Normalized defining polynomial
\( x^{12} - x^{11} + 2x^{10} - 3x^{9} + 3x^{8} - 4x^{7} + 3x^{6} - 4x^{5} + 3x^{4} - 3x^{3} + 2x^{2} - x + 1 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[2, 5]$ |
| |
| Discriminant: |
\(-266846228071\)
\(\medspace = -\,31\cdot 92779^{2}\)
|
| |
| Root discriminant: | \(8.96\) |
| |
| Galois root discriminant: | $31^{1/2}92779^{1/2}\approx 1695.9212835506253$ | ||
| Ramified primes: |
\(31\), \(92779\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-31}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$
| Monogenic: | Yes | |
| 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: | $6$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a^{11}-a^{10}+2a^{9}-3a^{8}+3a^{7}-4a^{6}+3a^{5}-4a^{4}+3a^{3}-3a^{2}+2a-1$, $a^{9}-a^{8}+a^{7}-2a^{6}+2a^{5}-2a^{4}+a^{3}-2a^{2}+a-1$, $a^{11}-a^{7}+a^{6}-2a^{5}+a^{4}-2a^{3}+a^{2}-2a+1$, $a^{10}-2a^{9}+3a^{8}-4a^{7}+4a^{6}-5a^{5}+4a^{4}-4a^{3}+4a^{2}-3a+2$, $a^{10}-a^{9}+2a^{8}-3a^{7}+3a^{6}-4a^{5}+3a^{4}-4a^{3}+2a^{2}-2a+1$, $a^{11}-2a^{10}+2a^{9}-3a^{8}+3a^{7}-3a^{6}+2a^{5}-2a^{4}+2a^{3}-2a^{2}+a+1$
|
| |
| Regulator: | \( 4.68688998752 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 4.68688998752 \cdot 1}{2\cdot\sqrt{266846228071}}\cr\approx \mathstrut & 0.177698418647 \end{aligned}\]
Galois group
$C_2^6.S_6$ (as 12T293):
| A non-solvable group of order 46080 |
| The 65 conjugacy class representatives for $C_2^6.S_6$ |
| Character table for $C_2^6.S_6$ |
Intermediate fields
| 6.4.92779.1 |
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 40 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 | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.12.0.1}{12} }$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(31\)
| 31.1.2.1a1.2 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 31.10.1.0a1.1 | $x^{10} + 30 x^{5} + 26 x^{4} + 13 x^{3} + 13 x^{2} + 13 x + 3$ | $1$ | $10$ | $0$ | $C_{10}$ | $$[\ ]^{10}$$ | |
|
\(92779\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |