Normalized defining polynomial
\( x^{8} - 4x^{7} + 4x^{6} - 2x^{4} - 4x^{2} + 8x - 2 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(708837376\)
\(\medspace = 2^{22}\cdot 13^{2}\)
|
| |
| Root discriminant: | \(12.77\) |
| |
| Galois root discriminant: | $2^{11/4}13^{1/2}\approx 24.255161140409015$ | ||
| Ramified primes: |
\(2\), \(13\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{79}a^{7}-\frac{10}{79}a^{6}-\frac{15}{79}a^{5}+\frac{11}{79}a^{4}+\frac{11}{79}a^{3}+\frac{13}{79}a^{2}-\frac{3}{79}a+\frac{26}{79}$
| 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: |
$\frac{37}{79}a^{7}-\frac{133}{79}a^{6}+\frac{77}{79}a^{5}+\frac{91}{79}a^{4}-\frac{67}{79}a^{3}-\frac{72}{79}a^{2}-\frac{111}{79}a+\frac{251}{79}$, $\frac{11}{79}a^{7}-\frac{31}{79}a^{6}-\frac{7}{79}a^{5}+\frac{42}{79}a^{4}-\frac{37}{79}a^{3}+\frac{64}{79}a^{2}-\frac{33}{79}a+\frac{49}{79}$, $a-1$, $\frac{36}{79}a^{7}-\frac{123}{79}a^{6}+\frac{92}{79}a^{5}+\frac{1}{79}a^{4}-\frac{78}{79}a^{3}-\frac{6}{79}a^{2}-\frac{108}{79}a+\frac{225}{79}$, $\frac{11}{79}a^{7}-\frac{31}{79}a^{6}-\frac{7}{79}a^{5}+\frac{42}{79}a^{4}+\frac{42}{79}a^{3}-\frac{94}{79}a^{2}-\frac{112}{79}a+\frac{49}{79}$
|
| |
| Regulator: | \( 42.9351678954 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 42.9351678954 \cdot 1}{2\cdot\sqrt{708837376}}\cr\approx \mathstrut & 0.509318656276 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 8T24):
| A solvable group of order 48 |
| The 10 conjugacy class representatives for $S_4\times C_2$ |
| Character table for $S_4\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.2.6656.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 6 siblings: | 6.2.1384448.1, 6.0.2249728.1 |
| Degree 8 sibling: | 8.0.7487094784.4 |
| Degree 12 siblings: | 12.2.199336411529216.1, 12.0.323921668734976.15, 12.0.1295686674939904.39, 12.0.323921668734976.5, 12.4.30667140235264.3, 12.0.5182746699759616.10 |
| Degree 16 sibling: | deg 16 |
| Degree 24 siblings: | deg 24, deg 24, deg 24, deg 24 |
| Minimal sibling: | 6.2.1384448.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.22d1.22 | $x^{8} + 4 x^{7} + 2 x^{4} + 4 x^{2} + 14$ | $8$ | $1$ | $22$ | $D_4$ | $$[2, 3, \frac{7}{2}]$$ |
|
\(13\)
| 13.2.1.0a1.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 13.2.1.0a1.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 13.2.2.2a1.2 | $x^{4} + 24 x^{3} + 148 x^{2} + 48 x + 17$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |