Normalized defining polynomial
\( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $(0, 6)$ |
| |
| Discriminant: |
\(1157018619904\)
\(\medspace = 2^{12}\cdot 7^{10}\)
|
| |
| Root discriminant: | \(10.12\) |
| |
| Galois root discriminant: | $2\cdot 7^{5/6}\approx 10.122280369592772$ | ||
| Ramified primes: |
\(2\), \(7\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_2\times C_6$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(28=2^{2}\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{28}(1,·)$, $\chi_{28}(3,·)$, $\chi_{28}(5,·)$, $\chi_{28}(9,·)$, $\chi_{28}(11,·)$, $\chi_{28}(13,·)$, $\chi_{28}(15,·)$, $\chi_{28}(17,·)$, $\chi_{28}(19,·)$, $\chi_{28}(23,·)$, $\chi_{28}(25,·)$, $\chi_{28}(27,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})\)$^{3}$, 6.0.153664.1$^{3}$, \(\Q(\zeta_{28})\)$^{24}$ | ||
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 $28$)
|
| |
| Fundamental units: |
$a^{4}+1$, $a^{10}-a^{8}$, $a^{3}-a^{2}$, $a^{3}-1$, $a^{5}-1$
|
| |
| Regulator: | \( 123.252731541 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 123.252731541 \cdot 1}{28\cdot\sqrt{1157018619904}}\cr\approx \mathstrut & 0.251795264265 \end{aligned}\]
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
| \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{7}) \), \(\Q(\zeta_{7})^+\), \(\Q(i, \sqrt{7})\), 6.0.153664.1, \(\Q(\zeta_{7})\), \(\Q(\zeta_{28})^+\) |
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 | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.1.0.1}{1} }^{12}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.6a1.1 | $x^{6} + 2 x^{4} + 4 x^{3} + x^{2} + 4 x + 5$ | $2$ | $3$ | $6$ | $C_6$ | $$[2]^{3}$$ |
| 2.3.2.6a1.1 | $x^{6} + 2 x^{4} + 4 x^{3} + x^{2} + 4 x + 5$ | $2$ | $3$ | $6$ | $C_6$ | $$[2]^{3}$$ | |
|
\(7\)
| 7.2.6.10a1.2 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 192996 x^{6} + 266328 x^{5} + 234495 x^{4} + 131220 x^{3} + 45198 x^{2} + 8748 x + 736$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $$[\ ]_{6}^{2}$$ |