Normalized defining polynomial
\( x^{12} + 44x^{6} - 2 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[2, 5]$ |
| |
| Discriminant: |
\(-164341563462254592\)
\(\medspace = -\,2^{35}\cdot 3^{14}\)
|
| |
| Root discriminant: | \(27.20\) |
| |
| Galois root discriminant: | $2^{35/12}3^{7/6}\approx 27.20480503537909$ | ||
| Ramified primes: |
\(2\), \(3\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{9}a^{6}+\frac{4}{9}$, $\frac{1}{9}a^{7}+\frac{4}{9}a$, $\frac{1}{9}a^{8}+\frac{4}{9}a^{2}$, $\frac{1}{9}a^{9}+\frac{4}{9}a^{3}$, $\frac{1}{27}a^{10}-\frac{1}{27}a^{8}+\frac{1}{27}a^{6}-\frac{5}{27}a^{4}+\frac{5}{27}a^{2}-\frac{5}{27}$, $\frac{1}{27}a^{11}-\frac{1}{27}a^{9}+\frac{1}{27}a^{7}-\frac{5}{27}a^{5}+\frac{5}{27}a^{3}-\frac{5}{27}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $6$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{5}{27}a^{10}+\frac{1}{27}a^{8}-\frac{1}{27}a^{6}+\frac{218}{27}a^{4}+\frac{49}{27}a^{2}+\frac{5}{27}$, $\frac{2}{9}a^{10}+\frac{89}{9}a^{4}-1$, $\frac{11}{27}a^{11}+\frac{11}{27}a^{10}+\frac{1}{27}a^{9}-\frac{5}{27}a^{8}-\frac{4}{27}a^{7}-\frac{1}{27}a^{6}+\frac{485}{27}a^{5}+\frac{485}{27}a^{4}+\frac{49}{27}a^{3}-\frac{218}{27}a^{2}-\frac{169}{27}a-\frac{49}{27}$, $\frac{1}{9}a^{9}+\frac{40}{9}a^{3}-1$, $\frac{1}{9}a^{9}+\frac{40}{9}a^{3}+1$, $\frac{2}{9}a^{11}+\frac{1}{27}a^{10}-\frac{1}{27}a^{8}+\frac{1}{9}a^{7}-\frac{5}{27}a^{6}+\frac{89}{9}a^{5}+\frac{49}{27}a^{4}-a^{3}+\frac{32}{27}a^{2}-\frac{5}{9}a-\frac{29}{27}$
|
| |
| Regulator: | \( 17436.4884271 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 17436.4884271 \cdot 1}{2\cdot\sqrt{164341563462254592}}\cr\approx \mathstrut & 0.842392819266 \end{aligned}\]
Galois group
| A solvable group of order 24 |
| The 9 conjugacy class representatives for $D_{12}$ |
| Character table for $D_{12}$ |
Intermediate fields
| \(\Q(\sqrt{6}) \), 3.1.108.1, 4.2.18432.1, 6.2.4478976.4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 24 |
| Degree 12 sibling: | 12.0.54780521154084864.58 |
| Minimal sibling: | 12.0.54780521154084864.58 |
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 | ${\href{/padicField/5.2.0.1}{2} }^{5}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.1.0.1}{1} }^{12}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.12.35a1.81 | $x^{12} + 8 x^{3} + 10$ | $12$ | $1$ | $35$ | $D_{12}$ | $$[3, 4]_{3}^{2}$$ |
|
\(3\)
| 3.2.6.14a1.3 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2624 x^{6} + 3264 x^{5} + 3123 x^{4} + 2252 x^{3} + 1176 x^{2} + 408 x + 79$ | $6$ | $2$ | $14$ | $D_6$ | $$[\frac{3}{2}]_{2}^{2}$$ |