Normalized defining polynomial
\( x^{8} - x^{7} + 4x^{6} - 4x^{5} - 2x^{4} - 4x^{3} - 5x^{2} - x + 1 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-182660427\)
\(\medspace = -\,3^{7}\cdot 17^{4}\)
|
| |
| Root discriminant: | \(10.78\) |
| |
| Galois root discriminant: | $3^{7/8}17^{2/3}\approx 17.289418182735098$ | ||
| Ramified primes: |
\(3\), \(17\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{3}-\frac{1}{3}a+\frac{1}{3}$
| 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: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+a^{4}-a^{3}-\frac{4}{3}a^{2}-a-\frac{1}{3}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{4}{3}a^{5}-\frac{5}{3}a^{4}-\frac{1}{3}a^{3}-\frac{8}{3}a^{2}-a$, $\frac{2}{3}a^{7}-a^{6}+\frac{10}{3}a^{5}-\frac{14}{3}a^{4}+\frac{5}{3}a^{3}-\frac{14}{3}a^{2}-\frac{4}{3}a+\frac{2}{3}$
|
| |
| Regulator: | \( 9.96103143264 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 9.96103143264 \cdot 1}{2\cdot\sqrt{182660427}}\cr\approx \mathstrut & 0.365638296583 \end{aligned}\]
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
| 4.2.7803.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 16 sibling: | 16.0.33364831591822329.1 |
| Degree 24 sibling: | deg 24 |
| Arithmetically equivalent sibling: | 8.2.182660427.1 |
| Minimal sibling: | 8.2.182660427.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }$ | R | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | R | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.1.8.7a1.2 | $x^{8} + 6$ | $8$ | $1$ | $7$ | $QD_{16}$ | $$[\ ]_{8}^{2}$$ |
|
\(17\)
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 17.2.3.4a1.2 | $x^{6} + 48 x^{5} + 777 x^{4} + 4384 x^{3} + 2331 x^{2} + 432 x + 44$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *48 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.867.3t2.b.a | $2$ | $ 3 \cdot 17^{2}$ | 3.1.867.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.2601.24t22.b.a | $2$ | $ 3^{2} \cdot 17^{2}$ | 8.2.182660427.2 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
| 2.2601.24t22.b.b | $2$ | $ 3^{2} \cdot 17^{2}$ | 8.2.182660427.2 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
| 3.2601.6t8.a.a | $3$ | $ 3^{2} \cdot 17^{2}$ | 4.2.7803.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
| *48 | 3.7803.4t5.a.a | $3$ | $ 3^{3} \cdot 17^{2}$ | 4.2.7803.1 | $S_4$ (as 4T5) | $1$ | $1$ |
| *48 | 4.23409.8t23.b.a | $4$ | $ 3^{4} \cdot 17^{2}$ | 8.2.182660427.2 | $\textrm{GL(2,3)}$ (as 8T23) | $1$ | $0$ |
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 *.