Normalized defining polynomial
\( x^{8} - 3x^{7} + 6x^{5} + 18x^{4} + 18x^{3} + 36x^{2} + 54x + 54 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(2048997570624\)
\(\medspace = 2^{6}\cdot 3^{8}\cdot 47^{4}\)
|
| |
| Root discriminant: | \(34.59\) |
| |
| Galois root discriminant: | $2^{6/7}3^{7/6}47^{2/3}\approx 84.99611583197739$ | ||
| Ramified primes: |
\(2\), \(3\), \(47\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{9}a^{5}+\frac{1}{3}a^{2}$, $\frac{1}{9}a^{6}$, $\frac{1}{27}a^{7}+\frac{1}{9}a^{4}+\frac{1}{3}a^{2}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{4}$, which has order $4$ |
| |
| Narrow class group: | $C_{4}$, which has order $4$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{27}a^{7}-\frac{2}{9}a^{6}+\frac{1}{3}a^{5}+\frac{1}{9}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+a-1$, $\frac{34}{27}a^{7}-5a^{6}+\frac{29}{9}a^{5}+\frac{109}{9}a^{4}+2a^{3}+8a^{2}+45a+21$, $\frac{29}{27}a^{7}-\frac{62}{9}a^{6}+\frac{158}{9}a^{5}-\frac{199}{9}a^{4}+\frac{100}{3}a^{3}-\frac{125}{3}a^{2}+55a-43$
|
| |
| Regulator: | \( 1150.90294062 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 1150.90294062 \cdot 4}{2\cdot\sqrt{2048997570624}}\cr\approx \mathstrut & 2.50620993387 \end{aligned}\]
Galois group
$\PSL(2,7)$ (as 8T37):
| A non-solvable group of order 168 |
| The 6 conjugacy class representatives for $\PSL(2,7)$ |
| Character table for $\PSL(2,7)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 7 siblings: | 7.3.227666396736.2, 7.3.227666396736.3 |
| Degree 14 siblings: | deg 14, deg 14 |
| Degree 21 sibling: | deg 21 |
| Degree 24 sibling: | deg 24 |
| Degree 28 sibling: | deg 28 |
| Degree 42 siblings: | deg 42, some data not computed |
| Minimal sibling: | 7.3.227666396736.2 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 2.1.7.6a1.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $$[\ ]_{7}^{3}$$ | |
|
\(3\)
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.1.6.7a1.3 | $x^{6} + 6 x^{2} + 6$ | $6$ | $1$ | $7$ | $S_3$ | $$[\frac{3}{2}]_{2}$$ | |
|
\(47\)
| 47.2.1.0a1.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 47.2.3.4a1.2 | $x^{6} + 135 x^{5} + 6090 x^{4} + 92475 x^{3} + 30450 x^{2} + 3375 x + 172$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ |