Normalized defining polynomial
\( x^{12} - 3x^{8} - 4x^{6} + 9x^{4} + 6x^{2} + 4 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[0, 6]$ |
| |
| Discriminant: |
\(320979616137216\)
\(\medspace = 2^{26}\cdot 3^{14}\)
|
| |
| Root discriminant: | \(16.18\) |
| |
| Galois root discriminant: | $2^{11/4}3^{7/6}\approx 24.23672593327708$ | ||
| Ramified primes: |
\(2\), \(3\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2^2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-3}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{3}$, $\frac{1}{18}a^{10}+\frac{1}{9}a^{8}+\frac{1}{18}a^{6}+\frac{2}{9}a^{4}-\frac{1}{18}a^{2}+\frac{2}{9}$, $\frac{1}{18}a^{11}+\frac{1}{9}a^{9}+\frac{1}{18}a^{7}+\frac{2}{9}a^{5}-\frac{1}{18}a^{3}+\frac{2}{9}a$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
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Unit group
| Rank: | $5$ |
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| Torsion generator: |
\( -\frac{1}{6} a^{10} + \frac{1}{2} a^{6} + \frac{1}{3} a^{4} - \frac{3}{2} a^{2} \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{1}{9}a^{10}-\frac{1}{9}a^{8}-\frac{2}{9}a^{6}-\frac{5}{9}a^{4}+\frac{14}{9}a^{2}+\frac{1}{9}$, $\frac{2}{9}a^{11}-\frac{2}{9}a^{10}-\frac{2}{9}a^{9}-\frac{1}{9}a^{8}-\frac{4}{9}a^{7}+\frac{7}{9}a^{6}-\frac{1}{9}a^{5}+\frac{10}{9}a^{4}+\frac{19}{9}a^{3}-\frac{13}{9}a^{2}-\frac{16}{9}a-\frac{17}{9}$, $\frac{1}{18}a^{10}+\frac{1}{9}a^{8}-\frac{5}{18}a^{6}+\frac{2}{9}a^{4}-\frac{1}{18}a^{2}-\frac{1}{9}$, $\frac{5}{18}a^{11}-\frac{1}{18}a^{10}-\frac{1}{9}a^{9}+\frac{2}{9}a^{8}-\frac{19}{18}a^{7}+\frac{5}{18}a^{6}-\frac{8}{9}a^{5}-\frac{2}{9}a^{4}+\frac{55}{18}a^{3}-\frac{11}{18}a^{2}+\frac{7}{9}a+\frac{10}{9}$, $\frac{2}{9}a^{11}+\frac{1}{18}a^{10}-\frac{2}{9}a^{9}+\frac{1}{9}a^{8}-\frac{4}{9}a^{7}+\frac{1}{18}a^{6}-\frac{1}{9}a^{5}-\frac{7}{9}a^{4}+\frac{19}{9}a^{3}-\frac{1}{18}a^{2}-\frac{7}{9}a+\frac{11}{9}$
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| Regulator: | \( 889.528343238 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 889.528343238 \cdot 1}{6\cdot\sqrt{320979616137216}}\cr\approx \mathstrut & 0.509153840263 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 12T24):
| A solvable group of order 48 |
| The 10 conjugacy class representatives for $C_2 \times S_4$ |
| Character table for $C_2 \times S_4$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.216.1, 6.0.139968.1, 6.2.5971968.1, 6.0.17915904.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 6 siblings: | 6.0.8957952.1, some data not computed |
| Degree 8 siblings: | 8.4.3057647616.6, 8.0.1719926784.2 |
| Degree 12 siblings: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| Minimal sibling: | 6.2.5971968.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.6a1.1 | $x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 3$ | $2$ | $2$ | $6$ | $C_2^2$ | $$[3]^{2}$$ |
| 2.2.4.20a1.4 | $x^{8} + 4 x^{7} + 14 x^{6} + 36 x^{5} + 59 x^{4} + 76 x^{3} + 58 x^{2} + 32 x + 7$ | $4$ | $2$ | $20$ | $D_4\times C_2$ | $$[2, 3, \frac{7}{2}]^{2}$$ | |
|
\(3\)
| 3.2.6.14a2.1 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2624 x^{6} + 3264 x^{5} + 3126 x^{4} + 2264 x^{3} + 1200 x^{2} + 432 x + 91$ | $6$ | $2$ | $14$ | $D_6$ | $$[\frac{3}{2}]_{2}^{2}$$ |