Normalized defining polynomial
\( x^{3} - x^{2} - 12x + 26 \)
Invariants
| Degree: | $3$ |
| |
| Signature: | $[1, 1]$ |
| |
| Discriminant: |
\(-5476\)
\(\medspace = -\,2^{2}\cdot 37^{2}\)
|
| |
| Root discriminant: | \(17.63\) |
| |
| Galois root discriminant: | $2\cdot 37^{2/3}\approx 22.20740493717357$ | ||
| Ramified primes: |
\(2\), \(37\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{6}$, which has order $6$ |
| |
| Narrow class group: | $C_{6}$, which has order $6$ |
|
Unit group
| Rank: | $1$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental unit: |
$a^{2}+a-11$
|
| |
| Regulator: | \( 3.71772475299 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{1}\cdot 3.71772475299 \cdot 6}{2\cdot\sqrt{5476}}\cr\approx \mathstrut & 1.89398542250 \end{aligned}\]
Galois group
| A solvable group of order 6 |
| The 3 conjugacy class representatives for $S_3$ |
| Character table for $S_3$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Galois closure: | 6.0.119946304.2 |
| Minimal sibling: | This field is its own minimal sibling |
Multiplicative Galois module structure
| $U_{K^{gal}}/\textrm{Tors}(U_{K^{gal}}) \cong$ $A$ |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | R | ${\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ | |
|
\(37\)
| 37.1.3.2a1.2 | $x^{3} + 74$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 2.5476.3t2.a.a | $2$ | $ 2^{2} \cdot 37^{2}$ | 3.1.5476.1 | $S_3$ (as 3T2) | $1$ | $0$ |
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 *.