Normalized defining polynomial
\( x^{12} - 26x^{10} + 272x^{8} - 1356x^{6} + 3036x^{4} - 2104x^{2} + 676 \)
Invariants
| Degree: | $12$ |
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| Signature: | $[0, 6]$ |
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| Discriminant: |
\(209678066018942976\)
\(\medspace = 2^{22}\cdot 3^{6}\cdot 7^{4}\cdot 13^{4}\)
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| Root discriminant: | \(27.76\) |
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| Galois root discriminant: | $2^{11/6}3^{1/2}7^{2/3}13^{2/3}\approx 124.8753008745792$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(13\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_6$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-2}, \sqrt{-3})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{28}a^{8}-\frac{1}{7}a^{6}+\frac{2}{7}a^{4}-\frac{1}{2}a^{2}+\frac{1}{7}$, $\frac{1}{364}a^{9}+\frac{6}{91}a^{7}+\frac{30}{91}a^{5}+\frac{3}{26}a^{3}+\frac{36}{91}a$, $\frac{1}{117572}a^{10}+\frac{71}{29393}a^{8}+\frac{335}{3094}a^{6}-\frac{22053}{58786}a^{4}-\frac{12067}{29393}a^{2}-\frac{645}{2261}$, $\frac{1}{117572}a^{11}-\frac{3}{9044}a^{9}+\frac{131}{3094}a^{7}+\frac{2479}{8398}a^{5}+\frac{27869}{58786}a^{3}+\frac{1340}{4199}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{3}$, which has order $3$ |
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| Narrow class group: | $C_{3}$, which has order $3$ |
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Unit group
| Rank: | $5$ |
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| Torsion generator: |
\( \frac{9}{3094} a^{10} - \frac{59}{884} a^{8} + \frac{930}{1547} a^{6} - \frac{3776}{1547} a^{4} + \frac{12671}{3094} a^{2} - \frac{135}{119} \)
(order $6$)
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| Fundamental units: |
$\frac{311}{58786}a^{10}+\frac{12307}{117572}a^{8}-\frac{1199}{1547}a^{6}+\frac{64501}{29393}a^{4}-\frac{126047}{58786}a^{2}+\frac{4223}{2261}$, $\frac{132}{29393}a^{10}+\frac{13809}{117572}a^{8}-\frac{3837}{3094}a^{6}+\frac{26105}{4199}a^{4}-\frac{807421}{58786}a^{2}+\frac{2647}{323}$, $\frac{129}{117572}a^{11}+\frac{9}{3094}a^{10}+\frac{122}{4199}a^{9}-\frac{59}{884}a^{8}-\frac{451}{1547}a^{7}+\frac{930}{1547}a^{6}+\frac{5753}{4522}a^{5}-\frac{3776}{1547}a^{4}-\frac{50928}{29393}a^{3}+\frac{12671}{3094}a^{2}-\frac{33654}{29393}a-\frac{16}{119}$, $\frac{3}{1292}a^{11}+\frac{61}{29393}a^{10}+\frac{7417}{117572}a^{9}-\frac{449}{8398}a^{8}-\frac{1111}{1547}a^{7}+\frac{1517}{3094}a^{6}+\frac{241999}{58786}a^{5}-\frac{49295}{29393}a^{4}-\frac{97219}{8398}a^{3}-\frac{5048}{29393}a^{2}+\frac{321066}{29393}a+\frac{13164}{2261}$, $\frac{743}{117572}a^{11}+\frac{12}{4199}a^{10}-\frac{2663}{16796}a^{9}-\frac{9551}{117572}a^{8}+\frac{4853}{3094}a^{7}+\frac{1472}{1547}a^{6}-\frac{418843}{58786}a^{5}-\frac{152550}{29393}a^{4}+\frac{760125}{58786}a^{3}+\frac{97069}{8398}a^{2}-\frac{65600}{29393}a-\frac{7739}{2261}$
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| Regulator: | \( 4699.920698268625 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 4699.920698268625 \cdot 3}{6\cdot\sqrt{209678066018942976}}\cr\approx \mathstrut & 0.315764457038560 \end{aligned}\]
Galois group
$C_6\times S_3$ (as 12T18):
| A solvable group of order 36 |
| The 18 conjugacy class representatives for $C_6\times S_3$ |
| Character table for $C_6\times S_3$ |
Intermediate fields
| \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}, \sqrt{-3})\), 6.0.3577392.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 18 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 | R | R | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{6}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 | $x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 126 x^{7} + 141 x^{6} + 126 x^{5} + 90 x^{4} + 50 x^{3} + 21 x^{2} + 6 x + 3$ | $6$ | $2$ | $22$ | $D_6$ | $$[3]_{3}^{2}$$ |
|
\(3\)
| 3.3.2.3a1.2 | $x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
| 3.3.2.3a1.2 | $x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
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\(7\)
| 7.2.1.0a1.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 7.2.1.0a1.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 7.2.1.0a1.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 7.2.3.4a1.1 | $x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 169 x + 27$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ | |
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\(13\)
| 13.2.1.0a1.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 13.2.1.0a1.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 13.2.1.0a1.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 13.2.3.4a1.2 | $x^{6} + 36 x^{5} + 438 x^{4} + 1872 x^{3} + 876 x^{2} + 144 x + 21$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ |