Normalized defining polynomial
\( x^{8} - 30x^{6} + 330x^{4} - 1595x^{2} + 4205 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(16820000000\)
\(\medspace = 2^{8}\cdot 5^{7}\cdot 29^{2}\)
|
| |
| Root discriminant: | \(18.98\) |
| |
| Galois root discriminant: | $2^{7/4}5^{7/8}29^{1/2}\approx 74.06282017462216$ | ||
| 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. | |||
| Maximal CM subfield: | \(\Q(\zeta_{5})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{67309}a^{6}-\frac{17140}{67309}a^{4}+\frac{417}{67309}a^{2}-\frac{59}{2321}$, $\frac{1}{67309}a^{7}-\frac{17140}{67309}a^{5}+\frac{417}{67309}a^{3}-\frac{59}{2321}a$
| 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: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -\frac{291}{67309} a^{6} + \frac{6874}{67309} a^{4} - \frac{54038}{67309} a^{2} + \frac{5564}{2321} \)
(order $10$)
|
| |
| Fundamental units: |
$\frac{291}{67309}a^{6}-\frac{6874}{67309}a^{4}+\frac{54038}{67309}a^{2}-\frac{3243}{2321}$, $\frac{456}{67309}a^{7}-\frac{2898}{67309}a^{6}-\frac{7996}{67309}a^{5}+\frac{64987}{67309}a^{4}-\frac{11775}{67309}a^{3}-\frac{400758}{67309}a^{2}+\frac{21837}{2321}a-\frac{3093}{2321}$, $\frac{602}{67309}a^{7}-\frac{67}{6119}a^{6}-\frac{20003}{67309}a^{5}+\frac{4127}{6119}a^{4}+\frac{116416}{67309}a^{3}-\frac{21820}{6119}a^{2}-\frac{19271}{2321}a+\frac{3953}{211}$
|
| |
| Regulator: | \( 166.761653125 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 166.761653125 \cdot 2}{10\cdot\sqrt{16820000000}}\cr\approx \mathstrut & 0.400804570281 \end{aligned}\]
Galois group
$\OD_{16}:C_2$ (as 8T16):
| A solvable group of order 32 |
| The 11 conjugacy class representatives for $(C_8:C_2):C_2$ |
| Character table for $(C_8:C_2):C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 8 sibling: | data not computed |
| Degree 16 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
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 | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }$ | R | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.1.0.1}{1} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.2.8a5.1 | $x^{8} + 2 x^{7} + 4 x^{5} + 6 x^{4} + 2 x^{3} + 3 x^{2} + 6 x + 5$ | $2$ | $4$ | $8$ | $(C_8:C_2):C_2$ | $$[2, 2, 2]^{4}$$ |
|
\(5\)
| 5.1.8.7a1.1 | $x^{8} + 5$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $$[\ ]_{8}^{2}$$ |
|
\(29\)
| 29.2.2.2a1.1 | $x^{4} + 48 x^{3} + 580 x^{2} + 125 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
| 29.4.1.0a1.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *32 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *32 | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.20.2t1.a.a | $1$ | $ 2^{2} \cdot 5 $ | \(\Q(\sqrt{-5}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.20.4t1.a.a | $1$ | $ 2^{2} \cdot 5 $ | \(\Q(\zeta_{20})^+\) | $C_4$ (as 4T1) | $0$ | $1$ | |
| *32 | 1.5.4t1.a.a | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
| 1.20.4t1.a.b | $1$ | $ 2^{2} \cdot 5 $ | \(\Q(\zeta_{20})^+\) | $C_4$ (as 4T1) | $0$ | $1$ | |
| *32 | 1.5.4t1.a.b | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
| 2.336400.4t3.b.a | $2$ | $ 2^{4} \cdot 5^{2} \cdot 29^{2}$ | 4.0.6728000.4 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| 2.67280.4t3.d.a | $2$ | $ 2^{4} \cdot 5 \cdot 29^{2}$ | 4.0.269120.4 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| *32 | 4.134560000.8t16.c.a | $4$ | $ 2^{8} \cdot 5^{4} \cdot 29^{2}$ | 8.0.16820000000.1 | $(C_8:C_2):C_2$ (as 8T16) | $1$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.