Normalized defining polynomial
\( x^{8} - 44x^{6} + 502x^{4} - 1672x^{2} + 980 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[8, 0]$ |
| |
| Discriminant: |
\(370818940928000\)
\(\medspace = 2^{22}\cdot 5^{3}\cdot 29^{4}\)
|
| |
| Root discriminant: | \(66.24\) |
| |
| Galois root discriminant: | $2^{11/4}5^{1/2}29^{1/2}\approx 81.00586972196002$ | ||
| Ramified primes: |
\(2\), \(5\), \(29\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2470}a^{6}-\frac{233}{1235}a^{4}-\frac{223}{1235}a^{2}-\frac{118}{247}$, $\frac{1}{17290}a^{7}+\frac{3239}{17290}a^{5}-\frac{3928}{8645}a^{3}+\frac{376}{1729}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
| |
| Narrow class group: | $C_{4}$, which has order $4$ |
|
Unit group
| Rank: | $7$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{95}a^{6}-\frac{77}{190}a^{4}+\frac{314}{95}a^{2}-\frac{46}{19}$, $\frac{16}{1235}a^{6}-\frac{1327}{2470}a^{4}+\frac{6449}{1235}a^{2}-\frac{2788}{247}$, $\frac{37}{17290}a^{7}-\frac{1187}{17290}a^{5}+\frac{1629}{8645}a^{3}+\frac{1809}{1729}a-1$, $\frac{37}{17290}a^{7}-\frac{1187}{17290}a^{5}+\frac{1629}{8645}a^{3}+\frac{1809}{1729}a+1$, $\frac{58}{8645}a^{7}+\frac{29}{2470}a^{6}-\frac{2328}{8645}a^{5}-\frac{582}{1235}a^{4}+\frac{19827}{8645}a^{3}+\frac{4648}{1235}a^{2}-\frac{4796}{1729}a-\frac{705}{247}$, $\frac{79}{17290}a^{7}-\frac{3}{2470}a^{6}-\frac{3469}{17290}a^{5}+\frac{163}{2470}a^{4}+\frac{18198}{8645}a^{3}-\frac{566}{1235}a^{2}-\frac{8334}{1729}a+\frac{107}{247}$, $\frac{174}{8645}a^{7}+\frac{1}{19}a^{6}-\frac{6984}{8645}a^{5}-\frac{77}{38}a^{4}+\frac{59481}{8645}a^{3}+\frac{295}{19}a^{2}-\frac{7472}{1729}a-\frac{154}{19}$
|
| |
| Regulator: | \( 42030.7472431 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 42030.7472431 \cdot 2}{2\cdot\sqrt{370818940928000}}\cr\approx \mathstrut & 0.558761046258 \end{aligned}\]
Galois group
| A solvable group of order 16 |
| The 7 conjugacy class representatives for $D_{8}$ |
| Character table for $D_{8}$ |
Intermediate fields
| \(\Q(\sqrt{29}) \), 4.4.269120.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }$ | R | ${\href{/padicField/7.2.0.1}{2} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.22a1.72 | $x^{8} + 12 x^{7} + 42 x^{6} + 88 x^{5} + 127 x^{4} + 128 x^{3} + 102 x^{2} + 76 x + 23$ | $4$ | $2$ | $22$ | $C_8$ | $$[3, 4]^{2}$$ |
|
\(5\)
| 5.1.2.1a1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(29\)
| 29.1.2.1a1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 29.1.2.1a1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 29.1.2.1a1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 29.1.2.1a1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |