Normalized defining polynomial
\( x^{8} - 3x^{7} - 22x^{6} + 20x^{5} + 175x^{4} + 386x^{3} + 682x^{2} + 516x + 372 \)
Invariants
| Degree: | $8$ |
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| Signature: | $(0, 4)$ |
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| Discriminant: |
\(112501209744\)
\(\medspace = 2^{4}\cdot 3^{4}\cdot 7^{2}\cdot 11^{6}\)
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| Root discriminant: | \(24.07\) |
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| Galois root discriminant: | $2\cdot 3^{1/2}7^{1/2}11^{3/4}\approx 55.35847998534994$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(11\)
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| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-3}, \sqrt{-11})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{41684714}a^{7}-\frac{3799317}{41684714}a^{6}+\frac{3540513}{20842357}a^{5}-\frac{9296771}{20842357}a^{4}-\frac{5223011}{41684714}a^{3}+\frac{8726998}{20842357}a^{2}-\frac{577435}{20842357}a-\frac{9617972}{20842357}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
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| Narrow class group: | $C_{2}$, which has order $2$ |
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Unit group
| Rank: | $3$ |
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| Torsion generator: |
\( \frac{1175}{133178} a^{7} - \frac{2163}{66589} a^{6} - \frac{23411}{133178} a^{5} + \frac{19758}{66589} a^{4} + \frac{203849}{133178} a^{3} + \frac{298491}{133178} a^{2} + \frac{188963}{66589} a + \frac{101624}{66589} \)
(order $6$)
|
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| Fundamental units: |
$\frac{54529}{41684714}a^{7}+\frac{71887}{41684714}a^{6}-\frac{2119514}{20842357}a^{5}+\frac{5023452}{20842357}a^{4}+\frac{26083943}{41684714}a^{3}-\frac{19063439}{20842357}a^{2}-\frac{14994045}{20842357}a-\frac{43850711}{20842357}$, $\frac{1407437}{41684714}a^{7}-\frac{2525483}{20842357}a^{6}-\frac{29329257}{41684714}a^{5}+\frac{24657460}{20842357}a^{4}+\frac{248804663}{41684714}a^{3}+\frac{368542125}{41684714}a^{2}+\frac{231908333}{20842357}a-\frac{37318118}{20842357}$, $\frac{1563780}{1226021}a^{7}-\frac{5528386}{1226021}a^{6}-\frac{32923919}{1226021}a^{5}+\frac{54660379}{1226021}a^{4}+\frac{273595192}{1226021}a^{3}+\frac{394274549}{1226021}a^{2}+\frac{619793593}{1226021}a+\frac{188758067}{1226021}$
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| Regulator: | \( 383.84675264 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 383.84675264 \cdot 2}{6\cdot\sqrt{112501209744}}\cr\approx \mathstrut & 0.59453509151 \end{aligned}\]
Galois group
$D_4:C_2^2$ (as 8T22):
| A solvable group of order 32 |
| The 17 conjugacy class representatives for $Q_8:C_2^2$ |
| Character table for $Q_8:C_2^2$ |
Intermediate fields
| \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-3}, \sqrt{-11})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 8 siblings: | data not computed |
| Degree 16 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 | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | R | R | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.4a1.2 | $x^{4} + 2 x^{3} + 5 x^{2} + 8 x + 5$ | $2$ | $2$ | $4$ | $C_4$ | $$[2]^{2}$$ |
| 2.4.1.0a1.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
|
\(3\)
| 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 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(7\)
| 7.2.1.0a1.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 7.2.1.0a1.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 7.2.2.2a1.2 | $x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(11\)
| 11.2.4.6a1.3 | $x^{8} + 28 x^{7} + 302 x^{6} + 1540 x^{5} + 3601 x^{4} + 3080 x^{3} + 1208 x^{2} + 268 x + 115$ | $4$ | $2$ | $6$ | $Q_8$ | $$[\ ]_{4}^{2}$$ |