This is the totally real cubic field with with the smallest absolute discriminant, and the cubic field with Galois group $C_3$ of smallest discriminant.
Normalized defining polynomial
\( x^{3} - x^{2} - 2x + 1 \)
Invariants
| Degree: | $3$ |
| |
| Signature: | $(3, 0)$ |
| |
| Discriminant: |
\(49\)
\(\medspace = 7^{2}\)
|
| |
| Root discriminant: | \(3.66\) |
| |
| Galois root discriminant: | $7^{2/3}\approx 3.6593057100229713$ | ||
| Ramified primes: |
\(7\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_3$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{7}(1,·)$, $\chi_{7}(2,·)$, $\chi_{7}(4,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$
| 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: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a^{2}-2$, $a^{2}-1$
|
| |
| Regulator: | \( 0.525454682123 \) |
| |
| Unit signature rank: | \( 3 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{0}\cdot 0.525454682123 \cdot 1}{2\cdot\sqrt{49}}\cr\approx \mathstrut & 0.3002598183560 \end{aligned}\]
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 | ${\href{/padicField/2.3.0.1}{3} }$ | ${\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.3.0.1}{3} }$ | R | ${\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.1.3.2a1.1 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
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 | 1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| *3 | 1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (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 *.