Normalized defining polynomial
\( x^{12} - 6x^{10} + 3x^{8} + 48x^{6} - 72x^{2} - 36 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[2, 5]$ |
| |
| Discriminant: |
\(-328683126924509184\)
\(\medspace = -\,2^{36}\cdot 3^{14}\)
|
| |
| Root discriminant: | \(28.82\) |
| |
| Galois root discriminant: | $2^{55/16}3^{25/18}\approx 49.82634369891186$ | ||
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{24}a^{8}-\frac{1}{12}a^{6}+\frac{3}{8}a^{4}+\frac{1}{4}$, $\frac{1}{24}a^{9}-\frac{1}{12}a^{7}+\frac{3}{8}a^{5}+\frac{1}{4}a$, $\frac{1}{48}a^{10}-\frac{1}{6}a^{7}+\frac{5}{48}a^{6}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}-\frac{3}{8}a^{2}+\frac{1}{4}$, $\frac{1}{48}a^{11}+\frac{5}{48}a^{7}-\frac{1}{8}a^{5}-\frac{3}{8}a^{3}+\frac{1}{4}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
| |
| Narrow class group: | $C_{2}\times C_{2}$, which has order $4$ |
|
Unit group
| Rank: | $6$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{4}a^{10}-\frac{5}{3}a^{8}+\frac{23}{12}a^{6}+\frac{21}{2}a^{4}-\frac{15}{2}a^{2}-12$, $\frac{1}{24}a^{10}-\frac{7}{24}a^{8}+\frac{11}{24}a^{6}+\frac{9}{8}a^{4}-\frac{3}{4}a^{2}-\frac{5}{4}$, $\frac{1}{4}a^{11}+\frac{5}{48}a^{10}-\frac{41}{24}a^{9}-\frac{17}{24}a^{8}+\frac{13}{6}a^{7}+\frac{15}{16}a^{6}+\frac{81}{8}a^{5}+4a^{4}-8a^{3}-\frac{23}{8}a^{2}-\frac{49}{4}a-4$, $\frac{1}{4}a^{11}-\frac{5}{48}a^{10}-\frac{41}{24}a^{9}+\frac{17}{24}a^{8}+\frac{13}{6}a^{7}-\frac{15}{16}a^{6}+\frac{81}{8}a^{5}-4a^{4}-8a^{3}+\frac{23}{8}a^{2}-\frac{49}{4}a+4$, $\frac{5}{4}a^{10}-\frac{69}{8}a^{8}+\frac{23}{2}a^{6}+\frac{399}{8}a^{4}-\frac{89}{2}a^{2}-\frac{203}{4}$, $\frac{5}{24}a^{11}-\frac{1}{8}a^{10}-\frac{5}{4}a^{9}+\frac{5}{8}a^{8}+\frac{7}{8}a^{7}+\frac{1}{8}a^{6}+\frac{17}{2}a^{5}-\frac{45}{8}a^{4}+\frac{1}{4}a^{3}-\frac{19}{4}a^{2}-4a+\frac{1}{4}$
|
| |
| Regulator: | \( 12241.4549258 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 12241.4549258 \cdot 2}{2\cdot\sqrt{328683126924509184}}\cr\approx \mathstrut & 0.836380051536 \end{aligned}\]
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: | 8.2.3057647616.2 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.4.10a1.1 | $x^{4} + 4 x^{3} + 2$ | $4$ | $1$ | $10$ | $D_{4}$ | $$[2, 3, \frac{7}{2}]$$ |
| 2.1.8.26c1.8 | $x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{4} + 8 x^{3} + 2$ | $8$ | $1$ | $26$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $$[2, 2, 3, \frac{7}{2}, 4]^{2}$$ | |
|
\(3\)
| 3.2.6.14a1.2 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2630 x^{6} + 3303 x^{5} + 3240 x^{4} + 2462 x^{3} + 1410 x^{2} + 567 x + 124$ | $6$ | $2$ | $14$ | $(C_3\times C_3):C_4$ | $$[\frac{3}{2}, \frac{3}{2}]_{2}^{2}$$ |