Q(\sqrt{-2 + \sqrt{2}})\), \(\Q(\zeta_{16})\), 8.0.2147483648.1, \(\Q(\zeta_{32})^+\) |
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 | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.16.64g1.8 | $x^{16} + 16 x^{13} + 16 x^{11} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 16 x + 2$ | $16$ | $1$ | $64$ | $C_8\times C_2$ | $$[2, 3, 4, 5]$$ |