Normalized defining polynomial
\( x^{8} - x^{7} - 14611 x^{6} + 167762 x^{5} + 54504499 x^{4} - 1420847358 x^{3} - 30406309452 x^{2} + \cdots - 5770386205141 \)
Invariants
| Degree: | $8$ |
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| Signature: | $(8, 0)$ |
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| Discriminant: |
\(1388804176712512646676107246129\)
\(\medspace = 97^{6}\cdot 401^{7}\)
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| Root discriminant: | \(5859.09\) |
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| Galois root discriminant: | $97^{3/4}401^{7/8}\approx 5859.089427923591$ | ||
| Ramified primes: |
\(97\), \(401\)
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| Discriminant root field: | \(\Q(\sqrt{401}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_8$ |
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| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(38897=97\cdot 401\) | ||
| Dirichlet character group: | not computed | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}-\frac{1}{7}$, $\frac{1}{30\cdots 71}a^{7}-\frac{13\cdots 37}{30\cdots 71}a^{6}-\frac{11\cdots 62}{43\cdots 53}a^{5}-\frac{16\cdots 10}{43\cdots 53}a^{4}+\frac{16\cdots 00}{43\cdots 53}a^{3}-\frac{21\cdots 34}{43\cdots 53}a^{2}-\frac{54\cdots 22}{30\cdots 71}a+\frac{14\cdots 08}{30\cdots 71}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $7$ |
Class group and class number
| Ideal class group: | $C_{2260}$, which has order $2260$ (assuming GRH) |
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| Narrow class group: | $C_{2260}$, which has order $2260$ (assuming GRH) |
|
Unit group
| Rank: | $7$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{12794509621722}{20\cdots 83}a^{7}+\frac{122707647619186}{20\cdots 83}a^{6}-\frac{18\cdots 12}{20\cdots 83}a^{5}+\frac{11\cdots 64}{20\cdots 83}a^{4}+\frac{70\cdots 64}{20\cdots 83}a^{3}-\frac{10\cdots 58}{20\cdots 83}a^{2}-\frac{62\cdots 90}{20\cdots 83}a+\frac{12\cdots 33}{20\cdots 83}$, $\frac{79\cdots 37}{43\cdots 53}a^{7}-\frac{50\cdots 38}{30\cdots 71}a^{6}-\frac{13\cdots 86}{43\cdots 53}a^{5}+\frac{42\cdots 23}{43\cdots 53}a^{4}+\frac{41\cdots 99}{43\cdots 53}a^{3}-\frac{25\cdots 29}{43\cdots 53}a^{2}+\frac{43\cdots 59}{43\cdots 53}a-\frac{13\cdots 27}{30\cdots 71}$, $\frac{14\cdots 38}{30\cdots 71}a^{7}+\frac{37\cdots 31}{30\cdots 71}a^{6}-\frac{29\cdots 41}{43\cdots 53}a^{5}-\frac{42\cdots 90}{43\cdots 53}a^{4}+\frac{10\cdots 11}{43\cdots 53}a^{3}-\frac{25\cdots 19}{43\cdots 53}a^{2}-\frac{48\cdots 00}{30\cdots 71}a+\frac{34\cdots 64}{30\cdots 71}$, $\frac{98\cdots 16}{43\cdots 53}a^{7}+\frac{19\cdots 85}{30\cdots 71}a^{6}-\frac{13\cdots 41}{43\cdots 53}a^{5}-\frac{22\cdots 99}{43\cdots 53}a^{4}+\frac{47\cdots 19}{43\cdots 53}a^{3}-\frac{70\cdots 32}{43\cdots 53}a^{2}-\frac{30\cdots 26}{43\cdots 53}a+\frac{16\cdots 92}{30\cdots 71}$, $\frac{74\cdots 60}{43\cdots 53}a^{7}-\frac{74\cdots 36}{30\cdots 71}a^{6}-\frac{11\cdots 87}{43\cdots 53}a^{5}+\frac{22\cdots 09}{43\cdots 53}a^{4}+\frac{49\cdots 55}{43\cdots 53}a^{3}-\frac{12\cdots 51}{43\cdots 53}a^{2}-\frac{34\cdots 79}{43\cdots 53}a+\frac{63\cdots 79}{30\cdots 71}$, $\frac{35\cdots 16}{30\cdots 71}a^{7}+\frac{14\cdots 21}{43\cdots 53}a^{6}-\frac{69\cdots 29}{43\cdots 53}a^{5}-\frac{12\cdots 74}{43\cdots 53}a^{4}+\frac{23\cdots 52}{43\cdots 53}a^{3}+\frac{88\cdots 16}{43\cdots 53}a^{2}-\frac{10\cdots 54}{30\cdots 71}a+\frac{95\cdots 87}{43\cdots 53}$, $\frac{43\cdots 91}{30\cdots 71}a^{7}+\frac{13\cdots 63}{30\cdots 71}a^{6}-\frac{85\cdots 91}{43\cdots 53}a^{5}-\frac{16\cdots 60}{43\cdots 53}a^{4}+\frac{29\cdots 15}{43\cdots 53}a^{3}+\frac{49\cdots 22}{43\cdots 53}a^{2}-\frac{13\cdots 59}{30\cdots 71}a+\frac{83\cdots 97}{30\cdots 71}$
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| Regulator: | \( 1611497881.93 \) (assuming GRH) |
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| Unit signature rank: | \( 8 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 1611497881.93 \cdot 2260}{2\cdot\sqrt{1388804176712512646676107246129}}\cr\approx \mathstrut & 0.395573910727 \end{aligned}\] (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
| \(\Q(\sqrt{401}) \), \(\Q(\sqrt{77794 +194 \sqrt{401}})\) |
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 | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.1.0.1}{1} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 | $x^{8} + 384 x^{7} + 55316 x^{6} + 3544704 x^{5} + 85487766 x^{4} + 17723520 x^{3} + 1382900 x^{2} + 48097 x + 625$ | $4$ | $2$ | $6$ | $C_8$ | $$[\ ]_{4}^{2}$$ |
|
\(401\)
| Deg $8$ | $8$ | $1$ | $7$ |