Normalized defining polynomial
\( x^{7} - 21x^{5} - 21x^{4} + 91x^{3} + 112x^{2} - 84x - 97 \)
Invariants
| Degree: | $7$ |
| |
| Signature: | $(7, 0)$ |
| |
| Discriminant: |
\(13841287201\)
\(\medspace = 7^{12}\)
|
| |
| Root discriminant: | \(28.10\) |
| |
| Galois root discriminant: | $7^{12/7}\approx 28.102146738487733$ | ||
| Ramified primes: |
\(7\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_7$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(49=7^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{49}(1,·)$, $\chi_{49}(36,·)$, $\chi_{49}(22,·)$, $\chi_{49}(8,·)$, $\chi_{49}(43,·)$, $\chi_{49}(29,·)$, $\chi_{49}(15,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{589}a^{6}+\frac{239}{589}a^{5}-\frac{33}{589}a^{4}-\frac{251}{589}a^{3}+\frac{180}{589}a^{2}+\frac{135}{589}a-\frac{214}{589}$
| Monogenic: | No | |
| 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: | $6$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{125}{589}a^{6}-\frac{164}{589}a^{5}-\frac{2358}{589}a^{4}+\frac{431}{589}a^{3}+\frac{10131}{589}a^{2}+\frac{972}{589}a-\frac{10258}{589}$, $\frac{315}{589}a^{6}-\frac{696}{589}a^{5}-\frac{5094}{589}a^{4}+\frac{4573}{589}a^{3}+\frac{19004}{589}a^{2}-\frac{5773}{589}a-\frac{14400}{589}$, $\frac{110}{589}a^{6}-\frac{215}{589}a^{5}-\frac{1863}{589}a^{4}+\frac{1251}{589}a^{3}+\frac{7431}{589}a^{2}-\frac{1642}{589}a-\frac{5870}{589}$, $\frac{10}{589}a^{6}+\frac{34}{589}a^{5}-\frac{330}{589}a^{4}-\frac{743}{589}a^{3}+\frac{2389}{589}a^{2}+\frac{3117}{589}a-\frac{5085}{589}$, $\frac{320}{589}a^{6}-\frac{679}{589}a^{5}-\frac{5259}{589}a^{4}+\frac{4496}{589}a^{3}+\frac{19315}{589}a^{2}-\frac{6276}{589}a-\frac{14881}{589}$, $\frac{379}{589}a^{6}-\frac{714}{589}a^{5}-\frac{6617}{589}a^{4}+\frac{4412}{589}a^{3}+\frac{26401}{589}a^{2}-\frac{5968}{589}a-\frac{20439}{589}$
|
| |
| Regulator: | \( 550.88577714 \) |
| |
| Unit signature rank: | \( 7 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 550.88577714 \cdot 1}{2\cdot\sqrt{13841287201}}\cr\approx \mathstrut & 0.2996769181 \end{aligned}\]
Galois group
| A cyclic group of order 7 |
| The 7 conjugacy class representatives for $C_7$ |
| Character table for $C_7$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | R | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.1.0.1}{1} }^{7}$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.1.0.1}{1} }^{7}$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.1.7.12a6.1 | $x^{7} + 42 x^{6} + 7$ | $7$ | $1$ | $12$ | $C_7$ | $$[2]$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *7 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *7 | 1.49.7t1.a.a | $1$ | $ 7^{2}$ | 7.7.13841287201.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| *7 | 1.49.7t1.a.b | $1$ | $ 7^{2}$ | 7.7.13841287201.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| *7 | 1.49.7t1.a.c | $1$ | $ 7^{2}$ | 7.7.13841287201.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| *7 | 1.49.7t1.a.d | $1$ | $ 7^{2}$ | 7.7.13841287201.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| *7 | 1.49.7t1.a.e | $1$ | $ 7^{2}$ | 7.7.13841287201.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| *7 | 1.49.7t1.a.f | $1$ | $ 7^{2}$ | 7.7.13841287201.1 | $C_7$ (as 7T1) | $0$ | $1$ |
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 *.