Normalized defining polynomial
\( x^{12} - 12x^{10} + 42x^{8} - 84x^{6} + 225x^{4} - 432x^{2} + 288 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[8, 2]$ |
| |
| Discriminant: |
\(657366253849018368\)
\(\medspace = 2^{37}\cdot 3^{14}\)
|
| |
| Root discriminant: | \(30.54\) |
| |
| Galois root discriminant: | $2^{27/8}3^{25/18}\approx 47.713870191292706$ | ||
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{6}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{12}a^{7}-\frac{1}{12}a^{6}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{12}a^{8}-\frac{1}{12}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{12}a^{9}-\frac{1}{12}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{2}$, $\frac{1}{372}a^{10}-\frac{7}{186}a^{8}+\frac{2}{93}a^{6}-\frac{27}{62}a^{4}-\frac{3}{124}a^{2}+\frac{12}{31}$, $\frac{1}{744}a^{11}-\frac{7}{372}a^{9}+\frac{1}{93}a^{7}-\frac{27}{124}a^{5}-\frac{3}{248}a^{3}-\frac{19}{62}a$
| 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: | $9$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{372}a^{10}-\frac{7}{186}a^{8}+\frac{35}{186}a^{6}-\frac{27}{62}a^{4}+\frac{59}{124}a^{2}-\frac{19}{31}$, $\frac{7}{124}a^{10}-\frac{67}{124}a^{8}+\frac{385}{372}a^{6}-\frac{235}{124}a^{4}+\frac{209}{31}a^{2}-\frac{213}{31}$, $\frac{1}{186}a^{11}+\frac{3}{62}a^{10}-\frac{7}{93}a^{9}-\frac{95}{186}a^{8}+\frac{109}{372}a^{7}+\frac{485}{372}a^{6}-\frac{23}{62}a^{5}-\frac{145}{62}a^{4}+\frac{149}{124}a^{3}+\frac{969}{124}a^{2}-\frac{169}{62}a-\frac{311}{31}$, $\frac{7}{744}a^{11}+\frac{35}{372}a^{10}-\frac{49}{372}a^{9}-\frac{335}{372}a^{8}+\frac{61}{124}a^{7}+\frac{163}{93}a^{6}-\frac{65}{124}a^{5}-\frac{433}{124}a^{4}+\frac{661}{248}a^{3}+\frac{1507}{124}a^{2}-\frac{257}{62}a-\frac{355}{31}$, $\frac{35}{248}a^{11}+\frac{8}{31}a^{10}-\frac{259}{186}a^{9}-\frac{81}{31}a^{8}+\frac{551}{186}a^{7}+\frac{2225}{372}a^{6}-\frac{174}{31}a^{5}-\frac{335}{31}a^{4}+\frac{5017}{248}a^{3}+\frac{4703}{124}a^{2}-\frac{1189}{62}a-\frac{1235}{31}$, $\frac{1}{186}a^{11}-\frac{43}{372}a^{10}-\frac{7}{93}a^{9}+\frac{149}{124}a^{8}+\frac{109}{372}a^{7}-\frac{272}{93}a^{6}-\frac{23}{62}a^{5}+\frac{617}{124}a^{4}+\frac{149}{124}a^{3}-\frac{2227}{124}a^{2}-\frac{169}{62}a+\frac{631}{31}$, $\frac{1}{12}a^{10}-\frac{5}{6}a^{8}+\frac{11}{6}a^{6}-\frac{7}{2}a^{4}+\frac{55}{4}a^{2}-15$, $\frac{1}{186}a^{11}+\frac{43}{372}a^{10}-\frac{7}{93}a^{9}-\frac{149}{124}a^{8}+\frac{109}{372}a^{7}+\frac{272}{93}a^{6}-\frac{23}{62}a^{5}-\frac{617}{124}a^{4}+\frac{149}{124}a^{3}+\frac{2227}{124}a^{2}-\frac{169}{62}a-\frac{631}{31}$, $\frac{55}{744}a^{11}+\frac{1}{372}a^{10}-\frac{73}{93}a^{9}-\frac{7}{186}a^{8}+\frac{249}{124}a^{7}+\frac{35}{186}a^{6}-\frac{100}{31}a^{5}-\frac{27}{62}a^{4}+\frac{2749}{248}a^{3}+\frac{59}{124}a^{2}-\frac{414}{31}a+\frac{43}{31}$
|
| |
| Regulator: | \( 153152.048963 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{2}\cdot 153152.048963 \cdot 1}{2\cdot\sqrt{657366253849018368}}\cr\approx \mathstrut & 0.954528313266 \end{aligned}\]
Galois group
$S_3^2:C_4$ (as 12T79):
| A solvable group of order 144 |
| The 18 conjugacy class representatives for $S_3^2:C_4$ |
| Character table for $S_3^2:C_4$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.4.18432.1, 6.4.5971968.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
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 | ${\href{/padicField/5.12.0.1}{12} }$ | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.4.11a1.10 | $x^{4} + 4 x^{2} + 18$ | $4$ | $1$ | $11$ | $C_4$ | $$[3, 4]$$ |
| 2.1.8.26c1.12 | $x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + 8 x^{4} + 8 x^{3} + 18$ | $8$ | $1$ | $26$ | $C_2^2:C_4$ | $$[2, 3, \frac{7}{2}, 4]$$ | |
|
\(3\)
| 3.2.6.14a1.2 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2630 x^{6} + 3303 x^{5} + 3240 x^{4} + 2462 x^{3} + 1410 x^{2} + 567 x + 124$ | $6$ | $2$ | $14$ | $(C_3\times C_3):C_4$ | $$[\frac{3}{2}, \frac{3}{2}]_{2}^{2}$$ |