Normalized defining polynomial
\( x^{12} - 14x^{6} + 64 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[0, 6]$ |
| |
| Discriminant: |
\(25389989167104000000\)
\(\medspace = 2^{22}\cdot 3^{18}\cdot 5^{6}\)
|
| |
| Root discriminant: | \(41.41\) |
| |
| Galois root discriminant: | $2^{11/6}3^{3/2}5^{1/2}\approx 41.40523078218084$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2^2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-15}) \), 6.0.39366000.1$^{3}$, 6.0.335923200.1$^{3}$, 12.0.1015599566684160000.1$^{6}$, 12.0.25389989167104000000.1$^{12}$, deg 12$^{6}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{9}+\frac{1}{4}a^{3}$, $\frac{1}{16}a^{10}+\frac{1}{8}a^{4}$, $\frac{1}{32}a^{11}-\frac{7}{16}a^{5}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{6}$, which has order $6$ |
| |
| Narrow class group: | $C_{6}$, which has order $6$ |
| |
| Relative class number: | $3$ |
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{16}a^{10}-\frac{1}{4}a^{8}+\frac{1}{8}a^{4}+\frac{5}{2}a^{2}-1$, $\frac{1}{32}a^{11}+\frac{1}{16}a^{10}-\frac{7}{16}a^{5}-\frac{7}{8}a^{4}-a^{2}-a-3$, $\frac{5}{32}a^{11}-\frac{1}{4}a^{10}+\frac{1}{8}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{19}{16}a^{5}+\frac{5}{2}a^{4}-\frac{11}{4}a^{3}+\frac{1}{2}a^{2}+2a+1$, $\frac{1}{32}a^{11}-\frac{1}{16}a^{10}-\frac{7}{16}a^{5}+\frac{7}{8}a^{4}+a^{2}-a+3$, $\frac{3}{32}a^{11}+\frac{1}{16}a^{10}-\frac{1}{2}a^{7}-\frac{5}{16}a^{5}-\frac{7}{8}a^{4}-a^{2}+4a+3$
|
| |
| Regulator: | \( 51657.6426592 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 51657.6426592 \cdot 6}{2\cdot\sqrt{25389989167104000000}}\cr\approx \mathstrut & 1.89236013634 \end{aligned}\]
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
| \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-2}) \), 3.3.1620.1, \(\Q(\sqrt{-2}, \sqrt{-15})\), 6.6.5038848000.2, 6.0.39366000.1, 6.0.335923200.1 |
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: | 12.0.1015599566684160000.1, deg 12, deg 12 |
| Minimal sibling: | 12.0.1015599566684160000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | ${\href{/padicField/7.2.0.1}{2} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.6.11a1.1 | $x^{6} + 2$ | $6$ | $1$ | $11$ | $D_{6}$ | $$[3]_{3}^{2}$$ |
| 2.1.6.11a1.1 | $x^{6} + 2$ | $6$ | $1$ | $11$ | $D_{6}$ | $$[3]_{3}^{2}$$ | |
|
\(3\)
| 3.1.6.9a2.3 | $x^{6} + 6 x^{4} + 6$ | $6$ | $1$ | $9$ | $D_{6}$ | $$[2]_{2}^{2}$$ |
| 3.1.6.9a2.3 | $x^{6} + 6 x^{4} + 6$ | $6$ | $1$ | $9$ | $D_{6}$ | $$[2]_{2}^{2}$$ | |
|
\(5\)
| 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 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |