Normalized defining polynomial
\( x^{12} - 6x^{10} + 3x^{8} + 12x^{6} + 9x^{4} - 18x^{2} - 9 \)
Invariants
| Degree: | $12$ |
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| Signature: | $[6, 3]$ |
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| Discriminant: |
\(-82170781731127296\)
\(\medspace = -\,2^{34}\cdot 3^{14}\)
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| Root discriminant: | \(25.68\) |
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| Galois root discriminant: | $2^{29/8}3^{25/18}\approx 56.74167391580153$ | ||
| Ramified primes: |
\(2\), \(3\)
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| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{6}a^{8}-\frac{1}{2}$, $\frac{1}{6}a^{9}-\frac{1}{2}a$, $\frac{1}{24}a^{10}-\frac{1}{12}a^{9}-\frac{1}{24}a^{8}+\frac{1}{12}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{3}{8}a^{2}-\frac{1}{4}a+\frac{3}{8}$, $\frac{1}{24}a^{11}-\frac{1}{24}a^{9}-\frac{1}{12}a^{8}+\frac{1}{12}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{3}{8}a^{3}+\frac{3}{8}a-\frac{1}{4}$
| 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: | $C_{2}$, which has order $2$ |
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Unit group
| Rank: | $8$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{1}{12}a^{10}-\frac{7}{12}a^{8}+\frac{5}{6}a^{6}+\frac{1}{2}a^{4}-\frac{3}{4}a^{2}-\frac{11}{4}$, $\frac{1}{6}a^{10}-\frac{5}{6}a^{8}-\frac{1}{3}a^{6}+2a^{4}+\frac{3}{2}a^{2}+\frac{1}{2}$, $\frac{1}{8}a^{11}+\frac{1}{8}a^{10}-\frac{7}{8}a^{9}-\frac{17}{24}a^{8}+\frac{5}{4}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{5}{4}a^{4}+\frac{7}{8}a^{3}-\frac{1}{8}a^{2}-\frac{17}{8}a-\frac{5}{8}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{7}{8}a^{9}+\frac{17}{24}a^{8}+\frac{5}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{5}{4}a^{4}+\frac{7}{8}a^{3}+\frac{1}{8}a^{2}-\frac{17}{8}a+\frac{5}{8}$, $\frac{1}{24}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}+\frac{7}{8}a^{8}-\frac{7}{12}a^{7}-\frac{5}{4}a^{6}+\frac{3}{4}a^{5}-\frac{1}{4}a^{4}+\frac{13}{8}a^{3}-\frac{7}{8}a^{2}-\frac{7}{8}a+\frac{25}{8}$, $\frac{1}{24}a^{11}+\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{7}{8}a^{8}-\frac{7}{12}a^{7}+\frac{5}{4}a^{6}+\frac{3}{4}a^{5}+\frac{1}{4}a^{4}+\frac{13}{8}a^{3}+\frac{7}{8}a^{2}-\frac{7}{8}a-\frac{25}{8}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{5}{4}a^{9}+\frac{5}{4}a^{8}-\frac{1}{2}a^{7}+\frac{1}{2}a^{6}+\frac{5}{2}a^{5}-\frac{5}{2}a^{4}+\frac{19}{4}a^{3}-\frac{19}{4}a^{2}+\frac{1}{4}a-\frac{5}{4}$, $\frac{1}{6}a^{11}-a^{9}+\frac{1}{6}a^{8}+\frac{2}{3}a^{7}-a^{6}+a^{5}+\frac{5}{2}a^{3}+3a^{2}-a+\frac{5}{2}$
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| Regulator: | \( 66877.8468166 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{3}\cdot 66877.8468166 \cdot 1}{2\cdot\sqrt{82170781731127296}}\cr\approx \mathstrut & 1.85188134268 \end{aligned}\]
Galois group
$S_4\wr C_2$ (as 12T201):
| 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: | 8.2.195689447424.6 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.3a1.3 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ |
| 2.1.2.3a1.3 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ | |
| 2.1.8.28a1.12 | $x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + 24 x^{4} + 16 x + 2$ | $8$ | $1$ | $28$ | $Z_8 : Z_8^\times$ | $$[2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$$ | |
|
\(3\)
| 3.2.6.14a1.2 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2630 x^{6} + 3303 x^{5} + 3240 x^{4} + 2462 x^{3} + 1410 x^{2} + 567 x + 124$ | $6$ | $2$ | $14$ | $(C_3\times C_3):C_4$ | $$[\frac{3}{2}, \frac{3}{2}]_{2}^{2}$$ |