Normalized defining polynomial
\( x^{8} - 4x^{7} + 4x^{6} - 10x^{5} + 40x^{4} - 40x^{3} + 10x^{2} - 10x - 5 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-139968000000\)
\(\medspace = -\,2^{12}\cdot 3^{7}\cdot 5^{6}\)
|
| |
| Root discriminant: | \(24.73\) |
| |
| Galois root discriminant: | $2^{19/12}3^{7/8}5^{5/6}\approx 29.963112711597162$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{1049}a^{7}+\frac{326}{1049}a^{6}-\frac{463}{1049}a^{5}+\frac{354}{1049}a^{4}+\frac{421}{1049}a^{3}+\frac{422}{1049}a^{2}-\frac{247}{1049}a+\frac{302}{1049}$
| 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: |
$\frac{4}{1049}a^{7}+\frac{255}{1049}a^{6}-\frac{803}{1049}a^{5}+\frac{367}{1049}a^{4}-\frac{2512}{1049}a^{3}+\frac{7982}{1049}a^{2}-\frac{4135}{1049}a+\frac{2257}{1049}$, $\frac{375}{1049}a^{7}-\frac{1532}{1049}a^{6}+\frac{1558}{1049}a^{5}-\frac{3620}{1049}a^{4}+\frac{15211}{1049}a^{3}-\frac{15884}{1049}a^{2}+\frac{2834}{1049}a-\frac{3189}{1049}$, $\frac{793}{1049}a^{7}-\frac{3732}{1049}a^{6}+\frac{5236}{1049}a^{5}-\frac{8802}{1049}a^{4}+\frac{34888}{1049}a^{3}-\frac{52435}{1049}a^{2}+\frac{18125}{1049}a+\frac{10804}{1049}$, $\frac{502}{1049}a^{7}-\frac{2090}{1049}a^{6}+\frac{1501}{1049}a^{5}-\frac{3769}{1049}a^{4}+\frac{20424}{1049}a^{3}-\frac{15789}{1049}a^{2}-\frac{7555}{1049}a-\frac{501}{1049}$
|
| |
| Regulator: | \( 486.909901635 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 486.909901635 \cdot 2}{2\cdot\sqrt{139968000000}}\cr\approx \mathstrut & 1.29131984167 \end{aligned}\]
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
| 4.2.10800.2 |
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: | 8.2.139968000000.2 |
| Minimal sibling: | 8.2.139968000000.2 |
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 | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.12b1.1 | $x^{8} + 2 x^{5} + 2 x^{2} + 2$ | $8$ | $1$ | $12$ | $\textrm{GL(2,3)}$ | $$[\frac{4}{3}, \frac{4}{3}, 2]_{3}^{2}$$ |
|
\(3\)
| 3.1.8.7a1.2 | $x^{8} + 6$ | $8$ | $1$ | $7$ | $QD_{16}$ | $$[\ ]_{8}^{2}$$ |
|
\(5\)
| 5.1.2.1a1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.1.6.5a1.1 | $x^{6} + 5$ | $6$ | $1$ | $5$ | $D_{6}$ | $$[\ ]_{6}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 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) | $1$ | $-1$ | |
| 2.300.3t2.a.a | $2$ | $ 2^{2} \cdot 3 \cdot 5^{2}$ | 3.1.300.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.3600.24t22.d.a | $2$ | $ 2^{4} \cdot 3^{2} \cdot 5^{2}$ | 8.2.139968000000.3 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
| 2.3600.24t22.d.b | $2$ | $ 2^{4} \cdot 3^{2} \cdot 5^{2}$ | 8.2.139968000000.3 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
| 3.3600.6t8.b.a | $3$ | $ 2^{4} \cdot 3^{2} \cdot 5^{2}$ | 4.2.10800.2 | $S_4$ (as 4T5) | $1$ | $-1$ | |
| * | 3.10800.4t5.d.a | $3$ | $ 2^{4} \cdot 3^{3} \cdot 5^{2}$ | 4.2.10800.2 | $S_4$ (as 4T5) | $1$ | $1$ |
| * | 4.12960000.8t23.c.a | $4$ | $ 2^{8} \cdot 3^{4} \cdot 5^{4}$ | 8.2.139968000000.3 | $\textrm{GL(2,3)}$ (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 *.