Normalized defining polynomial
\( x^{12} + 12x^{10} + 18x^{8} - 44x^{6} - 111x^{4} - 128 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[2, 5]$ |
| |
| Discriminant: |
\(-73040694872113152\)
\(\medspace = -\,2^{37}\cdot 3^{12}\)
|
| |
| Root discriminant: | \(25.43\) |
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| Galois root discriminant: | $2^{27/8}3^{25/18}\approx 47.713870191292706$ | ||
| Ramified primes: |
\(2\), \(3\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{24}a^{6}+\frac{1}{4}a^{4}-\frac{3}{8}a^{2}+\frac{1}{3}$, $\frac{1}{24}a^{7}-\frac{1}{4}a^{5}-\frac{3}{8}a^{3}-\frac{1}{6}a$, $\frac{1}{24}a^{8}+\frac{1}{8}a^{4}-\frac{5}{12}a^{2}$, $\frac{1}{48}a^{9}-\frac{1}{48}a^{7}-\frac{1}{16}a^{5}-\frac{1}{2}a^{4}+\frac{23}{48}a^{3}-\frac{1}{2}a^{2}-\frac{1}{6}a$, $\frac{1}{2064}a^{10}+\frac{29}{2064}a^{8}-\frac{5}{2064}a^{6}-\frac{1}{48}a^{4}-\frac{1}{2}a^{3}+\frac{95}{1032}a^{2}-\frac{1}{2}a-\frac{40}{129}$, $\frac{1}{4128}a^{11}-\frac{7}{2064}a^{9}-\frac{1}{86}a^{7}+\frac{7}{48}a^{5}-\frac{1}{2}a^{4}+\frac{2039}{4128}a^{3}-\frac{1}{2}a^{2}+\frac{1}{86}a$
| 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 $1$ |
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Unit group
| Rank: | $6$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{7}{1032}a^{10}+\frac{37}{516}a^{8}+\frac{1}{129}a^{6}-\frac{5}{12}a^{4}+\frac{169}{1032}a^{2}+\frac{257}{129}$, $\frac{31}{4128}a^{11}+\frac{85}{1032}a^{9}+\frac{73}{2064}a^{7}-\frac{13}{24}a^{5}-\frac{2581}{4128}a^{3}+\frac{68}{129}a+1$, $\frac{25}{4128}a^{11}-\frac{1}{48}a^{10}+\frac{169}{2064}a^{9}-\frac{11}{48}a^{8}+\frac{173}{1032}a^{7}-\frac{7}{48}a^{6}-\frac{29}{48}a^{5}+\frac{49}{48}a^{4}-\frac{3893}{4128}a^{3}+\frac{25}{24}a^{2}-\frac{11}{258}a+\frac{1}{3}$, $\frac{13}{1376}a^{11}-\frac{7}{516}a^{10}+\frac{157}{2064}a^{9}-\frac{191}{1032}a^{8}-\frac{37}{129}a^{7}-\frac{133}{258}a^{6}-\frac{19}{16}a^{5}+\frac{5}{24}a^{4}+\frac{2809}{4128}a^{3}+\frac{797}{258}a^{2}+\frac{682}{129}a+\frac{389}{129}$, $\frac{13}{1376}a^{11}+\frac{7}{516}a^{10}+\frac{157}{2064}a^{9}+\frac{191}{1032}a^{8}-\frac{37}{129}a^{7}+\frac{133}{258}a^{6}-\frac{19}{16}a^{5}-\frac{5}{24}a^{4}+\frac{2809}{4128}a^{3}-\frac{797}{258}a^{2}+\frac{682}{129}a-\frac{389}{129}$, $\frac{7}{344}a^{10}+\frac{179}{1032}a^{8}-\frac{40}{129}a^{6}-\frac{3}{8}a^{4}+\frac{937}{1032}a^{2}-\frac{89}{129}$
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| Regulator: | \( 15367.1418105 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 15367.1418105 \cdot 1}{2\cdot\sqrt{73040694872113152}}\cr\approx \mathstrut & 1.11362760638 \end{aligned}\]
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
| \(\Q(\sqrt{2}) \), 4.2.2048.1, 6.2.11943936.1 |
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 | R | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.2.0.1}{2} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.4.11a1.14 | $x^{4} + 4 x^{2} + 8 x + 18$ | $4$ | $1$ | $11$ | $D_{4}$ | $$[2, 3, 4]$$ |
| 2.1.8.26c1.10 | $x^{8} + 4 x^{6} + 8 x^{5} + 8 x^{4} + 8 x^{3} + 18$ | $8$ | $1$ | $26$ | $C_2^2:C_4$ | $$[2, 3, \frac{7}{2}, 4]$$ | |
|
\(3\)
| 3.2.3.6a3.1 | $x^{6} + 6 x^{5} + 18 x^{4} + 38 x^{3} + 48 x^{2} + 36 x + 11$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $$[\frac{3}{2}, \frac{3}{2}]_{2}^{2}$$ |
| 3.2.3.6a3.1 | $x^{6} + 6 x^{5} + 18 x^{4} + 38 x^{3} + 48 x^{2} + 36 x + 11$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $$[\frac{3}{2}, \frac{3}{2}]_{2}^{2}$$ |