Normalized defining polynomial
\( x^{8} - 10x^{6} + 37x^{4} - 70x^{2} + 61 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(39976960000\)
\(\medspace = 2^{20}\cdot 5^{4}\cdot 61\)
|
| |
| Root discriminant: | \(21.15\) |
| |
| Galois root discriminant: | $2^{3}5^{1/2}61^{1/2}\approx 139.71399357258383$ | ||
| Ramified primes: |
\(2\), \(5\), \(61\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{61}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{13}a^{6}-\frac{2}{13}a^{4}-\frac{5}{13}a^{2}-\frac{6}{13}$, $\frac{1}{13}a^{7}-\frac{2}{13}a^{5}-\frac{5}{13}a^{3}-\frac{6}{13}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{2}{13}a^{6}-\frac{17}{13}a^{4}+\frac{42}{13}a^{2}-\frac{51}{13}$, $\frac{5}{13}a^{6}-\frac{36}{13}a^{4}+\frac{79}{13}a^{2}-\frac{95}{13}$, $\frac{2}{13}a^{7}-\frac{4}{13}a^{6}-\frac{17}{13}a^{5}+\frac{34}{13}a^{4}+\frac{42}{13}a^{3}-\frac{97}{13}a^{2}-\frac{25}{13}a+\frac{89}{13}$, $\frac{2}{13}a^{7}+\frac{2}{13}a^{6}-\frac{17}{13}a^{5}-\frac{17}{13}a^{4}+\frac{42}{13}a^{3}+\frac{42}{13}a^{2}-\frac{51}{13}a-\frac{64}{13}$, $\frac{2}{13}a^{7}-\frac{8}{13}a^{6}-\frac{17}{13}a^{5}+\frac{55}{13}a^{4}+\frac{42}{13}a^{3}-\frac{129}{13}a^{2}-\frac{64}{13}a+\frac{152}{13}$
|
| |
| Regulator: | \( 226.870166322 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 226.870166322 \cdot 1}{2\cdot\sqrt{39976960000}}\cr\approx \mathstrut & 0.358362229910 \end{aligned}\]
Galois group
$C_2\wr D_4$ (as 8T35):
| A solvable group of order 128 |
| The 20 conjugacy class representatives for $C_2 \wr C_2\wr C_2$ |
| Character table for $C_2 \wr C_2\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.2.1600.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 32 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 | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.20a1.7 | $x^{8} + 4 x^{7} + 10 x^{6} + 20 x^{5} + 27 x^{4} + 28 x^{3} + 18 x^{2} + 8 x + 3$ | $4$ | $2$ | $20$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $$[2, 3, 3, \frac{7}{2}]^{4}$$ |
|
\(5\)
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(61\)
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 61.1.2.1a1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 61.2.1.0a1.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 61.2.1.0a1.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |