Normalized defining polynomial
\( x^{16} - x^{12} + x^{8} - x^{4} + 1 \)
Invariants
| Degree: | $16$ |
| |
| Signature: | $[0, 8]$ |
| |
| Discriminant: |
\(1048576000000000000\)
\(\medspace = 2^{32}\cdot 5^{12}\)
|
| |
| Root discriminant: | \(13.37\) |
| |
| Galois root discriminant: | $2^{2}5^{3/4}\approx 13.37480609952844$ | ||
| Ramified primes: |
\(2\), \(5\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_2^2\times C_4$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(40=2^{3}\cdot 5\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{40}(1,·)$, $\chi_{40}(3,·)$, $\chi_{40}(7,·)$, $\chi_{40}(9,·)$, $\chi_{40}(11,·)$, $\chi_{40}(13,·)$, $\chi_{40}(17,·)$, $\chi_{40}(19,·)$, $\chi_{40}(21,·)$, $\chi_{40}(23,·)$, $\chi_{40}(27,·)$, $\chi_{40}(29,·)$, $\chi_{40}(31,·)$, $\chi_{40}(33,·)$, $\chi_{40}(37,·)$, $\chi_{40}(39,·)$$\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 $40$)
|
| |
| Fundamental units: |
$a^{8}-a^{4}$, $a^{10}-a^{6}-a^{4}+1$, $a^{14}+a^{12}+a^{6}+a^{4}-a^{2}-1$, $a^{15}-a^{11}+a^{7}+a^{6}-a^{3}$, $a^{13}+a^{12}$, $a^{12}-a$, $a^{12}+a^{9}$
|
| |
| Regulator: | \( 3557.06767863 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3557.06767863 \cdot 1}{40\cdot\sqrt{1048576000000000000}}\cr\approx \mathstrut & 0.210945918718 \end{aligned}\]
Galois group
$C_2^2\times C_4$ (as 16T2):
| An abelian group of order 16 |
| The 16 conjugacy class representatives for $C_4\times C_2^2$ |
| Character table for $C_4\times C_2^2$ |
Intermediate fields
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 | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.1.0.1}{1} }^{16}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.32b1.1 | $x^{16} + 4 x^{13} + 4 x^{12} + 6 x^{10} + 12 x^{9} + 8 x^{8} + 4 x^{7} + 12 x^{6} + 16 x^{5} + 13 x^{4} + 4 x^{3} + 8 x^{2} + 12 x + 9$ | $4$ | $4$ | $32$ | $C_4\times C_2^2$ | $$[2, 3]^{4}$$ |
|
\(5\)
| 5.2.4.6a1.2 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ |
| 5.2.4.6a1.2 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ |