Normalized defining polynomial
\( x^{12} + 3x^{10} + 3x^{8} + 14x^{6} + 30x^{4} + 12x^{2} + 4 \)
Invariants
| Degree: | $12$ |
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| Signature: | $[0, 6]$ |
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| Discriminant: |
\(320979616137216\)
\(\medspace = 2^{26}\cdot 3^{14}\)
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| Root discriminant: | \(16.18\) |
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| Galois root discriminant: | $2^{3}3^{7/6}\approx 28.822486924224066$ | ||
| Ramified primes: |
\(2\), \(3\)
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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: | \(\Q(\sqrt{-3}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a$, $\frac{1}{6}a^{8}-\frac{1}{6}a^{6}-\frac{1}{2}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{6}a^{9}-\frac{1}{6}a^{7}-\frac{1}{2}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{18}a^{10}-\frac{1}{18}a^{8}+\frac{1}{18}a^{6}-\frac{1}{9}a^{4}+\frac{1}{9}a^{2}-\frac{1}{9}$, $\frac{1}{18}a^{11}-\frac{1}{18}a^{9}+\frac{1}{18}a^{7}-\frac{1}{9}a^{5}+\frac{1}{9}a^{3}-\frac{1}{9}a$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: |
\( -\frac{1}{6} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{13}{6} a^{4} - \frac{10}{3} a^{2} - \frac{1}{3} \)
(order $6$)
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| Fundamental units: |
$\frac{1}{18}a^{10}+\frac{5}{18}a^{8}+\frac{7}{18}a^{6}+\frac{8}{9}a^{4}+\frac{22}{9}a^{2}+\frac{11}{9}$, $\frac{1}{18}a^{11}-\frac{1}{9}a^{10}+\frac{5}{18}a^{9}-\frac{7}{18}a^{8}+\frac{7}{18}a^{7}-\frac{5}{18}a^{6}+\frac{8}{9}a^{5}-\frac{23}{18}a^{4}+\frac{22}{9}a^{3}-\frac{29}{9}a^{2}+\frac{20}{9}a-\frac{4}{9}$, $\frac{1}{18}a^{11}-\frac{1}{18}a^{10}+\frac{5}{18}a^{9}+\frac{2}{9}a^{8}+\frac{7}{18}a^{7}+\frac{1}{9}a^{6}+\frac{8}{9}a^{5}-\frac{7}{18}a^{4}+\frac{31}{9}a^{3}+\frac{23}{9}a^{2}+\frac{20}{9}a+\frac{7}{9}$, $\frac{5}{9}a^{11}-\frac{2}{9}a^{10}+\frac{29}{18}a^{9}-\frac{4}{9}a^{8}+\frac{25}{18}a^{7}-\frac{2}{9}a^{6}+\frac{133}{18}a^{5}-\frac{23}{9}a^{4}+\frac{142}{9}a^{3}-\frac{37}{9}a^{2}+\frac{29}{9}a+\frac{13}{9}$, $\frac{1}{9}a^{10}+\frac{1}{18}a^{8}-\frac{1}{18}a^{6}+\frac{23}{18}a^{4}+\frac{8}{9}a^{2}+\frac{1}{9}$
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| Regulator: | \( 746.133664789 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 746.133664789 \cdot 1}{6\cdot\sqrt{320979616137216}}\cr\approx \mathstrut & 0.427076690321 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 12T24):
| A solvable group of order 48 |
| The 10 conjugacy class representatives for $C_2 \times S_4$ |
| Character table for $C_2 \times S_4$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.216.1, 6.2.5971968.2, 6.0.139968.1, 6.0.17915904.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 6 siblings: | data not computed |
| Degree 8 siblings: | 8.0.27518828544.6, 8.4.12230590464.6 |
| Degree 12 siblings: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| Minimal sibling: | 6.2.5971968.2 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.4a1.1 | $x^{4} + 2 x^{3} + 5 x^{2} + 4 x + 5$ | $2$ | $2$ | $4$ | $C_2^2$ | $$[2]^{2}$$ |
| 2.2.4.22a1.62 | $x^{8} + 12 x^{7} + 34 x^{6} + 64 x^{5} + 79 x^{4} + 72 x^{3} + 54 x^{2} + 44 x + 15$ | $4$ | $2$ | $22$ | $D_4\times C_2$ | $$[2, 3, 4]^{2}$$ | |
|
\(3\)
| 3.2.6.14a2.1 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2624 x^{6} + 3264 x^{5} + 3126 x^{4} + 2264 x^{3} + 1200 x^{2} + 432 x + 91$ | $6$ | $2$ | $14$ | $D_6$ | $$[\frac{3}{2}]_{2}^{2}$$ |