Normalized defining polynomial
\( x^{8} - x^{7} - x^{6} - 4x^{5} + 6x^{4} - 4x^{3} - x^{2} - x + 1 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-1626347584\)
\(\medspace = -\,2^{6}\cdot 71^{4}\)
|
| |
| Root discriminant: | \(14.17\) |
| |
| Galois root discriminant: | $2\cdot 71^{2/3}\approx 34.292743267870684$ | ||
| Ramified primes: |
\(2\), \(71\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: |
$a^{7}-a^{6}-a^{5}-4a^{4}+6a^{3}-4a^{2}-a-1$, $\frac{3}{4}a^{7}-\frac{1}{2}a^{6}-\frac{3}{4}a^{5}-\frac{13}{4}a^{4}+\frac{11}{4}a^{3}-\frac{7}{4}a^{2}-2a-\frac{7}{4}$, $\frac{7}{4}a^{7}-\frac{1}{2}a^{6}-\frac{9}{4}a^{5}-\frac{35}{4}a^{4}+\frac{17}{4}a^{3}-\frac{13}{4}a^{2}-\frac{7}{2}a-\frac{13}{4}$, $\frac{3}{2}a^{7}-a^{6}-\frac{3}{2}a^{5}-\frac{13}{2}a^{4}+\frac{13}{2}a^{3}-\frac{9}{2}a^{2}-2a-\frac{5}{2}$
|
| |
| Regulator: | \( 76.3102907518 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 76.3102907518 \cdot 1}{2\cdot\sqrt{1626347584}}\cr\approx \mathstrut & 0.938741515019 \end{aligned}\]
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
| 4.2.20164.1 |
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: | 8.2.1626347584.2 |
| Minimal sibling: | 8.2.1626347584.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.2a1.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ |
| 2.2.1.0a1.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 2.2.2.4a1.1 | $x^{4} + 2 x^{3} + 5 x^{2} + 4 x + 5$ | $2$ | $2$ | $4$ | $C_2^2$ | $$[2]^{2}$$ | |
|
\(71\)
| 71.2.1.0a1.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 71.1.3.2a1.1 | $x^{3} + 71$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
| 71.1.3.2a1.1 | $x^{3} + 71$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *48 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.20164.3t2.a.a | $2$ | $ 2^{2} \cdot 71^{2}$ | 3.1.20164.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.20164.24t22.b.a | $2$ | $ 2^{2} \cdot 71^{2}$ | 8.2.1626347584.1 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
| 2.20164.24t22.b.b | $2$ | $ 2^{2} \cdot 71^{2}$ | 8.2.1626347584.1 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
| 3.80656.6t8.a.a | $3$ | $ 2^{4} \cdot 71^{2}$ | 4.2.20164.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
| *48 | 3.20164.4t5.a.a | $3$ | $ 2^{2} \cdot 71^{2}$ | 4.2.20164.1 | $S_4$ (as 4T5) | $1$ | $1$ |
| *48 | 4.80656.8t23.b.a | $4$ | $ 2^{4} \cdot 71^{2}$ | 8.2.1626347584.1 | $\textrm{GL(2,3)}$ (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 *.