Normalized defining polynomial
\( x^{8} - 4x^{7} - 8x^{6} + 32x^{5} + 18x^{4} - 56x^{3} - 20x^{2} + 16x - 2 \)
Invariants
| Degree: | $8$ |
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| Signature: | $[8, 0]$ |
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| Discriminant: |
\(41108373504\)
\(\medspace = 2^{22}\cdot 3^{4}\cdot 11^{2}\)
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| Root discriminant: | \(21.22\) |
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| Galois root discriminant: | $2^{11/4}3^{1/2}11^{1/2}\approx 38.644657093393754$ | ||
| Ramified primes: |
\(2\), \(3\), \(11\)
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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. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{39}a^{7}+\frac{1}{39}a^{6}+\frac{10}{39}a^{5}-\frac{3}{13}a^{4}+\frac{4}{13}a^{3}+\frac{4}{39}a^{2}-\frac{10}{39}$
| 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: | $C_{2}$, which has order $2$ |
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Unit group
| Rank: | $7$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{35}{39}a^{7}-\frac{134}{39}a^{6}-\frac{100}{13}a^{5}+\frac{350}{13}a^{4}+\frac{270}{13}a^{3}-\frac{1693}{39}a^{2}-\frac{82}{3}a+\frac{87}{13}$, $\frac{24}{13}a^{7}-\frac{93}{13}a^{6}-\frac{202}{13}a^{5}+\frac{733}{13}a^{4}+\frac{522}{13}a^{3}-\frac{1204}{13}a^{2}-52a+\frac{215}{13}$, $\frac{2}{13}a^{7}-\frac{7}{39}a^{6}-\frac{122}{39}a^{5}+\frac{34}{13}a^{4}+\frac{206}{13}a^{3}-\frac{122}{13}a^{2}-\frac{52}{3}a+\frac{161}{39}$, $\frac{24}{13}a^{7}-\frac{93}{13}a^{6}-\frac{202}{13}a^{5}+\frac{733}{13}a^{4}+\frac{522}{13}a^{3}-\frac{1204}{13}a^{2}-51a+\frac{215}{13}$, $\frac{116}{39}a^{7}-\frac{430}{39}a^{6}-\frac{1063}{39}a^{5}+\frac{1147}{13}a^{4}+\frac{1049}{13}a^{3}-\frac{5893}{39}a^{2}-103a+\frac{1141}{39}$, $\frac{1}{13}a^{7}-\frac{10}{39}a^{6}-\frac{35}{39}a^{5}+\frac{30}{13}a^{4}+\frac{51}{13}a^{3}-\frac{74}{13}a^{2}-\frac{16}{3}a+\frac{35}{39}$, $\frac{2}{13}a^{7}-\frac{20}{39}a^{6}-\frac{70}{39}a^{5}+\frac{60}{13}a^{4}+\frac{89}{13}a^{3}-\frac{122}{13}a^{2}-\frac{26}{3}a+\frac{109}{39}$
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| Regulator: | \( 711.54464646 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 711.54464646 \cdot 1}{2\cdot\sqrt{41108373504}}\cr\approx \mathstrut & 0.44920747883 \end{aligned}\]
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
| \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), 4.4.50688.2, 4.4.50688.1, \(\Q(\sqrt{2}, \sqrt{3})\) |
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: | 8.8.1243528298496.1, 8.8.552679243776.1, 8.8.310882074624.1 |
| 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 | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.22d1.22 | $x^{8} + 4 x^{7} + 2 x^{4} + 4 x^{2} + 14$ | $8$ | $1$ | $22$ | $D_4$ | $$[2, 3, \frac{7}{2}]$$ |
|
\(3\)
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(11\)
| 11.2.1.0a1.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 11.2.1.0a1.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 11.2.2.2a1.2 | $x^{4} + 14 x^{3} + 53 x^{2} + 28 x + 15$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |