Normalized defining polynomial
\( x^{12} - x^{11} - 3x^{10} + x^{9} + 5x^{8} + 3x^{7} - 3x^{6} - 7x^{5} - 2x^{4} + 3x^{3} + 2x^{2} + x + 1 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[0, 6]$ |
| |
| Discriminant: |
\(71939651121\)
\(\medspace = 3^{6}\cdot 13^{3}\cdot 44917\)
|
| |
| Root discriminant: | \(8.03\) |
| |
| Galois root discriminant: | $3^{1/2}13^{1/2}44917^{1/2}\approx 1323.5418391573423$ | ||
| Ramified primes: |
\(3\), \(13\), \(44917\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{583921}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$
| 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: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( a^{10} - 3 a^{8} - a^{7} + 2 a^{6} + 5 a^{5} + 3 a^{4} - 3 a^{3} - 4 a^{2} - 2 a - 1 \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{3}{2}a^{11}-\frac{3}{2}a^{10}-3a^{9}+\frac{1}{2}a^{8}+\frac{9}{2}a^{7}+4a^{6}-\frac{3}{2}a^{5}-\frac{11}{2}a^{4}-\frac{3}{2}a^{3}+\frac{1}{2}a^{2}+\frac{1}{2}a+\frac{1}{2}$, $2a^{11}-2a^{10}-5a^{9}+2a^{8}+7a^{7}+5a^{6}-4a^{5}-9a^{4}-a^{3}+3a^{2}$, $a^{11}-a^{10}-a^{9}-a^{8}+a^{7}+4a^{6}+2a^{5}-2a^{4}-3a^{3}-3a^{2}$, $2a^{11}-\frac{1}{2}a^{10}-5a^{9}-3a^{8}+\frac{11}{2}a^{7}+10a^{6}+4a^{5}-\frac{15}{2}a^{4}-8a^{3}-\frac{5}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-3a^{10}+\frac{13}{2}a^{8}+3a^{7}-5a^{6}-\frac{21}{2}a^{5}-5a^{4}+\frac{15}{2}a^{3}+7a^{2}+\frac{5}{2}a+2$
|
| |
| Regulator: | \( 4.93206182745 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 4.93206182745 \cdot 1}{6\cdot\sqrt{71939651121}}\cr\approx \mathstrut & 0.188569872878 \end{aligned}\]
Galois group
$S_3\wr D_4$ (as 12T274):
| A solvable group of order 10368 |
| The 54 conjugacy class representatives for $S_3\wr D_4$ |
| Character table for $S_3\wr D_4$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 4.0.117.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 sibling: | data not computed |
| Degree 18 sibling: | 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 | ${\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.4.0.1}{4} }$ | R | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.12.0.1}{12} }$ | R | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.6.2.6a1.2 | $x^{12} + 4 x^{10} + 6 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} + 9 x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $$[\ ]_{2}^{6}$$ |
|
\(13\)
| 13.2.1.0a1.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 13.4.1.0a1.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
| 13.3.2.3a1.1 | $x^{6} + 4 x^{4} + 22 x^{3} + 4 x^{2} + 57 x + 121$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
|
\(44917\)
| $\Q_{44917}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ |