Normalized defining polynomial
\( x^{12} + 3x^{10} + 9x^{8} + 14x^{6} + 18x^{4} + 12x^{2} + 4 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[0, 6]$ |
| |
| Discriminant: |
\(320979616137216\)
\(\medspace = 2^{26}\cdot 3^{14}\)
|
| |
| Root discriminant: | \(16.18\) |
| |
| Galois root discriminant: | $2^{55/16}3^{7/6}\approx 39.0330127936465$ | ||
| Ramified primes: |
\(2\), \(3\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$
| 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 $1$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -\frac{1}{2} a^{10} - a^{8} - 3 a^{6} - \frac{7}{2} a^{4} - 4 a^{2} - 1 \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{1}{2}a^{10}+a^{8}+3a^{6}+\frac{7}{2}a^{4}+3a^{2}+1$, $\frac{1}{2}a^{11}+a^{9}+3a^{7}+\frac{7}{2}a^{5}+4a^{3}+a-1$, $\frac{3}{2}a^{10}+\frac{7}{2}a^{8}+\frac{21}{2}a^{6}+13a^{4}+15a^{2}+5$, $\frac{1}{2}a^{11}+\frac{1}{2}a^{10}+a^{9}+\frac{3}{2}a^{8}+3a^{7}+\frac{7}{2}a^{6}+\frac{5}{2}a^{5}+5a^{4}+3a^{3}+5a^{2}+3$, $\frac{1}{2}a^{11}+\frac{1}{2}a^{10}+2a^{9}+\frac{3}{2}a^{8}+5a^{7}+\frac{7}{2}a^{6}+\frac{17}{2}a^{5}+5a^{4}+9a^{3}+4a^{2}+5a+1$
|
| |
| Regulator: | \( 727.506577153 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 727.506577153 \cdot 1}{6\cdot\sqrt{320979616137216}}\cr\approx \mathstrut & 0.416414800484 \end{aligned}\]
Galois group
$C_2^3:S_4$ (as 12T108):
| A solvable group of order 192 |
| The 14 conjugacy class representatives for $C_2^3:S_4$ |
| Character table for $C_2^3:S_4$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.216.1, 6.0.139968.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 siblings: | data not computed |
| Degree 12 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 8.0.27518828544.5 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.4a2.2 | $x^{4} + 4 x^{3} + 5 x^{2} + 4 x + 7$ | $2$ | $2$ | $4$ | $D_{4}$ | $$[2, 2]^{2}$$ |
| 2.2.4.22a1.58 | $x^{8} + 4 x^{7} + 18 x^{6} + 40 x^{5} + 71 x^{4} + 88 x^{3} + 78 x^{2} + 60 x + 15$ | $4$ | $2$ | $22$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $$[2, 2, 3, \frac{7}{2}, 4]^{2}$$ | |
|
\(3\)
| 3.2.6.14a2.1 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2624 x^{6} + 3264 x^{5} + 3126 x^{4} + 2264 x^{3} + 1200 x^{2} + 432 x + 91$ | $6$ | $2$ | $14$ | $D_6$ | $$[\frac{3}{2}]_{2}^{2}$$ |