Normalized defining polynomial
\( x^{8} - 4x^{7} + 8x^{6} - 8x^{5} + 6x^{3} - 2x^{2} - 2 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-81415168\)
\(\medspace = -\,2^{10}\cdot 43^{3}\)
|
| |
| Root discriminant: | \(9.75\) |
| |
| Galois root discriminant: | $2^{4/3}43^{1/2}\approx 16.52370966031933$ | ||
| Ramified primes: |
\(2\), \(43\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-43}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{19}a^{7}+\frac{7}{19}a^{6}+\frac{9}{19}a^{5}-\frac{4}{19}a^{4}-\frac{6}{19}a^{3}-\frac{3}{19}a^{2}+\frac{3}{19}a-\frac{5}{19}$
| 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: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{19}a^{7}-\frac{12}{19}a^{6}+\frac{28}{19}a^{5}-\frac{42}{19}a^{4}+\frac{13}{19}a^{3}+\frac{35}{19}a^{2}-\frac{16}{19}a-\frac{5}{19}$, $\frac{13}{19}a^{7}-\frac{42}{19}a^{6}+\frac{79}{19}a^{5}-\frac{71}{19}a^{4}-\frac{2}{19}a^{3}+\frac{18}{19}a^{2}+\frac{1}{19}a+\frac{11}{19}$, $a-1$, $\frac{7}{19}a^{7}-\frac{27}{19}a^{6}+\frac{44}{19}a^{5}-\frac{28}{19}a^{4}-\frac{42}{19}a^{3}+\frac{55}{19}a^{2}+\frac{21}{19}a-\frac{35}{19}$
|
| |
| Regulator: | \( 8.25075445397 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 8.25075445397 \cdot 1}{2\cdot\sqrt{81415168}}\cr\approx \mathstrut & 0.453639226823 \end{aligned}\]
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
| 4.2.688.1 |
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: | 8.2.81415168.3 |
| 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 | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | R | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.10b1.1 | $x^{8} + 2 x^{3} + 2 x^{2} + 2$ | $8$ | $1$ | $10$ | $\textrm{GL(2,3)}$ | $$[\frac{4}{3}, \frac{4}{3}, \frac{3}{2}]_{3}^{2}$$ |
|
\(43\)
| 43.2.1.0a1.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 43.1.2.1a1.2 | $x^{2} + 129$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 43.2.2.2a1.2 | $x^{4} + 84 x^{3} + 1770 x^{2} + 252 x + 52$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *48 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.43.2t1.a.a | $1$ | $ 43 $ | \(\Q(\sqrt{-43}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.172.3t2.a.a | $2$ | $ 2^{2} \cdot 43 $ | 3.1.172.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.344.24t22.a.a | $2$ | $ 2^{3} \cdot 43 $ | 8.2.81415168.2 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
| 2.344.24t22.a.b | $2$ | $ 2^{3} \cdot 43 $ | 8.2.81415168.2 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
| 3.29584.6t8.c.a | $3$ | $ 2^{4} \cdot 43^{2}$ | 4.2.688.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
| *48 | 3.688.4t5.b.a | $3$ | $ 2^{4} \cdot 43 $ | 4.2.688.1 | $S_4$ (as 4T5) | $1$ | $1$ |
| *48 | 4.118336.8t23.a.a | $4$ | $ 2^{6} \cdot 43^{2}$ | 8.2.81415168.2 | $\textrm{GL(2,3)}$ (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 *.