Normalized defining polynomial
\( x^{8} - 1764x^{4} - 32928x^{2} - 259308 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-1783963098021888\)
\(\medspace = -\,2^{22}\cdot 3^{11}\cdot 7^{4}\)
|
| |
| Root discriminant: | \(80.62\) |
| |
| Galois root discriminant: | $2^{11/4}3^{11/6}7^{1/2}\approx 133.3840218795417$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $\frac{1}{7}a^{2}$, $\frac{1}{7}a^{3}$, $\frac{1}{392}a^{4}-\frac{1}{14}a^{2}+\frac{1}{4}$, $\frac{1}{392}a^{5}-\frac{1}{14}a^{3}+\frac{1}{4}a$, $\frac{1}{8232}a^{6}-\frac{1}{28}a^{2}$, $\frac{1}{24696}a^{7}+\frac{1}{28}a^{3}-\frac{1}{3}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{4}$, which has order $4$ |
| |
| Narrow class group: | $C_{2}\times C_{4}$, which has order $8$ |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{4116}a^{6}-\frac{1}{392}a^{4}-\frac{3}{7}a^{2}-\frac{13}{4}$, $\frac{1}{8232}a^{6}-\frac{5}{28}a^{2}-2$, $\frac{3861229}{686}a^{7}-\frac{41363480}{1029}a^{6}+\frac{13966719}{49}a^{5}-\frac{98765852}{49}a^{4}+\frac{30554631}{7}a^{3}-\frac{212952718}{7}a^{2}+29773436a-208678771$, $\frac{457371220325263}{4116}a^{7}-\frac{34235590264027}{98}a^{6}-\frac{51232648354781}{98}a^{5}+\frac{40929903938396}{7}a^{4}-\frac{28\cdots 55}{14}a^{3}+562952613995275a^{2}-24\!\cdots\!25a+12\!\cdots\!27$
|
| |
| Regulator: | \( 31279.9046433 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 31279.9046433 \cdot 4}{2\cdot\sqrt{1783963098021888}}\cr\approx \mathstrut & 1.46960993484 \end{aligned}\]
Galois group
$\GL(2,3)$ (as 8T23):
| A solvable group of order 48 |
| The 8 conjugacy class representatives for $\textrm{GL(2,3)}$ |
| Character table for $\textrm{GL(2,3)}$ |
Intermediate fields
| 4.2.15552.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 16 sibling: | deg 16 |
| Degree 24 sibling: | deg 24 |
| Arithmetically equivalent sibling: | 8.2.1783963098021888.55 |
| 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 | ${\href{/padicField/5.8.0.1}{8} }$ | R | ${\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.22a1.72 | $x^{8} + 12 x^{7} + 42 x^{6} + 88 x^{5} + 127 x^{4} + 128 x^{3} + 102 x^{2} + 76 x + 23$ | $4$ | $2$ | $22$ | $C_8$ | $$[3, 4]^{2}$$ |
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 3.1.6.11a1.1 | $x^{6} + 3$ | $6$ | $1$ | $11$ | $S_3$ | $$[\frac{5}{2}]_{2}$$ | |
|
\(7\)
| 7.1.2.1a1.2 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 7.3.2.3a1.1 | $x^{6} + 12 x^{5} + 36 x^{4} + 8 x^{3} + 48 x^{2} + 7 x + 16$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |