Normalized defining polynomial
\( x^{8} - x^{7} - 26x^{6} - 231x^{5} + 651x^{4} + 3114x^{3} + 5544x^{2} + 18360x + 32400 \)
Invariants
| Degree: | $8$ |
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| Signature: | $[0, 4]$ |
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| Discriminant: |
\(259403074940025\)
\(\medspace = 3^{4}\cdot 5^{2}\cdot 71^{6}\)
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| Root discriminant: | \(63.35\) |
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| Galois root discriminant: | $3^{1/2}5^{1/2}71^{3/4}\approx 94.73040435062453$ | ||
| Ramified primes: |
\(3\), \(5\), \(71\)
|
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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: | \(\Q(\sqrt{-3}, \sqrt{-71})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{6}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{2}a$, $\frac{1}{180}a^{5}-\frac{1}{45}a^{4}+\frac{13}{45}a^{3}+\frac{23}{60}a^{2}-\frac{7}{15}a$, $\frac{1}{180}a^{6}+\frac{1}{30}a^{4}+\frac{37}{180}a^{3}+\frac{2}{5}a^{2}-\frac{11}{30}a$, $\frac{1}{358197120}a^{7}-\frac{672811}{358197120}a^{6}+\frac{18425}{17909856}a^{5}+\frac{212917}{7959936}a^{4}+\frac{4392019}{13266560}a^{3}+\frac{49583}{103645}a^{2}-\frac{350095}{994992}a-\frac{8045}{331664}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{14}$, which has order $14$ |
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| Narrow class group: | $C_{14}$, which has order $14$ |
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Unit group
| Rank: | $3$ |
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| Torsion generator: |
\( -\frac{25747}{358197120} a^{7} + \frac{54097}{358197120} a^{6} + \frac{118553}{89549280} a^{5} + \frac{1835519}{119399040} a^{4} - \frac{6904849}{119399040} a^{3} - \frac{33379}{310935} a^{2} - \frac{2331119}{4974960} a - \frac{155385}{331664} \)
(order $6$)
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| Fundamental units: |
$\frac{2213}{14924880}a^{7}-\frac{8131}{14924880}a^{6}-\frac{17819}{3731220}a^{5}-\frac{271739}{14924880}a^{4}+\frac{1037827}{4974960}a^{3}+\frac{40136}{103645}a^{2}-\frac{297319}{207290}a-\frac{137225}{41458}$, $\frac{12796009}{358197120}a^{7}-\frac{43751971}{358197120}a^{6}-\frac{95881979}{89549280}a^{5}+\frac{525145073}{39799680}a^{4}+\frac{5584427107}{119399040}a^{3}+\frac{26995897}{310935}a^{2}+\frac{1233841973}{4974960}a+\frac{125670555}{331664}$, $\frac{195228446489}{358197120}a^{7}-\frac{491765544659}{358197120}a^{6}-\frac{224001517199}{17909856}a^{5}-\frac{2502264707321}{23879808}a^{4}+\frac{61764362424979}{119399040}a^{3}+\frac{309903219016}{310935}a^{2}+\frac{282888232003}{331664}a+\frac{3191437668091}{331664}$
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| Regulator: | \( 10525.9475375 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 10525.9475375 \cdot 14}{6\cdot\sqrt{259403074940025}}\cr\approx \mathstrut & 2.37667555824 \end{aligned}\]
Galois group
$C_2\times D_4$ (as 8T9):
| A solvable group of order 16 |
| The 10 conjugacy class representatives for $D_4\times C_2$ |
| Character table for $D_4\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{-71}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{213}) \), 4.0.5368665.2, 4.0.5368665.1, \(\Q(\sqrt{-3}, \sqrt{-71})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 16 |
| Degree 8 siblings: | 8.0.720564097055625.1, 8.0.6485076873500625.8, 8.4.6485076873500625.2 |
| 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 | ${\href{/padicField/2.2.0.1}{2} }^{4}$ | R | R | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.1.0.1}{1} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(5\)
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(71\)
| 71.2.4.6a1.2 | $x^{8} + 276 x^{7} + 28594 x^{6} + 1319832 x^{5} + 23067339 x^{4} + 9238824 x^{3} + 1401106 x^{2} + 94668 x + 2472$ | $4$ | $2$ | $6$ | $D_4$ | $$[\ ]_{4}^{2}$$ |