Normalized defining polynomial
\( x^{12} + 9x^{10} - 2x^{9} + 36x^{8} + 6x^{7} + 128x^{6} - 27x^{5} + 126x^{4} + 41x^{3} + 471x^{2} + 249x + 37 \)
Invariants
| Degree: | $12$ |
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| Signature: | $[0, 6]$ |
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| Discriminant: |
\(351694699925625\)
\(\medspace = 3^{14}\cdot 5^{4}\cdot 7^{6}\)
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| Root discriminant: | \(16.30\) |
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| Galois root discriminant: | $3^{7/6}5^{1/2}7^{1/2}\approx 21.314516523881878$ | ||
| Ramified primes: |
\(3\), \(5\), \(7\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2^2$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-3}, \sqrt{-7})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5571136537885}a^{11}+\frac{307211680796}{5571136537885}a^{10}+\frac{416958003225}{1114227307577}a^{9}-\frac{1511382611002}{5571136537885}a^{8}-\frac{2492277457486}{5571136537885}a^{7}-\frac{228827965067}{1114227307577}a^{6}+\frac{729701042138}{5571136537885}a^{5}-\frac{1046044952349}{5571136537885}a^{4}+\frac{1803700294347}{5571136537885}a^{3}+\frac{39813118444}{118534819955}a^{2}+\frac{2259284799169}{5571136537885}a+\frac{320172027818}{5571136537885}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
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| Narrow class group: | Trivial group, which has order $1$ |
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Unit group
| Rank: | $5$ |
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| Torsion generator: |
\( \frac{257854}{15958615} a^{11} + \frac{130664}{15958615} a^{10} + \frac{419500}{3191723} a^{9} + \frac{559302}{15958615} a^{8} + \frac{7778776}{15958615} a^{7} + \frac{881687}{3191723} a^{6} + \frac{33026407}{15958615} a^{5} - \frac{6382916}{15958615} a^{4} + \frac{26854183}{15958615} a^{3} + \frac{123586}{339545} a^{2} + \frac{151157511}{15958615} a + \frac{47737367}{15958615} \)
(order $6$)
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| Fundamental units: |
$\frac{6918137308}{1114227307577}a^{11}+\frac{1091442034}{1114227307577}a^{10}+\frac{56773841960}{1114227307577}a^{9}-\frac{6417412737}{1114227307577}a^{8}+\frac{228793657924}{1114227307577}a^{7}+\frac{43739570585}{1114227307577}a^{6}+\frac{1011203786677}{1114227307577}a^{5}-\frac{420627904170}{1114227307577}a^{4}+\frac{1272961870901}{1114227307577}a^{3}+\frac{1036649602}{23706963991}a^{2}+\frac{4852041449940}{1114227307577}a-\frac{32228355277}{1114227307577}$, $\frac{28057554522}{5571136537885}a^{11}-\frac{86999148493}{5571136537885}a^{10}+\frac{75264343559}{1114227307577}a^{9}-\frac{797420904519}{5571136537885}a^{8}+\frac{1884063376278}{5571136537885}a^{7}-\frac{444868799560}{1114227307577}a^{6}+\frac{3522344555286}{5571136537885}a^{5}-\frac{4520067029863}{5571136537885}a^{4}+\frac{7408112345749}{5571136537885}a^{3}-\frac{38732444022}{118534819955}a^{2}-\frac{5564865470617}{5571136537885}a+\frac{1246770325046}{5571136537885}$, $\frac{80205358276}{5571136537885}a^{11}-\frac{72456799244}{5571136537885}a^{10}+\frac{131830825158}{1114227307577}a^{9}-\frac{557120788517}{5571136537885}a^{8}+\frac{2142849098409}{5571136537885}a^{7}+\frac{45212419266}{1114227307577}a^{6}+\frac{5258196826443}{5571136537885}a^{5}-\frac{4070459356529}{5571136537885}a^{4}+\frac{4701886042777}{5571136537885}a^{3}+\frac{225845584019}{118534819955}a^{2}+\frac{18511147519079}{5571136537885}a+\frac{2079381540298}{5571136537885}$, $\frac{270190660571}{5571136537885}a^{11}-\frac{209650320014}{5571136537885}a^{10}+\frac{564386339001}{1114227307577}a^{9}-\frac{2912189227397}{5571136537885}a^{8}+\frac{13632844929929}{5571136537885}a^{7}-\frac{1935965642186}{1114227307577}a^{6}+\frac{45559852161653}{5571136537885}a^{5}-\frac{35721798941789}{5571136537885}a^{4}+\frac{65378111373667}{5571136537885}a^{3}-\frac{776572256746}{118534819955}a^{2}+\frac{149976626932704}{5571136537885}a+\frac{15841602607503}{5571136537885}$, $\frac{4990473749}{1114227307577}a^{11}-\frac{11986227193}{1114227307577}a^{10}+\frac{62827835383}{1114227307577}a^{9}-\frac{136041037132}{1114227307577}a^{8}+\frac{331190846825}{1114227307577}a^{7}-\frac{482924351732}{1114227307577}a^{6}+\frac{853893413170}{1114227307577}a^{5}-\frac{1045567784541}{1114227307577}a^{4}+\frac{1591838273002}{1114227307577}a^{3}-\frac{25331331043}{23706963991}a^{2}+\frac{849271788794}{1114227307577}a+\frac{379699592120}{1114227307577}$
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| Regulator: | \( 578.612933855 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 578.612933855 \cdot 1}{6\cdot\sqrt{351694699925625}}\cr\approx \mathstrut & 0.316397603471 \end{aligned}\]
Galois group
$C_2\times D_6$ (as 12T10):
| A solvable group of order 24 |
| The 12 conjugacy class representatives for $S_3 \times C_2^2$ |
| Character table for $S_3 \times C_2^2$ |
Intermediate fields
| \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), 3.1.135.1, \(\Q(\sqrt{-3}, \sqrt{-7})\), 6.2.18753525.2, 6.0.54675.1, 6.0.6251175.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 24 |
| Degree 12 siblings: | 12.4.8792367498140625.1, 12.0.8792367498140625.3, 12.0.976929722015625.1 |
| 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 | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | R | R | R | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.6.14a2.1 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2624 x^{6} + 3264 x^{5} + 3126 x^{4} + 2264 x^{3} + 1200 x^{2} + 432 x + 91$ | $6$ | $2$ | $14$ | $D_6$ | $$[\frac{3}{2}]_{2}^{2}$$ |
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\(5\)
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(7\)
| 7.1.2.1a1.1 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 7.1.2.1a1.1 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 7.2.2.2a1.2 | $x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 7.2.2.2a1.2 | $x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |