Normalized defining polynomial
\( x^{8} - 8x^{6} - 20x^{4} + 4 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(10173546496\)
\(\medspace = 2^{18}\cdot 197^{2}\)
|
| |
| Root discriminant: | \(17.82\) |
| |
| Galois root discriminant: | $2^{7/3}197^{1/2}\approx 70.73533852187154$ | ||
| Ramified primes: |
\(2\), \(197\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{8}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{8}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{8}a^{6}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{7}-\frac{1}{4}a^{3}$
| 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: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{8}a^{7}-\frac{1}{8}a^{6}-\frac{9}{8}a^{5}+\frac{9}{8}a^{4}-\frac{5}{4}a^{3}+\frac{5}{4}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{15}{8}a^{5}-\frac{1}{8}a^{4}-6a^{3}+a^{2}-\frac{11}{4}a+\frac{9}{4}$, $\frac{1}{4}a^{6}-\frac{9}{4}a^{4}-\frac{5}{2}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{7}-\frac{5}{4}a^{5}-\frac{1}{4}a^{3}+\frac{5}{2}a$, $\frac{1}{8}a^{7}-a^{5}-\frac{1}{8}a^{4}-\frac{9}{4}a^{3}+\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{5}{4}$
|
| |
| Regulator: | \( 176.977224939 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 176.977224939 \cdot 1}{2\cdot\sqrt{10173546496}}\cr\approx \mathstrut & 0.554154567941 \end{aligned}\]
Galois group
$C_2^3:S_4$ (as 8T39):
| A solvable group of order 192 |
| The 13 conjugacy class representatives for $C_2^3:S_4$ |
| Character table for $C_2^3:S_4$ |
Intermediate fields
| 4.4.50432.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 8.0.158961664.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.18c1.4 | $x^{8} + 2 x^{4} + 4 x^{3} + 6$ | $8$ | $1$ | $18$ | $S_4\times C_2$ | $$[2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$$ |
|
\(197\)
| 197.2.1.0a1.1 | $x^{2} + 192 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 197.2.1.0a1.1 | $x^{2} + 192 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 197.2.2.2a1.2 | $x^{4} + 384 x^{3} + 36868 x^{2} + 768 x + 201$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.197.2t1.a.a | $1$ | $ 197 $ | \(\Q(\sqrt{197}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 2.788.3t2.a.a | $2$ | $ 2^{2} \cdot 197 $ | 3.3.788.1 | $S_3$ (as 3T2) | $1$ | $2$ | |
| 3.50432.4t5.b.a | $3$ | $ 2^{8} \cdot 197 $ | 4.0.50432.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
| 3.155236.6t8.a.a | $3$ | $ 2^{2} \cdot 197^{2}$ | 4.0.788.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
| * | 3.50432.4t5.a.a | $3$ | $ 2^{8} \cdot 197 $ | 4.4.50432.1 | $S_4$ (as 4T5) | $1$ | $3$ |
| 3.9935104.6t8.a.a | $3$ | $ 2^{8} \cdot 197^{2}$ | 4.4.50432.1 | $S_4$ (as 4T5) | $1$ | $3$ | |
| 3.788.4t5.a.a | $3$ | $ 2^{2} \cdot 197 $ | 4.0.788.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
| 3.9935104.6t8.b.a | $3$ | $ 2^{8} \cdot 197^{2}$ | 4.0.50432.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
| * | 4.201728.8t39.e.a | $4$ | $ 2^{10} \cdot 197 $ | 8.4.10173546496.1 | $C_2^3:S_4$ (as 8T39) | $1$ | $0$ |
| 4.7828861952.8t39.e.a | $4$ | $ 2^{10} \cdot 197^{3}$ | 8.4.10173546496.1 | $C_2^3:S_4$ (as 8T39) | $1$ | $0$ | |
| 6.501047164928.8t34.a.a | $6$ | $ 2^{16} \cdot 197^{3}$ | 8.0.98706291490816.7 | $V_4^2:S_3$ (as 8T34) | $1$ | $-2$ | |
| 8.157...056.24t333.e.a | $8$ | $ 2^{20} \cdot 197^{4}$ | 8.4.10173546496.1 | $C_2^3:S_4$ (as 8T39) | $1$ | $0$ |
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 *.