Normalized defining polynomial
\( x^{12} + 12x^{10} + 54x^{8} + 108x^{6} + 81x^{4} + 48 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[0, 6]$ |
| |
| Discriminant: |
\(10968475320188928\)
\(\medspace = 2^{20}\cdot 3^{21}\)
|
| |
| Root discriminant: | \(21.71\) |
| |
| Galois root discriminant: | $2^{9/4}3^{25/12}\approx 46.91590565166763$ | ||
| Ramified primes: |
\(2\), \(3\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{3}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{6}-\frac{1}{4}a^{4}+\frac{1}{8}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{7}-\frac{1}{4}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{8}+\frac{1}{8}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{16}a^{9}-\frac{1}{16}a^{7}-\frac{1}{16}a^{5}-\frac{3}{16}a^{3}-\frac{1}{4}a$, $\frac{1}{16}a^{10}-\frac{1}{16}a^{8}-\frac{1}{16}a^{6}-\frac{3}{16}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{32}a^{11}-\frac{1}{32}a^{10}-\frac{1}{32}a^{9}+\frac{1}{32}a^{8}-\frac{1}{32}a^{7}+\frac{1}{32}a^{6}-\frac{3}{32}a^{5}+\frac{3}{32}a^{4}-\frac{1}{8}a^{3}+\frac{1}{8}a^{2}$
| 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: |
\( -\frac{1}{8} a^{6} - \frac{3}{4} a^{4} - \frac{9}{8} a^{2} + \frac{1}{2} \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{1}{16}a^{9}+\frac{9}{16}a^{7}-\frac{1}{8}a^{6}+\frac{27}{16}a^{5}-\frac{3}{4}a^{4}+\frac{31}{16}a^{3}-\frac{9}{8}a^{2}+\frac{3}{4}a+\frac{1}{2}$, $\frac{1}{32}a^{11}-\frac{1}{32}a^{10}+\frac{13}{32}a^{9}-\frac{11}{32}a^{8}+\frac{65}{32}a^{7}-\frac{43}{32}a^{6}+\frac{151}{32}a^{5}-\frac{73}{32}a^{4}+\frac{69}{16}a^{3}-\frac{7}{4}a^{2}-\frac{3}{4}a-\frac{1}{2}$, $\frac{1}{32}a^{11}+\frac{1}{32}a^{10}+\frac{13}{32}a^{9}+\frac{11}{32}a^{8}+\frac{65}{32}a^{7}+\frac{43}{32}a^{6}+\frac{151}{32}a^{5}+\frac{73}{32}a^{4}+\frac{69}{16}a^{3}+\frac{7}{4}a^{2}-\frac{3}{4}a+\frac{1}{2}$, $\frac{3}{32}a^{11}+\frac{3}{32}a^{10}+\frac{37}{32}a^{9}+\frac{33}{32}a^{8}+\frac{169}{32}a^{7}+\frac{125}{32}a^{6}+\frac{327}{32}a^{5}+\frac{171}{32}a^{4}+6a^{3}+\frac{3}{8}a^{2}-3a-4$, $\frac{5}{16}a^{10}+\frac{39}{16}a^{8}+\frac{103}{16}a^{6}+\frac{93}{16}a^{4}-\frac{3}{2}a^{2}+2$
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| Regulator: | \( 8232.85983222 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 8232.85983222 \cdot 1}{6\cdot\sqrt{10968475320188928}}\cr\approx \mathstrut & 0.806130865631 \end{aligned}\]
Galois group
$S_3^2:C_2^2$ (as 12T78):
| A solvable group of order 144 |
| The 18 conjugacy class representatives for $S_3^2:C_2^2$ |
| Character table for $S_3^2:C_2^2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 4.0.432.1, 6.0.3779136.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.14624633760251904.38 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{6}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.4a2.1 | $x^{4} + 4 x^{3} + 5 x^{2} + 4 x + 3$ | $2$ | $2$ | $4$ | $D_{4}$ | $$[2, 2]^{2}$$ |
| 2.2.4.16b5.1 | $x^{8} + 4 x^{7} + 12 x^{6} + 24 x^{5} + 35 x^{4} + 40 x^{3} + 34 x^{2} + 20 x + 9$ | $4$ | $2$ | $16$ | $D_4\times C_2$ | $$[2, 2, 3]^{2}$$ | |
|
\(3\)
| 3.1.12.21a2.5 | $x^{12} + 3 x^{10} + 18 x^{2} + 3$ | $12$ | $1$ | $21$ | 12T35 | $$[\frac{9}{4}, \frac{9}{4}]_{4}^{2}$$ |