Normalized defining polynomial
\( x^{8} - 4x^{7} + 4x^{6} - 12x^{5} + 20x^{4} - 44x^{3} + 36x^{2} - 20x - 15 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-95693045760\)
\(\medspace = -\,2^{22}\cdot 3^{3}\cdot 5\cdot 13^{2}\)
|
| |
| Root discriminant: | \(23.58\) |
| |
| Galois root discriminant: | $2^{11/4}3^{1/2}5^{1/2}13^{1/2}\approx 93.9398351563814$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(13\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-15}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{14339}a^{7}+\frac{1419}{14339}a^{6}-\frac{2558}{14339}a^{5}+\frac{2060}{14339}a^{4}+\frac{6244}{14339}a^{3}-\frac{5012}{14339}a^{2}-\frac{5557}{14339}a-\frac{6842}{14339}$
| 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: | $C_{2}\times C_{2}$, which has order $4$ |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{554}{14339}a^{7}-\frac{2519}{14339}a^{6}+\frac{2429}{14339}a^{5}-\frac{5880}{14339}a^{4}+\frac{17816}{14339}a^{3}-\frac{23560}{14339}a^{2}+\frac{32985}{14339}a-\frac{4972}{14339}$, $\frac{3014}{14339}a^{7}-\frac{10495}{14339}a^{6}+\frac{4570}{14339}a^{5}-\frac{28625}{14339}a^{4}+\frac{49665}{14339}a^{3}-\frac{78896}{14339}a^{2}+\frac{70849}{14339}a+\frac{40711}{14339}$, $\frac{5755}{14339}a^{7}-\frac{21224}{14339}a^{6}+\frac{19202}{14339}a^{5}-\frac{74748}{14339}a^{4}+\frac{101059}{14339}a^{3}-\frac{252094}{14339}a^{2}+\frac{196181}{14339}a-\frac{144206}{14339}$, $\frac{211}{14339}a^{7}-\frac{1710}{14339}a^{6}+\frac{5144}{14339}a^{5}-\frac{9849}{14339}a^{4}+\frac{12635}{14339}a^{3}-\frac{25124}{14339}a^{2}+\frac{31949}{14339}a-\frac{24101}{14339}$
|
| |
| Regulator: | \( 499.929118855 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 499.929118855 \cdot 2}{2\cdot\sqrt{95693045760}}\cr\approx \mathstrut & 1.60349585246 \end{aligned}\]
Galois group
$C_2\wr S_4$ (as 8T44):
| A solvable group of order 384 |
| The 20 conjugacy class representatives for $C_2 \wr S_4$ |
| Character table for $C_2 \wr S_4$ |
Intermediate fields
| 4.2.6656.1 |
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 24 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 | R | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }$ | R | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.22d1.22 | $x^{8} + 4 x^{7} + 2 x^{4} + 4 x^{2} + 14$ | $8$ | $1$ | $22$ | $D_4$ | $$[2, 3, \frac{7}{2}]$$ |
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 3.3.2.3a1.1 | $x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 7 x + 1$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
|
\(5\)
| 5.1.2.1a1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.3.1.0a1.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 5.3.1.0a1.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
|
\(13\)
| 13.1.2.1a1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 13.1.2.1a1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 13.4.1.0a1.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |