Normalized defining polynomial
\( x^{8} - x^{7} + 5513 x^{6} + 806255 x^{5} + 51051896 x^{4} + 2060468911 x^{3} + 78159964013 x^{2} + \cdots + 32330330838121 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(0, 4)$ |
| |
| Discriminant: |
\(3467718266061334766148954927097\)
\(\medspace = 97^{6}\cdot 457^{7}\)
|
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| Root discriminant: | \(6569.09\) |
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| Galois root discriminant: | $97^{3/4}457^{7/8}\approx 6569.093990039318$ | ||
| Ramified primes: |
\(97\), \(457\)
|
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| Discriminant root field: | \(\Q(\sqrt{457}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_8$ |
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| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(44329=97\cdot 457\) | ||
| Dirichlet character group: | not computed | ||
| This is a CM field. | |||
| Reflex fields: | 8.0.3467718266061334766148954927097.1$^{8}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{2}a^{3}+\frac{1}{6}a^{2}+\frac{1}{6}a+\frac{1}{6}$, $\frac{1}{46\cdots 42}a^{7}+\frac{18\cdots 89}{46\cdots 42}a^{6}+\frac{81\cdots 79}{23\cdots 71}a^{5}-\frac{81\cdots 55}{46\cdots 42}a^{4}-\frac{60\cdots 04}{23\cdots 71}a^{3}-\frac{14\cdots 67}{76\cdots 57}a^{2}-\frac{66\cdots 41}{76\cdots 57}a-\frac{15\cdots 39}{46\cdots 42}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{5}\times C_{469940}$, which has order $2349700$ (assuming GRH) |
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| Narrow class group: | $C_{5}\times C_{469940}$, which has order $2349700$ (assuming GRH) |
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| Relative class number: | $1174850$ (assuming GRH) |
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{13\cdots 70}{17\cdots 99}a^{7}-\frac{25\cdots 61}{17\cdots 99}a^{6}+\frac{67\cdots 24}{17\cdots 99}a^{5}+\frac{97\cdots 94}{17\cdots 99}a^{4}+\frac{46\cdots 26}{17\cdots 99}a^{3}+\frac{98\cdots 85}{17\cdots 99}a^{2}+\frac{23\cdots 32}{17\cdots 99}a-\frac{18\cdots 85}{17\cdots 99}$, $\frac{37\cdots 42}{17\cdots 99}a^{7}-\frac{19\cdots 93}{17\cdots 99}a^{6}+\frac{28\cdots 57}{17\cdots 99}a^{5}+\frac{17\cdots 48}{17\cdots 99}a^{4}+\frac{72\cdots 87}{17\cdots 99}a^{3}+\frac{41\cdots 01}{17\cdots 99}a^{2}+\frac{21\cdots 27}{17\cdots 99}a+\frac{19\cdots 29}{17\cdots 99}$, $\frac{36\cdots 06}{17\cdots 99}a^{7}-\frac{53\cdots 87}{17\cdots 99}a^{6}+\frac{26\cdots 45}{17\cdots 99}a^{5}+\frac{15\cdots 60}{17\cdots 99}a^{4}+\frac{63\cdots 39}{17\cdots 99}a^{3}+\frac{32\cdots 51}{17\cdots 99}a^{2}+\frac{20\cdots 35}{17\cdots 99}a+\frac{12\cdots 74}{17\cdots 99}$
|
| |
| Regulator: | \( 156867.695289 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 156867.695289 \cdot 2349700}{2\cdot\sqrt{3467718266061334766148954927097}}\cr\approx \mathstrut & 0.154245856888 \end{aligned}\] (assuming GRH)
Galois group
| A cyclic group of order 8 |
| The 8 conjugacy class representatives for $C_8$ |
| Character table for $C_8$ |
Intermediate fields
| \(\Q(\sqrt{457}) \), \(\Q(\sqrt{88658 +4074 \sqrt{457}})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(97\)
| 97.2.4.6a1.1 | $x^{8} + 384 x^{7} + 55316 x^{6} + 3544704 x^{5} + 85487766 x^{4} + 17723520 x^{3} + 1382900 x^{2} + 48097 x + 625$ | $4$ | $2$ | $6$ | $C_8$ | $$[\ ]_{4}^{2}$$ |
|
\(457\)
| Deg $8$ | $8$ | $1$ | $7$ |