Normalized defining polynomial
\( x^{8} - 2x^{7} - 20x^{6} - 320x^{5} + 1089x^{4} + 1184x^{3} + 29032x^{2} - 116902x + 64669 \)
Invariants
| Degree: | $8$ |
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| Signature: | $[4, 2]$ |
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| Discriminant: |
\(4579455597305625\)
\(\medspace = 3^{4}\cdot 5^{4}\cdot 67^{6}\)
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| Root discriminant: | \(90.70\) |
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| Galois root discriminant: | $3^{1/2}5^{1/2}67^{3/4}\approx 90.69883977059361$ | ||
| Ramified primes: |
\(3\), \(5\), \(67\)
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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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{3}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{1680}a^{6}+\frac{79}{1680}a^{5}-\frac{5}{56}a^{4}+\frac{33}{560}a^{3}-\frac{37}{280}a^{2}-\frac{409}{1680}a-\frac{299}{1680}$, $\frac{1}{2280904080}a^{7}+\frac{2459}{47518835}a^{6}+\frac{231430097}{2280904080}a^{5}-\frac{32263957}{760301360}a^{4}+\frac{1415927}{11696944}a^{3}+\frac{108507121}{456180816}a^{2}+\frac{686873}{81460860}a+\frac{418339193}{2280904080}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
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| Narrow class group: | $C_{2}$, which has order $2$ (assuming GRH) |
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Unit group
| Rank: | $5$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{19221}{380150680}a^{7}+\frac{8121}{380150680}a^{6}-\frac{88273}{95037670}a^{5}-\frac{6278191}{380150680}a^{4}+\frac{44064}{3655295}a^{3}+\frac{16931077}{380150680}a^{2}+\frac{484830923}{380150680}a-\frac{70853896}{47518835}$, $\frac{58465}{152060272}a^{7}-\frac{351643}{152060272}a^{6}-\frac{168187}{9503767}a^{5}-\frac{138725}{152060272}a^{4}+\frac{701217}{731059}a^{3}-\frac{13142489}{152060272}a^{2}-\frac{1919538269}{152060272}a+\frac{198023367}{10861448}$, $\frac{357949}{13576810}a^{7}-\frac{1417157}{65168688}a^{6}-\frac{232377511}{325843440}a^{5}-\frac{552293041}{54307240}a^{4}+\frac{128319923}{8354960}a^{3}+\frac{5540261079}{54307240}a^{2}+\frac{368234613217}{325843440}a-\frac{261936638209}{325843440}$, $\frac{2944699}{95037670}a^{7}-\frac{122867573}{570226020}a^{6}-\frac{249713351}{570226020}a^{5}-\frac{201528329}{95037670}a^{4}+\frac{879587479}{14621180}a^{3}-\frac{18182651873}{95037670}a^{2}+\frac{2831863585}{16292172}a-\frac{27467136149}{570226020}$, $\frac{13\cdots 41}{570226020}a^{7}+\frac{23\cdots 93}{2280904080}a^{6}+\frac{13\cdots 81}{760301360}a^{5}-\frac{24\cdots 69}{380150680}a^{4}-\frac{13\cdots 75}{11696944}a^{3}-\frac{98\cdots 91}{228090408}a^{2}+\frac{96\cdots 11}{2280904080}a-\frac{19\cdots 41}{760301360}$
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| Regulator: | \( 301201.561089 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 301201.561089 \cdot 1}{2\cdot\sqrt{4579455597305625}}\cr\approx \mathstrut & 1.40572385853 \end{aligned}\] (assuming GRH)
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
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{201}) \), \(\Q(\sqrt{1005}) \), 4.2.13534335.1, 4.2.13534335.2, \(\Q(\sqrt{5}, \sqrt{201})\) |
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: | 8.0.508828399700625.1, 8.0.183178223892225.1, 8.0.4579455597305625.15 |
| Minimal sibling: | 8.0.183178223892225.1 |
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 | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 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 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(5\)
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(67\)
| 67.2.4.6a1.2 | $x^{8} + 252 x^{7} + 23822 x^{6} + 1001700 x^{5} + 15848241 x^{4} + 2003400 x^{3} + 95288 x^{2} + 2016 x + 83$ | $4$ | $2$ | $6$ | $D_4$ | $$[\ ]_{4}^{2}$$ |