Normalized defining polynomial
\( x^{12} - 2x^{11} + 5x^{10} - 6x^{9} + 20x^{8} - 2x^{7} + 29x^{6} + 2x^{5} + 20x^{4} + 6x^{3} + 5x^{2} + 2x + 1 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[0, 6]$ |
| |
| Discriminant: |
\(2494869834563584\)
\(\medspace = 2^{22}\cdot 29^{6}\)
|
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| Root discriminant: | \(19.19\) |
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| Galois root discriminant: | $2^{11/6}29^{3/4}\approx 44.533499491161095$ | ||
| Ramified primes: |
\(2\), \(29\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{15}a^{11}-\frac{1}{15}a^{8}-\frac{2}{15}a^{7}-\frac{1}{15}a^{6}+\frac{2}{15}a^{5}-\frac{4}{15}a^{4}-\frac{1}{5}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{2}{15}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$a^{11}-2a^{10}+5a^{9}-6a^{8}+20a^{7}-2a^{6}+29a^{5}+2a^{4}+20a^{3}+6a^{2}+5a+2$, $\frac{2}{15}a^{11}-a^{10}+\frac{5}{3}a^{9}-\frac{52}{15}a^{8}+\frac{76}{15}a^{7}-\frac{187}{15}a^{6}-\frac{41}{15}a^{5}-\frac{101}{5}a^{4}-\frac{76}{15}a^{3}-\frac{25}{3}a^{2}-\frac{5}{3}a-\frac{11}{15}$, $\frac{4}{5}a^{11}-a^{10}+3a^{9}-\frac{14}{5}a^{8}+\frac{211}{15}a^{7}+\frac{113}{15}a^{6}+\frac{404}{15}a^{5}+\frac{152}{15}a^{4}+\frac{214}{15}a^{3}+\frac{10}{3}a^{2}+\frac{10}{3}a+\frac{3}{5}$, $\frac{6}{5}a^{11}-\frac{8}{3}a^{10}+\frac{20}{3}a^{9}-\frac{128}{15}a^{8}+\frac{379}{15}a^{7}-\frac{98}{15}a^{6}+\frac{536}{15}a^{5}-\frac{22}{15}a^{4}+\frac{346}{15}a^{3}+\frac{16}{3}a^{2}+\frac{8}{3}a+\frac{46}{15}$, $\frac{16}{15}a^{11}-2a^{10}+\frac{14}{3}a^{9}-\frac{86}{15}a^{8}+\frac{298}{15}a^{7}+\frac{3}{5}a^{6}+\frac{362}{15}a^{5}-\frac{18}{5}a^{4}+\frac{19}{5}a^{3}-\frac{4}{3}a-\frac{13}{15}$
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| Regulator: | \( 2189.03641237 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 2189.03641237 \cdot 1}{2\cdot\sqrt{2494869834563584}}\cr\approx \mathstrut & 1.34827429322 \end{aligned}\]
Galois group
$C_2\times S_5$ (as 12T123):
| A non-solvable group of order 240 |
| The 14 conjugacy class representatives for $C_2\times S_5$ |
| Character table for $C_2\times S_5$ |
Intermediate fields
| \(\Q(\sqrt{-2}) \), 6.0.49948672.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 10 siblings: | data not computed |
| Degree 12 sibling: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.6.17663873340416.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 | $x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 126 x^{7} + 141 x^{6} + 126 x^{5} + 90 x^{4} + 50 x^{3} + 21 x^{2} + 6 x + 3$ | $6$ | $2$ | $22$ | $D_6$ | $$[3]_{3}^{2}$$ |
|
\(29\)
| 29.2.1.0a1.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 29.2.1.0a1.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 29.2.4.6a1.4 | $x^{8} + 96 x^{7} + 3464 x^{6} + 55872 x^{5} + 345624 x^{4} + 111744 x^{3} + 13856 x^{2} + 913 x + 799$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ |