Normalized defining polynomial
\( x^{12} - 6 x^{11} + 29 x^{10} - 90 x^{9} + 242 x^{8} - 494 x^{7} + 893 x^{6} - 1262 x^{5} + 1594 x^{4} + \cdots + 387 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[0, 6]$ |
| |
| Discriminant: |
\(1721085137518592\)
\(\medspace = 2^{22}\cdot 17^{7}\)
|
| |
| Root discriminant: | \(18.61\) |
| |
| Galois root discriminant: | $2^{11/6}17^{5/6}\approx 37.780041142444276$ | ||
| Ramified primes: |
\(2\), \(17\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\Aut(K/\Q)$: | $S_3$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-2}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{159}a^{10}-\frac{5}{159}a^{9}-\frac{2}{53}a^{8}+\frac{18}{53}a^{7}-\frac{1}{159}a^{6}-\frac{16}{53}a^{5}-\frac{79}{159}a^{4}-\frac{20}{53}a^{3}-\frac{71}{159}a^{2}+\frac{56}{159}a-\frac{1}{53}$, $\frac{1}{12879}a^{11}+\frac{35}{12879}a^{10}+\frac{2}{4293}a^{9}-\frac{380}{4293}a^{8}+\frac{887}{12879}a^{7}-\frac{4222}{12879}a^{6}-\frac{1108}{4293}a^{5}+\frac{5419}{12879}a^{4}-\frac{1495}{4293}a^{3}-\frac{3473}{12879}a^{2}-\frac{358}{1431}a+\frac{22}{1431}$
| 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: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{466}{12879}a^{11}-\frac{2077}{12879}a^{10}+\frac{2957}{4293}a^{9}-\frac{7223}{4293}a^{8}+\frac{51515}{12879}a^{7}-\frac{81610}{12879}a^{6}+\frac{42599}{4293}a^{5}-\frac{126281}{12879}a^{4}+\frac{45992}{4293}a^{3}-\frac{59654}{12879}a^{2}+\frac{5918}{1431}a+\frac{2071}{1431}$, $\frac{97}{12879}a^{11}-\frac{574}{12879}a^{10}+\frac{1085}{4293}a^{9}-\frac{3164}{4293}a^{8}+\frac{26261}{12879}a^{7}-\frac{44953}{12879}a^{6}+\frac{26147}{4293}a^{5}-\frac{79508}{12879}a^{4}+\frac{31673}{4293}a^{3}-\frac{37910}{12879}a^{2}+\frac{4019}{1431}a+\frac{2026}{1431}$, $\frac{97}{12879}a^{11}-\frac{493}{12879}a^{10}+\frac{950}{4293}a^{9}-\frac{3326}{4293}a^{8}+\frac{30635}{12879}a^{7}-\frac{70792}{12879}a^{6}+\frac{46316}{4293}a^{5}-\frac{214697}{12879}a^{4}+\frac{90155}{4293}a^{3}-\frac{262604}{12879}a^{2}+\frac{20264}{1431}a-\frac{9449}{1431}$, $\frac{326}{12879}a^{11}-\frac{902}{12879}a^{10}+\frac{1138}{4293}a^{9}-\frac{517}{4293}a^{8}-\frac{2195}{12879}a^{7}+\frac{39751}{12879}a^{6}-\frac{28271}{4293}a^{5}+\frac{193493}{12879}a^{4}-\frac{76565}{4293}a^{3}+\frac{291458}{12879}a^{2}-\frac{19211}{1431}a+\frac{17000}{1431}$, $\frac{326}{12879}a^{11}-\frac{2684}{12879}a^{10}+\frac{4108}{4293}a^{9}-\frac{14125}{4293}a^{8}+\frac{107641}{12879}a^{7}-\frac{228926}{12879}a^{6}+\frac{124738}{4293}a^{5}-\frac{9974}{243}a^{4}+\frac{186604}{4293}a^{3}-\frac{522187}{12879}a^{2}+\frac{34096}{1431}a-\frac{21043}{1431}$
|
| |
| Regulator: | \( 554.099176051 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 554.099176051 \cdot 1}{2\cdot\sqrt{1721085137518592}}\cr\approx \mathstrut & 0.410899432753 \end{aligned}\]
Galois group
$\SOPlus(4,2)$ (as 12T35):
| A solvable group of order 72 |
| The 9 conjugacy class representatives for $\SOPlus(4,2)$ |
| Character table for $\SOPlus(4,2)$ |
Intermediate fields
| \(\Q(\sqrt{-2}) \), 4.0.1088.1, 6.0.10061824.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 6 siblings: | 6.0.10061824.1, some data not computed |
| Degree 9 sibling: | data not computed |
| Degree 12 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 6.0.10061824.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{3}$ | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.4.0.1}{4} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.6.22a1.1 | $x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 126 x^{7} + 141 x^{6} + 126 x^{5} + 90 x^{4} + 50 x^{3} + 21 x^{2} + 6 x + 3$ | $6$ | $2$ | $22$ | $D_6$ | $$[3]_{3}^{2}$$ |
|
\(17\)
| 17.1.2.1a1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 17.1.2.1a1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 17.1.2.1a1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 17.2.3.4a1.2 | $x^{6} + 48 x^{5} + 777 x^{4} + 4384 x^{3} + 2331 x^{2} + 432 x + 44$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ |