Normalized defining polynomial
\( x^{12} - 3 x^{11} + 8 x^{10} - 15 x^{9} + 18 x^{8} - 15 x^{7} - x^{6} + 15 x^{5} - 23 x^{4} + 20 x^{3} + \cdots - 1 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[2, 5]$ |
| |
| Discriminant: |
\(-11183262109375\)
\(\medspace = -\,5^{8}\cdot 31^{5}\)
|
| |
| Root discriminant: | \(12.23\) |
| |
| Galois root discriminant: | $5^{3/4}31^{5/6}\approx 58.4829020029174$ | ||
| Ramified primes: |
\(5\), \(31\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{512}a^{11}-\frac{13}{32}a^{10}+\frac{19}{64}a^{9}+\frac{57}{512}a^{8}+\frac{109}{512}a^{7}+\frac{21}{64}a^{6}-\frac{137}{512}a^{5}-\frac{15}{128}a^{4}-\frac{11}{512}a^{3}+\frac{227}{512}a^{2}+\frac{45}{512}a-\frac{5}{512}$
| 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: |
$\frac{49}{512}a^{11}+\frac{3}{32}a^{10}-\frac{29}{64}a^{9}+\frac{745}{512}a^{8}-\frac{1827}{512}a^{7}+\frac{261}{64}a^{6}-\frac{1593}{512}a^{5}-\frac{223}{128}a^{4}+\frac{2533}{512}a^{3}-\frac{2189}{512}a^{2}+\frac{1693}{512}a-\frac{245}{512}$, $a$, $\frac{315}{512}a^{11}-\frac{31}{32}a^{10}+\frac{161}{64}a^{9}-\frac{1501}{512}a^{8}+\frac{31}{512}a^{7}+\frac{151}{64}a^{6}-\frac{4243}{512}a^{5}+\frac{395}{128}a^{4}-\frac{393}{512}a^{3}-\frac{1199}{512}a^{2}+\frac{863}{512}a-\frac{551}{512}$, $\frac{13}{512}a^{11}-\frac{9}{32}a^{10}+\frac{55}{64}a^{9}-\frac{795}{512}a^{8}+\frac{1417}{512}a^{7}-\frac{111}{64}a^{6}-\frac{245}{512}a^{5}+\frac{317}{128}a^{4}-\frac{2191}{512}a^{3}-\frac{121}{512}a^{2}-\frac{439}{512}a-\frac{65}{512}$, $\frac{403}{256}a^{11}-\frac{71}{16}a^{10}+\frac{361}{32}a^{9}-\frac{5189}{256}a^{8}+\frac{5527}{256}a^{7}-\frac{465}{32}a^{6}-\frac{2219}{256}a^{5}+\frac{1507}{64}a^{4}-\frac{6993}{256}a^{3}+\frac{5209}{256}a^{2}-\frac{2601}{256}a+\frac{545}{256}$, $\frac{615}{512}a^{11}-\frac{91}{32}a^{10}+\frac{485}{64}a^{9}-\frac{6417}{512}a^{8}+\frac{6107}{512}a^{7}-\frac{461}{64}a^{6}-\frac{4895}{512}a^{5}+\frac{1911}{128}a^{4}-\frac{9325}{512}a^{3}+\frac{5461}{512}a^{2}-\frac{2533}{512}a-\frac{3}{512}$
|
| |
| Regulator: | \( 56.7780106185 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 56.7780106185 \cdot 1}{2\cdot\sqrt{11183262109375}}\cr\approx \mathstrut & 0.332525731280 \end{aligned}\]
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
| \(\Q(\sqrt{5}) \), 4.2.775.1, 6.2.600625.1 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
| 5.2.4.6a1.2 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ | |
|
\(31\)
| 31.6.1.0a1.1 | $x^{6} + 19 x^{3} + 16 x^{2} + 8 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ |
| 31.1.6.5a1.6 | $x^{6} + 837$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ |