Normalized defining polynomial
\( x^{8} - x^{7} + 3x^{6} - 6x^{5} + 7x^{4} - 6x^{3} + 11x^{2} - 2x + 4 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(216825625\)
\(\medspace = 5^{4}\cdot 19^{2}\cdot 31^{2}\)
|
| |
| Root discriminant: | \(11.02\) |
| |
| Galois root discriminant: | $5^{1/2}19^{2/3}31^{2/3}\approx 157.11893693679158$ | ||
| Ramified primes: |
\(5\), \(19\), \(31\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{14}a^{7}-\frac{3}{14}a^{6}-\frac{5}{14}a^{5}+\frac{2}{7}a^{4}-\frac{1}{14}a^{3}-\frac{2}{7}a^{2}+\frac{5}{14}a+\frac{1}{7}$
| 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: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{14}a^{7}-\frac{3}{14}a^{6}+\frac{9}{14}a^{5}-\frac{5}{7}a^{4}+\frac{13}{14}a^{3}-\frac{9}{7}a^{2}+\frac{5}{14}a+\frac{1}{7}$, $\frac{3}{14}a^{7}-\frac{9}{14}a^{6}+\frac{13}{14}a^{5}-\frac{15}{7}a^{4}+\frac{39}{14}a^{3}-\frac{20}{7}a^{2}+\frac{29}{14}a-\frac{11}{7}$, $\frac{5}{14}a^{7}-\frac{1}{14}a^{6}+\frac{3}{14}a^{5}-\frac{4}{7}a^{4}-\frac{5}{14}a^{3}+\frac{18}{7}a^{2}-\frac{3}{14}a+\frac{12}{7}$
|
| |
| Regulator: | \( 7.38106061983 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 7.38106061983 \cdot 1}{2\cdot\sqrt{216825625}}\cr\approx \mathstrut & 0.390618624567 \end{aligned}\]
Galois group
$A_4\wr C_2$ (as 8T42):
| A solvable group of order 288 |
| The 14 conjugacy class representatives for $A_4\wr C_2$ |
| Character table for $A_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 siblings: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| Degree 36 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 | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.3.2.3a1.2 | $x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
|
\(19\)
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 19.3.1.0a1.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 19.1.3.2a1.1 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
|
\(31\)
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 31.3.1.0a1.1 | $x^{3} + x + 28$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 31.1.3.2a1.2 | $x^{3} + 93$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |