Normalized defining polynomial
\( x^{7} - 7x - 3 \)
Invariants
| Degree: | $7$ |
| |
| Signature: | $[3, 2]$ |
| |
| Discriminant: |
\(4202539929\)
\(\medspace = 3^{6}\cdot 7^{8}\)
|
| |
| Root discriminant: | \(23.70\) |
| |
| Galois root discriminant: | $3^{7/6}7^{26/21}\approx 40.08236355961002$ | ||
| Ramified primes: |
\(3\), \(7\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$
| 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: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{3}a^{6}-\frac{2}{3}a^{5}+\frac{1}{3}a^{4}-\frac{2}{3}a^{3}+\frac{4}{3}a^{2}-\frac{2}{3}a-1$, $\frac{1}{3}a^{6}-\frac{2}{3}a^{5}+\frac{4}{3}a^{4}-\frac{5}{3}a^{3}+\frac{4}{3}a^{2}-\frac{2}{3}a-1$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{2}{3}a^{3}-\frac{5}{3}a^{2}-\frac{2}{3}a$, $a^{6}-a^{2}-7$
|
| |
| Regulator: | \( 328.357654884 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{2}\cdot 328.357654884 \cdot 1}{2\cdot\sqrt{4202539929}}\cr\approx \mathstrut & 0.799854420109 \end{aligned}\]
Galois group
$\PSL(2,7)$ (as 7T5):
| A non-solvable group of order 168 |
| The 6 conjugacy class representatives for $\GL(3,2)$ |
| Character table for $\GL(3,2)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 8 sibling: | 8.0.37822859361.1 |
| 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, deg 42, some data not computed |
| Arithmetically equivalent sibling: | 7.3.4202539929.3 |
| Minimal sibling: | 7.3.4202539929.3 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.7.0.1}{7} }$ | R | ${\href{/padicField/5.7.0.1}{7} }$ | R | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 3.1.3.3a1.2 | $x^{3} + 6 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $$[\frac{3}{2}]_{2}$$ | |
| 3.1.3.3a1.2 | $x^{3} + 6 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $$[\frac{3}{2}]_{2}$$ | |
|
\(7\)
| 7.1.7.8a2.1 | $x^{7} + 7 x^{2} + 7$ | $7$ | $1$ | $8$ | $C_7:C_3$ | $$[\frac{4}{3}]_{3}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *168 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 3.194481.42t37.b.a | $3$ | $ 3^{4} \cdot 7^{4}$ | 7.3.4202539929.1 | $\GL(3,2)$ (as 7T5) | $0$ | $-1$ | |
| 3.194481.42t37.b.b | $3$ | $ 3^{4} \cdot 7^{4}$ | 7.3.4202539929.1 | $\GL(3,2)$ (as 7T5) | $0$ | $-1$ | |
| *168 | 6.4202539929.7t5.b.a | $6$ | $ 3^{6} \cdot 7^{8}$ | 7.3.4202539929.1 | $\GL(3,2)$ (as 7T5) | $1$ | $2$ |
| 7.37822859361.8t37.b.a | $7$ | $ 3^{8} \cdot 7^{8}$ | 7.3.4202539929.1 | $\GL(3,2)$ (as 7T5) | $1$ | $-1$ | |
| 8.166...201.21t14.b.a | $8$ | $ 3^{10} \cdot 7^{10}$ | 7.3.4202539929.1 | $\GL(3,2)$ (as 7T5) | $1$ | $0$ |
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 *.