Normalized defining polynomial
\( x^{12} - 12x^{10} + 60x^{8} - 160x^{6} + 228x^{4} - 144x^{2} + 8 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[4, 4]$ |
| |
| Discriminant: |
\(369768517790072832\)
\(\medspace = 2^{33}\cdot 3^{16}\)
|
| |
| Root discriminant: | \(29.11\) |
| |
| Galois root discriminant: | $2^{115/32}3^{4/3}\approx 52.23820193327616$ | ||
| Ramified primes: |
\(2\), \(3\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $7$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{4}a^{8}-2a^{6}+\frac{11}{2}a^{4}-6a^{2}+1$, $\frac{1}{2}a^{4}-2a^{2}+2$, $\frac{1}{4}a^{8}-2a^{6}+\frac{11}{2}a^{4}-6a^{2}+a$, $\frac{1}{4}a^{8}-2a^{6}+\frac{11}{2}a^{4}-6a^{2}-a$, $\frac{1}{4}a^{8}+\frac{1}{2}a^{7}-\frac{3}{2}a^{6}-\frac{7}{2}a^{5}+\frac{5}{2}a^{4}+8a^{3}-a^{2}-8a-3$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{3}{2}a^{6}+3a^{5}+\frac{7}{2}a^{4}-7a^{3}-3a^{2}+6a$, $\frac{1}{2}a^{11}-6a^{9}+\frac{59}{2}a^{7}-\frac{1}{2}a^{6}-76a^{5}+4a^{4}+102a^{3}-11a^{2}-56a+13$
|
| |
| Regulator: | \( 11867.2866149 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 11867.2866149 \cdot 1}{2\cdot\sqrt{369768517790072832}}\cr\approx \mathstrut & 0.243330151011 \end{aligned}\]
Galois group
$C_2^4:C_{12}$ (as 12T99):
| A solvable group of order 192 |
| The 20 conjugacy class representatives for $C_2^4:C_{12}$ |
| Character table for $C_2^4:C_{12}$ |
Intermediate fields
| \(\Q(\zeta_{9})^+\), 6.2.3359232.2 |
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 16 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
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 | ${\href{/padicField/5.12.0.1}{12} }$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.12.0.1}{12} }$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.4.33a1.233 | $x^{12} + 4 x^{10} + 4 x^{9} + 10 x^{8} + 12 x^{7} + 22 x^{6} + 20 x^{5} + 25 x^{4} + 24 x^{3} + 14 x^{2} + 12 x + 7$ | $4$ | $3$ | $33$ | 12T99 | $$[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$$ |
|
\(3\)
| 3.4.3.16a2.1 | $x^{12} + 6 x^{11} + 12 x^{10} + 8 x^{9} + 12 x^{8} + 48 x^{7} + 48 x^{6} + 36 x^{4} + 72 x^{3} + 35$ | $3$ | $4$ | $16$ | $C_{12}$ | $$[2]^{4}$$ |