Normalized defining polynomial
\( x^{12} - 2x^{11} + 3x^{10} - 2x^{9} + 3x^{8} - 4x^{7} - 2x^{6} - 4x^{5} + 3x^{4} - 2x^{3} + 3x^{2} - 2x + 1 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[2, 5]$ |
| |
| Discriminant: |
\(-4511594708992\)
\(\medspace = -\,2^{28}\cdot 7^{5}\)
|
| |
| Root discriminant: | \(11.34\) |
| |
| Galois root discriminant: | $2^{11/4}7^{5/6}\approx 34.04715710793443$ | ||
| Ramified primes: |
\(2\), \(7\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}+\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}+\frac{1}{4}a^{3}-\frac{1}{4}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a-\frac{1}{2}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: | $6$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}+a^{8}-\frac{7}{4}a^{5}-\frac{5}{2}a^{4}-\frac{3}{4}a^{3}-a+\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{5}{4}a^{10}+\frac{3}{2}a^{9}-\frac{3}{4}a^{8}+\frac{1}{2}a^{7}-2a^{6}-a^{5}-\frac{1}{4}a^{4}+4a^{3}-\frac{1}{4}a^{2}+\frac{3}{2}a-1$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}+\frac{1}{2}a^{6}-a^{5}-\frac{1}{4}a^{4}-\frac{9}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a$, $\frac{3}{4}a^{11}-\frac{5}{4}a^{10}+\frac{5}{4}a^{9}+\frac{1}{4}a^{8}+\frac{1}{2}a^{7}-2a^{6}-\frac{11}{4}a^{5}-\frac{11}{4}a^{4}+\frac{13}{4}a^{3}-\frac{1}{4}a^{2}+\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-a^{10}+\frac{3}{2}a^{9}-\frac{3}{4}a^{8}+a^{7}-\frac{5}{4}a^{6}-\frac{3}{2}a^{5}-\frac{3}{2}a^{4}+\frac{1}{2}a^{3}-\frac{7}{4}a^{2}-\frac{5}{4}$
|
| |
| Regulator: | \( 35.0362288854 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 35.0362288854 \cdot 1}{2\cdot\sqrt{4511594708992}}\cr\approx \mathstrut & 0.323058857587 \end{aligned}\]
Galois group
$C_6^2:C_4$ (as 12T82):
| A solvable group of order 144 |
| The 18 conjugacy class representatives for $C_6^2:C_4$ |
| Character table for $C_6^2:C_4$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.2.448.1, 6.2.802816.1 |
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 24 siblings: | data not computed |
| Degree 36 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 | ${\href{/padicField/3.4.0.1}{4} }^{3}$ | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{5}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.6a1.5 | $x^{4} + 2 x^{3} + 7 x^{2} + 6 x + 7$ | $2$ | $2$ | $6$ | $C_2^2$ | $$[3]^{2}$$ |
| 2.2.4.22a1.49 | $x^{8} + 4 x^{7} + 10 x^{6} + 16 x^{5} + 23 x^{4} + 24 x^{3} + 22 x^{2} + 12 x + 7$ | $4$ | $2$ | $22$ | $C_4\times C_2$ | $$[3, 4]^{2}$$ | |
|
\(7\)
| 7.1.6.5a1.5 | $x^{6} + 35$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ |
| 7.6.1.0a1.1 | $x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ |