Normalized defining polynomial
\( x^{8} - x^{7} + 4209 x^{6} - 17071 x^{5} + 4138978 x^{4} - 34427564 x^{3} + 142938536 x^{2} + \cdots + 5334611456 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(1058724315790630924064057\)
\(\medspace = 73^{4}\cdot 233^{7}\)
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| |
| Root discriminant: | \(1007.16\) |
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| Galois root discriminant: | $73^{1/2}233^{7/8}\approx 1007.1585895352947$ | ||
| Ramified primes: |
\(73\), \(233\)
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| Discriminant root field: | \(\Q(\sqrt{233}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_8$ |
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| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(17009=73\cdot 233\) | ||
| Dirichlet character group: | not computed | ||
| This is a CM field. | |||
| Reflex fields: | 8.0.1058724315790630924064057.1$^{8}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{4}+\frac{1}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{16}a^{5}+\frac{1}{16}a^{3}-\frac{1}{8}a^{2}$, $\frac{1}{1216}a^{6}+\frac{5}{608}a^{5}-\frac{39}{1216}a^{4}+\frac{3}{152}a^{3}-\frac{75}{304}a^{2}+\frac{29}{76}a$, $\frac{1}{42\cdots 32}a^{7}+\frac{73\cdots 97}{42\cdots 32}a^{6}-\frac{22\cdots 85}{42\cdots 32}a^{5}+\frac{15\cdots 23}{42\cdots 32}a^{4}-\frac{84\cdots 85}{52\cdots 04}a^{3}+\frac{14\cdots 61}{10\cdots 08}a^{2}-\frac{94\cdots 59}{26\cdots 52}a+\frac{35\cdots 32}{86\cdots 13}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{3}\times C_{3939366}$, which has order $11818098$ (assuming GRH) |
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| Narrow class group: | $C_{3}\times C_{3939366}$, which has order $11818098$ (assuming GRH) |
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| Relative class number: | $11818098$ (assuming GRH) |
Unit group
| Rank: | $3$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{10\cdots 59}{21\cdots 16}a^{7}-\frac{13\cdots 65}{21\cdots 16}a^{6}+\frac{36\cdots 05}{21\cdots 16}a^{5}-\frac{36\cdots 55}{21\cdots 16}a^{4}+\frac{30\cdots 15}{26\cdots 52}a^{3}-\frac{22\cdots 69}{52\cdots 04}a^{2}+\frac{72\cdots 15}{13\cdots 76}a-\frac{19\cdots 33}{86\cdots 13}$, $\frac{86\cdots 05}{10\cdots 08}a^{7}-\frac{11\cdots 25}{10\cdots 08}a^{6}+\frac{36\cdots 35}{10\cdots 08}a^{5}-\frac{15\cdots 63}{10\cdots 08}a^{4}+\frac{43\cdots 17}{13\cdots 76}a^{3}-\frac{75\cdots 29}{26\cdots 52}a^{2}+\frac{42\cdots 77}{65\cdots 88}a+\frac{28\cdots 59}{86\cdots 13}$, $\frac{86\cdots 57}{10\cdots 08}a^{7}-\frac{73\cdots 55}{10\cdots 08}a^{6}+\frac{35\cdots 95}{10\cdots 08}a^{5}-\frac{14\cdots 69}{10\cdots 08}a^{4}+\frac{43\cdots 71}{13\cdots 76}a^{3}-\frac{74\cdots 19}{26\cdots 52}a^{2}+\frac{14\cdots 91}{65\cdots 88}a-\frac{12\cdots 97}{86\cdots 13}$
|
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| Regulator: | \( 13813.693434 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 13813.693434 \cdot 11818098}{2\cdot\sqrt{1058724315790630924064057}}\cr\approx \mathstrut & 123.63898595 \end{aligned}\] (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
| \(\Q(\sqrt{233}) \), 4.4.12649337.1 |
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 | ${\href{/padicField/2.1.0.1}{1} }^{8}$ | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.1.0.1}{1} }^{8}$ | ${\href{/padicField/23.1.0.1}{1} }^{8}$ | ${\href{/padicField/29.1.0.1}{1} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.1.0.1}{1} }^{8}$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(73\)
| 73.4.2.4a1.1 | $x^{8} + 32 x^{6} + 112 x^{5} + 266 x^{4} + 1792 x^{3} + 3296 x^{2} + 633 x + 25$ | $2$ | $4$ | $4$ | $C_8$ | $$[\ ]_{2}^{4}$$ |
|
\(233\)
| Deg $8$ | $8$ | $1$ | $7$ |