Normalized defining polynomial
\( x^{12} + 12x^{10} + 48x^{8} + 96x^{6} + 108x^{4} - 144x^{2} - 144 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[2, 5]$ |
| |
| Discriminant: |
\(-328683126924509184\)
\(\medspace = -\,2^{36}\cdot 3^{14}\)
|
| |
| Root discriminant: | \(28.82\) |
| |
| Galois root discriminant: | $2^{59/16}3^{25/18}\approx 59.25384244131698$ | ||
| Ramified primes: |
\(2\), \(3\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{12}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{12}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{12}a^{8}$, $\frac{1}{24}a^{9}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{1344}a^{10}+\frac{3}{224}a^{8}+\frac{11}{336}a^{6}-\frac{1}{4}a^{5}-\frac{13}{56}a^{4}-\frac{1}{2}a^{3}+\frac{3}{16}a^{2}-\frac{1}{2}a+\frac{1}{56}$, $\frac{1}{1344}a^{11}+\frac{3}{224}a^{9}+\frac{11}{336}a^{7}-\frac{13}{56}a^{5}+\frac{3}{16}a^{3}+\frac{1}{56}a$
| 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: |
$\frac{1}{96}a^{10}+\frac{5}{48}a^{8}+\frac{7}{24}a^{6}+\frac{1}{4}a^{4}-\frac{3}{8}a^{2}-\frac{11}{4}$, $\frac{1}{84}a^{10}+\frac{11}{84}a^{8}+\frac{37}{84}a^{6}+\frac{11}{14}a^{4}+\frac{3}{2}a^{2}+\frac{2}{7}$, $\frac{1}{42}a^{10}+\frac{11}{42}a^{8}+\frac{37}{42}a^{6}+\frac{11}{7}a^{4}+2a^{2}-\frac{31}{7}$, $\frac{29}{336}a^{10}+\frac{163}{168}a^{8}+\frac{71}{21}a^{6}+\frac{39}{7}a^{4}+\frac{21}{4}a^{2}-\frac{223}{14}$, $\frac{13}{1344}a^{11}+\frac{1}{64}a^{10}+\frac{89}{672}a^{9}+\frac{19}{96}a^{8}+\frac{227}{336}a^{7}+\frac{15}{16}a^{6}+\frac{111}{56}a^{5}+\frac{21}{8}a^{4}+\frac{63}{16}a^{3}+\frac{71}{16}a^{2}+\frac{153}{56}a+\frac{11}{8}$, $\frac{29}{336}a^{11}+\frac{1}{14}a^{10}+\frac{163}{168}a^{9}+\frac{15}{28}a^{8}+\frac{83}{28}a^{7}-\frac{5}{14}a^{6}+\frac{11}{7}a^{5}-\frac{81}{14}a^{4}-\frac{5}{4}a^{3}-\frac{27}{14}a+\frac{19}{7}$
|
| |
| Regulator: | \( 23300.3648249 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 23300.3648249 \cdot 1}{2\cdot\sqrt{328683126924509184}}\cr\approx \mathstrut & 0.795982195383 \end{aligned}\]
Galois group
$S_4\wr C_2$ (as 12T200):
| A solvable group of order 1152 |
| The 20 conjugacy class representatives for $S_4\wr C_2$ |
| Character table for $S_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 6.4.5971968.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 sibling: | data not computed |
| Degree 12 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 8.2.195689447424.7 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{5}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.4.8b1.6 | $x^{4} + 4 x^{3} + 2 x^{2} + 4 x + 14$ | $4$ | $1$ | $8$ | $C_2^2$ | $$[2, 3]$$ |
| 2.1.8.28a1.7 | $x^{8} + 4 x^{6} + 8 x^{5} + 8 x^{4} + 2$ | $8$ | $1$ | $28$ | $(C_4^2 : C_2):C_2$ | $$[2, 2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$$ | |
|
\(3\)
| 3.2.6.14a1.2 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2630 x^{6} + 3303 x^{5} + 3240 x^{4} + 2462 x^{3} + 1410 x^{2} + 567 x + 124$ | $6$ | $2$ | $14$ | $(C_3\times C_3):C_4$ | $$[\frac{3}{2}, \frac{3}{2}]_{2}^{2}$$ |