Normalized defining polynomial
\( x^{12} - 6 x^{11} + 21 x^{10} - 50 x^{9} + 90 x^{8} - 126 x^{7} + 177 x^{6} - 234 x^{5} - 180 x^{4} + \cdots + 361 \)
Invariants
| Degree: | $12$ |
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| Signature: | $[0, 6]$ |
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| Discriminant: |
\(131621703842267136\)
\(\medspace = 2^{22}\cdot 3^{22}\)
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| Root discriminant: | \(26.71\) |
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| Galois root discriminant: | $2^{11/6}3^{11/6}\approx 26.706109521254483$ | ||
| Ramified primes: |
\(2\), \(3\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $D_6$ |
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| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-2}, \sqrt{-3})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{2}$, $\frac{1}{9}a^{6}-\frac{1}{9}a^{3}-\frac{2}{9}$, $\frac{1}{9}a^{7}-\frac{1}{9}a^{4}-\frac{2}{9}a$, $\frac{1}{27}a^{8}-\frac{1}{27}a^{7}+\frac{1}{27}a^{6}+\frac{2}{27}a^{5}-\frac{2}{27}a^{4}+\frac{2}{27}a^{3}-\frac{8}{27}a^{2}+\frac{8}{27}a-\frac{8}{27}$, $\frac{1}{27}a^{9}-\frac{1}{9}a^{3}-\frac{2}{27}$, $\frac{1}{513}a^{10}-\frac{5}{513}a^{9}-\frac{1}{171}a^{8}-\frac{5}{171}a^{7}+\frac{2}{57}a^{6}-\frac{17}{171}a^{5}+\frac{4}{171}a^{4}+\frac{2}{171}a^{3}+\frac{56}{171}a^{2}-\frac{188}{513}a-\frac{8}{27}$, $\frac{1}{192375}a^{11}+\frac{182}{192375}a^{10}+\frac{487}{192375}a^{9}-\frac{1123}{64125}a^{8}+\frac{781}{64125}a^{7}+\frac{2036}{64125}a^{6}-\frac{547}{7125}a^{5}+\frac{1466}{21375}a^{4}+\frac{71}{7125}a^{3}+\frac{94441}{192375}a^{2}-\frac{77716}{192375}a+\frac{1462}{10125}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{3}\times C_{3}$, which has order $9$ |
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| Narrow class group: | $C_{3}\times C_{3}$, which has order $9$ |
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Unit group
| Rank: | $5$ |
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| Torsion generator: |
\( -\frac{308}{64125} a^{11} + \frac{1694}{64125} a^{10} - \frac{6496}{64125} a^{9} + \frac{5509}{21375} a^{8} - \frac{11548}{21375} a^{7} + \frac{18662}{21375} a^{6} - \frac{30716}{21375} a^{5} + \frac{2114}{1125} a^{4} - \frac{14812}{21375} a^{3} - \frac{34328}{64125} a^{2} + \frac{140528}{64125} a - \frac{1546}{3375} \)
(order $6$)
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| Fundamental units: |
$\frac{91}{64125}a^{11}-\frac{313}{64125}a^{10}+\frac{442}{64125}a^{9}+\frac{182}{21375}a^{8}-\frac{1304}{21375}a^{7}+\frac{364}{2375}a^{6}-\frac{1456}{7125}a^{5}+\frac{552}{2375}a^{4}-\frac{23101}{21375}a^{3}+\frac{9256}{64125}a^{2}+\frac{140219}{64125}a+\frac{3292}{3375}$, $\frac{2296}{192375}a^{11}-\frac{10003}{192375}a^{10}+\frac{29902}{192375}a^{9}-\frac{17783}{64125}a^{8}+\frac{25676}{64125}a^{7}-\frac{20594}{64125}a^{6}+\frac{43792}{64125}a^{5}-\frac{25942}{64125}a^{4}-\frac{288731}{64125}a^{3}+\frac{658036}{192375}a^{2}+\frac{707189}{192375}a-\frac{13748}{10125}$, $\frac{91}{192375}a^{11}-\frac{1063}{192375}a^{10}+\frac{4192}{192375}a^{9}-\frac{3818}{64125}a^{8}+\frac{7196}{64125}a^{7}-\frac{10724}{64125}a^{6}+\frac{13132}{64125}a^{5}-\frac{17032}{64125}a^{4}-\frac{7976}{64125}a^{3}+\frac{424756}{192375}a^{2}-\frac{22699}{10125}a+\frac{20167}{10125}$, $\frac{1568}{192375}a^{11}-\frac{5999}{192375}a^{10}+\frac{18866}{192375}a^{9}-\frac{11239}{64125}a^{8}+\frac{19108}{64125}a^{7}-\frac{18802}{64125}a^{6}+\frac{43736}{64125}a^{5}-\frac{21686}{64125}a^{4}-\frac{176923}{64125}a^{3}+\frac{137738}{192375}a^{2}-\frac{41063}{192375}a-\frac{19834}{10125}$, $\frac{3227}{192375}a^{11}-\frac{13061}{192375}a^{10}+\frac{41924}{192375}a^{9}-\frac{1384}{3375}a^{8}+\frac{45412}{64125}a^{7}-\frac{48328}{64125}a^{6}+\frac{100604}{64125}a^{5}-\frac{60404}{64125}a^{4}-\frac{319072}{64125}a^{3}+\frac{315482}{192375}a^{2}-\frac{128657}{192375}a-\frac{33001}{10125}$
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| Regulator: | \( 3703.01164497 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 3703.01164497 \cdot 9}{6\cdot\sqrt{131621703842267136}}\cr\approx \mathstrut & 0.942023621049 \end{aligned}\]
Galois group
| A solvable group of order 12 |
| The 6 conjugacy class representatives for $D_6$ |
| Character table for $D_6$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), 3.1.972.1 x3, \(\Q(\sqrt{-2}, \sqrt{-3})\), 6.0.2834352.1, 6.0.120932352.2 x3, 6.2.362797056.11 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
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 | ${\href{/padicField/5.2.0.1}{2} }^{6}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.1.0.1}{1} }^{12}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 | $x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 126 x^{7} + 141 x^{6} + 126 x^{5} + 90 x^{4} + 50 x^{3} + 21 x^{2} + 6 x + 3$ | $6$ | $2$ | $22$ | $D_6$ | $$[3]_{3}^{2}$$ |
|
\(3\)
| 3.1.6.11a1.3 | $x^{6} + 9 x^{2} + 3$ | $6$ | $1$ | $11$ | $S_3$ | $$[\frac{5}{2}]_{2}$$ |
| 3.1.6.11a1.3 | $x^{6} + 9 x^{2} + 3$ | $6$ | $1$ | $11$ | $S_3$ | $$[\frac{5}{2}]_{2}$$ |