Normalized defining polynomial
\( x^{12} - 2x^{11} + 9x^{10} + 10x^{9} - 10x^{8} - 2x^{7} + 9x^{6} + 2x^{5} - 10x^{4} - 10x^{3} + 9x^{2} + 2x + 1 \)
Invariants
| Degree: | $12$ |
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| Signature: | $[0, 6]$ |
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| Discriminant: |
\(25600000000000000\)
\(\medspace = 2^{22}\cdot 5^{14}\)
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| Root discriminant: | \(23.30\) |
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| Galois root discriminant: | $2^{11/6}5^{31/20}\approx 43.18091413946323$ | ||
| Ramified primes: |
\(2\), \(5\)
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| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{60}a^{10}-\frac{1}{12}a^{9}-\frac{1}{12}a^{8}-\frac{1}{6}a^{7}-\frac{1}{4}a^{6}-\frac{9}{20}a^{5}+\frac{1}{4}a^{4}-\frac{1}{6}a^{3}+\frac{1}{12}a^{2}-\frac{1}{12}a-\frac{1}{60}$, $\frac{1}{3180}a^{11}-\frac{5}{636}a^{10}+\frac{85}{636}a^{9}+\frac{19}{106}a^{8}+\frac{79}{636}a^{7}+\frac{151}{1060}a^{6}+\frac{101}{212}a^{5}-\frac{17}{159}a^{4}+\frac{131}{636}a^{3}+\frac{55}{212}a^{2}+\frac{303}{1060}a-\frac{103}{318}$
| 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: |
$a^{11}-2a^{10}+9a^{9}+10a^{8}-10a^{7}-2a^{6}+9a^{5}+2a^{4}-10a^{3}-10a^{2}+9a+2$, $\frac{31}{106}a^{11}-\frac{119}{212}a^{10}+\frac{539}{212}a^{9}+\frac{731}{212}a^{8}-\frac{369}{106}a^{7}+\frac{155}{212}a^{6}+\frac{1233}{212}a^{5}-\frac{251}{212}a^{4}-\frac{103}{53}a^{3}+\frac{111}{212}a^{2}+\frac{337}{212}a+\frac{111}{212}$, $\frac{423}{212}a^{11}-\frac{2363}{636}a^{10}+\frac{10915}{636}a^{9}+\frac{7307}{318}a^{8}-\frac{12421}{636}a^{7}-\frac{1831}{212}a^{6}+\frac{4315}{212}a^{5}+\frac{350}{53}a^{4}-\frac{13837}{636}a^{3}-\frac{14665}{636}a^{2}+\frac{11371}{636}a+\frac{2605}{318}$, $\frac{227}{318}a^{11}-\frac{303}{212}a^{10}+\frac{1335}{212}a^{9}+\frac{4645}{636}a^{8}-\frac{1304}{159}a^{7}-\frac{717}{212}a^{6}+\frac{1423}{212}a^{5}+\frac{1301}{636}a^{4}-\frac{753}{106}a^{3}-\frac{3817}{636}a^{2}+\frac{4639}{636}a+\frac{847}{636}$, $\frac{1037}{636}a^{11}-\frac{638}{159}a^{10}+\frac{2578}{159}a^{9}+\frac{2043}{212}a^{8}-\frac{15233}{636}a^{7}+\frac{675}{106}a^{6}+\frac{1745}{106}a^{5}-\frac{4211}{636}a^{4}-\frac{10189}{636}a^{3}-\frac{402}{53}a^{2}+\frac{1080}{53}a-\frac{2333}{636}$
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| Regulator: | \( 8678.54836379 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 8678.54836379 \cdot 1}{2\cdot\sqrt{25600000000000000}}\cr\approx \mathstrut & 1.66869252257 \end{aligned}\]
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
| \(\Q(\sqrt{-2}) \), 6.0.160000000.1 |
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: | 10.6.31250000000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 | $x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 126 x^{7} + 141 x^{6} + 126 x^{5} + 90 x^{4} + 50 x^{3} + 21 x^{2} + 6 x + 3$ | $6$ | $2$ | $22$ | $D_6$ | $$[3]_{3}^{2}$$ |
|
\(5\)
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 5.2.5.14a2.1 | $x^{10} + 20 x^{9} + 170 x^{8} + 800 x^{7} + 2285 x^{6} + 4124 x^{5} + 4830 x^{4} + 3760 x^{3} + 1900 x^{2} + 560 x + 77$ | $5$ | $2$ | $14$ | $F_{5}\times C_2$ | $$[\frac{7}{4}]_{4}^{2}$$ |