Normalized defining polynomial
\( x^{16} - 12x^{12} + 84x^{8} - 432x^{4} + 1296 \)
Invariants
| Degree: | $16$ |
| |
| Signature: | $(0, 8)$ |
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| Discriminant: |
\(344649238497994142121984\)
\(\medspace = 2^{56}\cdot 3^{14}\)
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| Root discriminant: | \(29.59\) |
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| Galois root discriminant: | $2^{29/8}3^{7/8}\approx 32.263749133641326$ | ||
| Ramified primes: |
\(2\), \(3\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2^2$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | 8.0.764411904.5 | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{12}a^{8}$, $\frac{1}{24}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{24}a^{10}-\frac{1}{4}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{72}a^{11}+\frac{1}{12}a^{7}+\frac{1}{6}a^{3}$, $\frac{1}{360}a^{12}+\frac{1}{30}a^{8}+\frac{1}{30}a^{4}-\frac{2}{5}$, $\frac{1}{720}a^{13}+\frac{1}{60}a^{9}-\frac{1}{4}a^{7}-\frac{7}{30}a^{5}+\frac{3}{10}a$, $\frac{1}{2160}a^{14}+\frac{7}{360}a^{10}-\frac{29}{180}a^{6}-\frac{2}{5}a^{2}$, $\frac{1}{2160}a^{15}+\frac{1}{180}a^{11}-\frac{11}{45}a^{7}+\frac{13}{30}a^{3}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
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| Narrow class group: | Trivial group, which has order $1$ |
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Unit group
| Rank: | $7$ |
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| Torsion generator: |
\( \frac{1}{180} a^{12} - \frac{1}{60} a^{8} + \frac{1}{15} a^{4} + \frac{1}{5} \)
(order $6$)
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| Fundamental units: |
$\frac{1}{120}a^{12}-\frac{1}{15}a^{8}+\frac{1}{10}a^{4}+\frac{4}{5}$, $\frac{1}{216}a^{14}-\frac{1}{360}a^{12}+\frac{1}{18}a^{10}-\frac{1}{30}a^{8}-\frac{7}{18}a^{6}+\frac{7}{15}a^{4}+a^{2}-\frac{8}{5}$, $\frac{1}{270}a^{14}+\frac{1}{90}a^{12}+\frac{1}{90}a^{10}-\frac{1}{30}a^{8}-\frac{19}{90}a^{6}+\frac{19}{30}a^{4}+\frac{1}{5}a^{2}-\frac{8}{5}$, $\frac{1}{1080}a^{15}-\frac{1}{1080}a^{14}+\frac{1}{120}a^{13}-\frac{1}{72}a^{12}-\frac{1}{360}a^{11}+\frac{1}{360}a^{10}-\frac{1}{15}a^{9}+\frac{1}{6}a^{8}-\frac{13}{180}a^{7}+\frac{13}{180}a^{6}+\frac{1}{10}a^{5}-\frac{2}{3}a^{4}+\frac{7}{10}a^{3}-\frac{7}{10}a^{2}-\frac{1}{5}a+1$, $\frac{1}{360}a^{15}+\frac{7}{1080}a^{14}-\frac{1}{180}a^{13}-\frac{1}{72}a^{12}+\frac{13}{360}a^{11}-\frac{11}{180}a^{10}+\frac{1}{60}a^{9}+\frac{1}{12}a^{8}-\frac{7}{60}a^{7}+\frac{11}{45}a^{6}-\frac{1}{15}a^{5}-\frac{1}{6}a^{4}+\frac{7}{30}a^{3}-\frac{3}{5}a^{2}-\frac{1}{5}a+1$, $\frac{1}{432}a^{15}-\frac{1}{216}a^{14}-\frac{1}{180}a^{13}+\frac{1}{360}a^{12}-\frac{1}{72}a^{11}+\frac{1}{18}a^{10}+\frac{1}{10}a^{9}+\frac{1}{30}a^{8}+\frac{1}{36}a^{7}-\frac{7}{18}a^{6}-\frac{17}{30}a^{5}+\frac{1}{30}a^{4}-\frac{1}{3}a^{3}+a^{2}+\frac{9}{5}a-\frac{7}{5}$, $\frac{1}{360}a^{15}+\frac{1}{216}a^{14}-\frac{1}{60}a^{13}-\frac{7}{180}a^{12}-\frac{13}{360}a^{11}-\frac{1}{72}a^{10}+\frac{2}{15}a^{9}+\frac{1}{5}a^{8}+\frac{7}{60}a^{7}+\frac{5}{36}a^{6}-\frac{7}{10}a^{5}-\frac{22}{15}a^{4}-\frac{37}{30}a^{3}-\frac{1}{2}a^{2}+\frac{22}{5}a+\frac{28}{5}$
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| Regulator: | \( 1211326.708742033 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1211326.708742033 \cdot 1}{6\cdot\sqrt{344649238497994142121984}}\cr\approx \mathstrut & 0.835334325959123 \end{aligned}\]
Galois group
| A solvable group of order 32 |
| The 11 conjugacy class representatives for $D_8:C_2$ |
| Character table for $D_8:C_2$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
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 | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.8.56a1.873 | $x^{16} + 16 x^{15} + 100 x^{14} + 392 x^{13} + 1110 x^{12} + 2432 x^{11} + 4292 x^{10} + 6240 x^{9} + 7591 x^{8} + 7776 x^{7} + 6756 x^{6} + 4976 x^{5} + 3126 x^{4} + 1664 x^{3} + 752 x^{2} + 264 x + 67$ | $8$ | $2$ | $56$ | 16T38 | $$[2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$$ |
|
\(3\)
| 3.1.8.7a1.1 | $x^{8} + 3$ | $8$ | $1$ | $7$ | $QD_{16}$ | $$[\ ]_{8}^{2}$$ |
| 3.1.8.7a1.1 | $x^{8} + 3$ | $8$ | $1$ | $7$ | $QD_{16}$ | $$[\ ]_{8}^{2}$$ |