Normalized defining polynomial
\( x^{8} - 23x^{6} + 230x^{4} - 736x^{2} + 736 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(6973082515456\)
\(\medspace = 2^{11}\cdot 23^{7}\)
|
| |
| Root discriminant: | \(40.31\) |
| |
| Galois root discriminant: | $2^{2}23^{7/8}\approx 62.16870885296182$ | ||
| Ramified primes: |
\(2\), \(23\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{46}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-23}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{6}a^{4}-\frac{1}{6}a^{2}+\frac{1}{3}$, $\frac{1}{12}a^{5}-\frac{1}{12}a^{3}-\frac{1}{2}a^{2}+\frac{1}{6}a$, $\frac{1}{48}a^{6}-\frac{1}{16}a^{4}+\frac{5}{24}a^{2}+\frac{1}{6}$, $\frac{1}{192}a^{7}-\frac{1}{96}a^{6}-\frac{1}{64}a^{5}+\frac{1}{32}a^{4}+\frac{5}{96}a^{3}+\frac{19}{48}a^{2}-\frac{11}{24}a-\frac{1}{12}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{6}$, which has order $6$ |
| |
| Narrow class group: | $C_{6}$, which has order $6$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{64}a^{7}+\frac{1}{32}a^{6}-\frac{19}{64}a^{5}-\frac{19}{32}a^{4}+\frac{77}{32}a^{3}+\frac{77}{16}a^{2}-\frac{15}{8}a-\frac{19}{4}$, $\frac{1}{24}a^{6}-\frac{115}{24}a^{4}+\frac{253}{12}a^{2}-24$, $\frac{3113}{192}a^{7}-\frac{2893}{96}a^{6}-\frac{61243}{192}a^{5}+\frac{57575}{96}a^{4}+\frac{84799}{32}a^{3}-\frac{80715}{16}a^{2}-\frac{66563}{24}a+\frac{72691}{12}$
|
| |
| Regulator: | \( 2026.13648654 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 2026.13648654 \cdot 6}{2\cdot\sqrt{6973082515456}}\cr\approx \mathstrut & 3.58754224098 \end{aligned}\]
Galois group
| A solvable group of order 32 |
| The 11 conjugacy class representatives for $Z_8 : Z_8^\times$ |
| Character table for $Z_8 : Z_8^\times$ |
Intermediate fields
| \(\Q(\sqrt{-23}) \), 4.0.97336.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 32 |
| Degree 16 siblings: | deg 16, deg 16, deg 16, deg 16 |
| Arithmetically equivalent sibling: | 8.0.6973082515456.2 |
| 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 | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.8.0.1}{8} }$ | R | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 | $$[\ ]$$ | |
| 2.1.2.3a1.1 | $x^{2} + 2$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ | |
| 2.1.4.8b1.1 | $x^{4} + 2 x^{2} + 4 x + 2$ | $4$ | $1$ | $8$ | $C_2^2$ | $$[2, 3]$$ | |
|
\(23\)
| 23.1.8.7a1.1 | $x^{8} + 23$ | $8$ | $1$ | $7$ | $D_{8}$ | $$[\ ]_{8}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *32 | 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) | $1$ | $-1$ | |
| *32 | 1.23.2t1.a.a | $1$ | $ 23 $ | \(\Q(\sqrt{-23}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.92.2t1.a.a | $1$ | $ 2^{2} \cdot 23 $ | \(\Q(\sqrt{23}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.184.2t1.a.a | $1$ | $ 2^{3} \cdot 23 $ | \(\Q(\sqrt{46}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.184.2t1.b.a | $1$ | $ 2^{3} \cdot 23 $ | \(\Q(\sqrt{-46}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *32 | 2.4232.4t3.b.a | $2$ | $ 2^{3} \cdot 23^{2}$ | 4.0.97336.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
| 2.16928.4t3.a.a | $2$ | $ 2^{5} \cdot 23^{2}$ | 4.0.389344.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| *32 | 4.71639296.8t15.a.a | $4$ | $ 2^{8} \cdot 23^{4}$ | 8.0.6973082515456.1 | $Z_8 : Z_8^\times$ (as 8T15) | $1$ | $0$ |
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 *.