Normalized defining polynomial
\( x^{7} - x^{6} - x^{4} + 3x^{2} - 1 \)
Invariants
| Degree: | $7$ |
| |
| Signature: | $[3, 2]$ |
| |
| Discriminant: |
\(649177\)
\(\medspace = 59\cdot 11003\)
|
| |
| Root discriminant: | \(6.77\) |
| |
| Galois root discriminant: | $59^{1/2}11003^{1/2}\approx 805.715210232499$ | ||
| Ramified primes: |
\(59\), \(11003\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{649177}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$
| Monogenic: | Yes | |
| 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: |
$a$, $2a^{6}-3a^{5}+2a^{4}-4a^{3}+2a^{2}+4a-2$, $2a^{6}-3a^{5}+2a^{4}-3a^{3}+2a^{2}+4a-2$, $2a^{6}-3a^{5}+2a^{4}-4a^{3}+3a^{2}+4a-2$
|
| |
| Regulator: | \( 1.04457859743 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{2}\cdot 1.04457859743 \cdot 1}{2\cdot\sqrt{649177}}\cr\approx \mathstrut & 0.204728964111 \end{aligned}\]
Galois group
| A non-solvable group of order 5040 |
| The 15 conjugacy class representatives for $S_7$ |
| Character table for $S_7$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 14 sibling: | deg 14 |
| Degree 21 sibling: | deg 21 |
| Degree 30 sibling: | deg 30 |
| Degree 35 sibling: | deg 35 |
| Degree 42 siblings: | deg 42, deg 42, some data not computed |
| 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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | R |
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 |
|---|---|---|---|---|---|---|---|
|
\(59\)
| 59.1.2.1a1.1 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 59.2.1.0a1.1 | $x^{2} + 58 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 59.3.1.0a1.1 | $x^{3} + 5 x + 57$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
|
\(11003\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $$[\ ]^{5}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *5040 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.649177.2t1.a.a | $1$ | $ 59 \cdot 11003 $ | \(\Q(\sqrt{649177}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 6.115...657.14t46.a.a | $6$ | $ 59^{5} \cdot 11003^{5}$ | 7.3.649177.1 | $S_7$ (as 7T7) | $1$ | $2$ | |
| *5040 | 6.649177.7t7.a.a | $6$ | $ 59 \cdot 11003 $ | 7.3.649177.1 | $S_7$ (as 7T7) | $1$ | $2$ |
| 14.177...241.21t38.a.a | $14$ | $ 59^{4} \cdot 11003^{4}$ | 7.3.649177.1 | $S_7$ (as 7T7) | $1$ | $2$ | |
| 14.132...649.42t413.a.a | $14$ | $ 59^{10} \cdot 11003^{10}$ | 7.3.649177.1 | $S_7$ (as 7T7) | $1$ | $2$ | |
| 14.204...337.30t565.a.a | $14$ | $ 59^{9} \cdot 11003^{9}$ | 7.3.649177.1 | $S_7$ (as 7T7) | $1$ | $2$ | |
| 14.115...657.30t565.a.a | $14$ | $ 59^{5} \cdot 11003^{5}$ | 7.3.649177.1 | $S_7$ (as 7T7) | $1$ | $2$ | |
| 15.115...657.42t412.a.a | $15$ | $ 59^{5} \cdot 11003^{5}$ | 7.3.649177.1 | $S_7$ (as 7T7) | $1$ | $-1$ | |
| 15.132...649.42t411.a.a | $15$ | $ 59^{10} \cdot 11003^{10}$ | 7.3.649177.1 | $S_7$ (as 7T7) | $1$ | $-1$ | |
| 20.132...649.70.a.a | $20$ | $ 59^{10} \cdot 11003^{10}$ | 7.3.649177.1 | $S_7$ (as 7T7) | $1$ | $-4$ | |
| 21.132...649.84.a.a | $21$ | $ 59^{10} \cdot 11003^{10}$ | 7.3.649177.1 | $S_7$ (as 7T7) | $1$ | $1$ | |
| 21.862...873.42t418.a.a | $21$ | $ 59^{11} \cdot 11003^{11}$ | 7.3.649177.1 | $S_7$ (as 7T7) | $1$ | $1$ | |
| 35.176...201.126.a.a | $35$ | $ 59^{20} \cdot 11003^{20}$ | 7.3.649177.1 | $S_7$ (as 7T7) | $1$ | $-1$ | |
| 35.153...393.70.a.a | $35$ | $ 59^{15} \cdot 11003^{15}$ | 7.3.649177.1 | $S_7$ (as 7T7) | $1$ | $-1$ |
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 *.