Normalized defining polynomial
\( x^{3} - x^{2} - 22x - 5 \)
Invariants
| Degree: | $3$ |
| |
| Signature: | $[3, 0]$ |
| |
| Discriminant: |
\(4489\)
\(\medspace = 67^{2}\)
|
| |
| Root discriminant: | \(16.50\) |
| |
| Galois root discriminant: | $67^{2/3}\approx 16.49617297223391$ | ||
| Ramified primes: |
\(67\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_3$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(67\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{67}(1,·)$, $\chi_{67}(29,·)$, $\chi_{67}(37,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{3}a^{2}-\frac{1}{3}$
| 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: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a^{2}-4a-7$, $\frac{4}{3}a^{2}-a-\frac{79}{3}$
|
| |
| Regulator: | \( 19.7033299804 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{0}\cdot 19.7033299804 \cdot 1}{2\cdot\sqrt{4489}}\cr\approx \mathstrut & 1.17631820779 \end{aligned}\]
Galois group
| A cyclic group of order 3 |
| The 3 conjugacy class representatives for $C_3$ |
| Character table for $C_3$ |
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 | ${\href{/padicField/2.3.0.1}{3} }$ | ${\href{/padicField/3.1.0.1}{1} }^{3}$ | ${\href{/padicField/5.1.0.1}{1} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(67\)
| 67.1.3.2a1.1 | $x^{3} + 67$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *3 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *3 | 1.67.3t1.a.a | $1$ | $ 67 $ | 3.3.4489.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| *3 | 1.67.3t1.a.b | $1$ | $ 67 $ | 3.3.4489.1 | $C_3$ (as 3T1) | $0$ | $1$ |
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 *.