Normalized defining polynomial
\( x^{8} - 4x^{7} + 3x^{6} - x^{5} + 11x^{4} - 9x^{3} + 2x^{2} - 9x - 2 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-2288890672\)
\(\medspace = -\,2^{4}\cdot 523^{3}\)
|
| |
| Root discriminant: | \(14.79\) |
| |
| Galois root discriminant: | $2^{2/3}523^{1/2}\approx 36.30258142598178$ | ||
| Ramified primes: |
\(2\), \(523\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-523}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$
| Monogenic: | Yes | |
| 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: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a^{7}-2a^{6}-2a^{5}-2a^{4}+7a^{3}+4a^{2}+4a+1$, $2a^{7}-5a^{6}-a^{5}-4a^{4}+14a^{3}+a^{2}+9a+1$, $9a^{7}-21a^{6}-7a^{5}-23a^{4}+59a^{3}+17a^{2}+54a+11$, $a^{7}-2a^{6}-a^{5}-3a^{4}+5a^{3}+a^{2}+5a+1$
|
| |
| Regulator: | \( 111.452286611 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 111.452286611 \cdot 1}{2\cdot\sqrt{2288890672}}\cr\approx \mathstrut & 1.15570251725 \end{aligned}\]
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
| 4.2.2092.1 |
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: | 8.2.2288890672.3 |
| 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 | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 2.2.3.4a1.2 | $x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
|
\(523\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $4$ | $2$ | $2$ | $2$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *48 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.523.2t1.a.a | $1$ | $ 523 $ | \(\Q(\sqrt{-523}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.2092.3t2.a.a | $2$ | $ 2^{2} \cdot 523 $ | 3.1.2092.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.2092.24t22.a.a | $2$ | $ 2^{2} \cdot 523 $ | 8.2.2288890672.2 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
| 2.2092.24t22.a.b | $2$ | $ 2^{2} \cdot 523 $ | 8.2.2288890672.2 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
| 3.1094116.6t8.a.a | $3$ | $ 2^{2} \cdot 523^{2}$ | 4.2.2092.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
| *48 | 3.2092.4t5.a.a | $3$ | $ 2^{2} \cdot 523 $ | 4.2.2092.1 | $S_4$ (as 4T5) | $1$ | $1$ |
| *48 | 4.1094116.8t23.a.a | $4$ | $ 2^{2} \cdot 523^{2}$ | 8.2.2288890672.2 | $\textrm{GL(2,3)}$ (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 *.