Normalized defining polynomial
\( x^{12} - x^{6} + 1 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $(0, 6)$ |
| |
| Discriminant: |
\(1586874322944\)
\(\medspace = 2^{12}\cdot 3^{18}\)
|
| |
| Root discriminant: | \(10.39\) |
| |
| Galois root discriminant: | $2\cdot 3^{3/2}\approx 10.392304845413264$ | ||
| Ramified primes: |
\(2\), \(3\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_2\times C_6$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(36=2^{2}\cdot 3^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{36}(1,·)$, $\chi_{36}(35,·)$, $\chi_{36}(5,·)$, $\chi_{36}(7,·)$, $\chi_{36}(11,·)$, $\chi_{36}(13,·)$, $\chi_{36}(17,·)$, $\chi_{36}(19,·)$, $\chi_{36}(23,·)$, $\chi_{36}(25,·)$, $\chi_{36}(29,·)$, $\chi_{36}(31,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{9})\)$^{3}$, 6.0.419904.1$^{3}$, \(\Q(\zeta_{36})\)$^{24}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$
| 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: | $5$ |
| |
| Torsion generator: |
\( a \)
(order $36$)
|
| |
| Fundamental units: |
$a^{4}+1$, $a^{4}-a^{2}$, $a^{3}+1$, $a-1$, $a^{11}-a^{10}+a^{4}$
|
| |
| Regulator: | \( 162.837701397 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 162.837701397 \cdot 1}{36\cdot\sqrt{1586874322944}}\cr\approx \mathstrut & 0.22093290661 \end{aligned}\]
Galois group
$C_2\times C_6$ (as 12T2):
| An abelian group of order 12 |
| The 12 conjugacy class representatives for $C_6\times C_2$ |
| Character table for $C_6\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{12})\), 6.0.419904.1, \(\Q(\zeta_{9})\), \(\Q(\zeta_{36})^+\) |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.1.0.1}{1} }^{12}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.2.12a1.1 | $x^{12} + 2 x^{10} + 2 x^{9} + x^{8} + 4 x^{7} + 5 x^{6} + 2 x^{5} + 6 x^{4} + 4 x^{3} + x^{2} + 4 x + 5$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $$[2]^{6}$$ |
|
\(3\)
| 3.2.6.18a1.16 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 786 x^{8} + 1680 x^{7} + 2816 x^{6} + 3744 x^{5} + 3936 x^{4} + 3200 x^{3} + 1920 x^{2} + 768 x + 163$ | $6$ | $2$ | $18$ | $C_6\times C_2$ | $$[2]_{2}^{2}$$ |