Normalized defining polynomial
\( x^{8} - 2x^{7} - 27x^{6} - 8x^{5} + 277x^{4} - 144x^{3} + 900x^{2} - 2016x - 1152 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
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| Discriminant: |
\(1875745012656384\)
\(\medspace = 2^{8}\cdot 3^{4}\cdot 67^{6}\)
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| Root discriminant: | \(81.12\) |
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| Galois root discriminant: | $2\cdot 3^{1/2}67^{3/4}\approx 81.12350848296349$ | ||
| Ramified primes: |
\(2\), \(3\), \(67\)
|
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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. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{6}a^{5}-\frac{1}{2}a^{3}-\frac{1}{3}a^{2}-\frac{1}{2}a$, $\frac{1}{12}a^{6}+\frac{1}{12}a^{4}-\frac{1}{2}a^{3}+\frac{1}{12}a^{2}-\frac{1}{2}a$, $\frac{1}{12559248}a^{7}+\frac{5069}{2093208}a^{6}-\frac{13971}{1395472}a^{5}+\frac{238349}{1569906}a^{4}+\frac{284989}{1395472}a^{3}-\frac{27916}{87217}a^{2}-\frac{142219}{348868}a-\frac{29707}{87217}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{4}$, which has order $4$ |
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| Narrow class group: | $C_{2}\times C_{8}$, which has order $16$ |
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Unit group
| Rank: | $5$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{111}{1395472}a^{7}+\frac{5345}{2093208}a^{6}-\frac{2309}{1395472}a^{5}-\frac{27241}{348868}a^{4}-\frac{614567}{4186416}a^{3}+\frac{172641}{348868}a^{2}+\frac{86929}{348868}a+\frac{325355}{87217}$, $\frac{30181}{2093208}a^{7}+\frac{8871}{348868}a^{6}-\frac{640391}{2093208}a^{5}-\frac{704501}{523302}a^{4}-\frac{611461}{697736}a^{3}-\frac{1556123}{523302}a^{2}+\frac{1004001}{174434}a+\frac{284409}{87217}$, $\frac{444967}{12559248}a^{7}-\frac{22777}{697736}a^{6}-\frac{5000479}{4186416}a^{5}-\frac{3231875}{3139812}a^{4}+\frac{20207651}{1395472}a^{3}+\frac{7509673}{1046604}a^{2}-\frac{13631199}{348868}a-\frac{1596055}{87217}$, $\frac{67025}{174434}a^{7}-\frac{80495}{261651}a^{6}-\frac{2145407}{174434}a^{5}-\frac{1133528}{87217}a^{4}+\frac{71142491}{523302}a^{3}+\frac{4625024}{87217}a^{2}-\frac{18342794}{87217}a-\frac{7824541}{87217}$, $\frac{135116603}{6279624}a^{7}+\frac{1076779}{1046604}a^{6}-\frac{1103851747}{2093208}a^{5}-\frac{960826673}{784953}a^{4}+\frac{1537611095}{697736}a^{3}-\frac{556818098}{261651}a^{2}+\frac{3233991109}{174434}a-\frac{764537285}{87217}$
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| Regulator: | \( 117724.584417 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 117724.584417 \cdot 4}{2\cdot\sqrt{1875745012656384}}\cr\approx \mathstrut & 3.43391847912 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 8T24):
| A solvable group of order 48 |
| The 10 conjugacy class representatives for $S_4\times C_2$ |
| Character table for $S_4\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), 4.2.3609156.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 6 siblings: | 6.0.43309872.2, 6.2.34821137088.1 |
| Degree 8 sibling: | 8.0.13026007032336.1 |
| Degree 12 siblings: | deg 12, deg 12, deg 12, deg 12, deg 12, deg 12 |
| Degree 16 sibling: | deg 16 |
| Degree 24 siblings: | deg 24, deg 24, deg 24, deg 24 |
| Minimal sibling: | 6.0.43309872.2 |
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 | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.1.0.1}{1} }^{8}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.2a1.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ |
| 2.1.2.2a1.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ | |
| 2.2.2.4a1.1 | $x^{4} + 2 x^{3} + 5 x^{2} + 4 x + 5$ | $2$ | $2$ | $4$ | $C_2^2$ | $$[2]^{2}$$ | |
|
\(3\)
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(67\)
| 67.2.4.6a1.2 | $x^{8} + 252 x^{7} + 23822 x^{6} + 1001700 x^{5} + 15848241 x^{4} + 2003400 x^{3} + 95288 x^{2} + 2016 x + 83$ | $4$ | $2$ | $6$ | $D_4$ | $$[\ ]_{4}^{2}$$ |