This is the number field with the smallest positive regulator [MR:1022309].
Normalized defining polynomial
\( x^{6} - x^{5} + 2x^{4} - 2x^{3} + 2x^{2} - 2x + 1 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(0, 3)$ |
| |
| Discriminant: |
\(-10051\)
\(\medspace = -\,19\cdot 23^{2}\)
|
| |
| Root discriminant: | \(4.65\) |
| |
| Galois root discriminant: | $19^{1/2}23^{1/2}\approx 20.904544960366874$ | ||
| Ramified primes: |
\(19\), \(23\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-19}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$
| 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: |
$a^{5}+2a^{3}-a^{2}+a-1$, $a$
|
| |
| Regulator: | \( 0.205216461048 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 0.205216461048 \cdot 1}{2\cdot\sqrt{10051}}\cr\approx \mathstrut & 0.2538733810104 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 6T11):
| A solvable group of order 48 |
| The 10 conjugacy class representatives for $S_4\times C_2$ |
| Character table for $S_4\times C_2$ |
Intermediate fields
| 3.1.23.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 4.2.8303.1 $\times$ \(\Q(\sqrt{437}) \) |
| Degree 6 sibling: | 6.2.231173.1 |
| Degree 8 siblings: | 8.0.68939809.1, 8.4.36469158961.1 |
| Degree 12 siblings: | 12.4.6964478817623209.1, 12.0.53440955929.1, 12.2.838790656103.1, 12.0.13165366384921.1, 12.0.6964478817623209.3, 12.0.6964478817623209.1 |
| Degree 16 sibling: | deg 16 |
| Degree 24 siblings: | deg 24, deg 24, deg 24, deg 24 |
| 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 | ${\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | R | R | ${\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(19\)
| 19.1.2.1a1.2 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 19.4.1.0a1.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
|
\(23\)
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 23.2.2.2a1.2 | $x^{4} + 42 x^{3} + 451 x^{2} + 210 x + 48$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *48 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.437.2t1.a.a | $1$ | $ 19 \cdot 23 $ | \(\Q(\sqrt{437}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.23.2t1.a.a | $1$ | $ 23 $ | \(\Q(\sqrt{-23}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.19.2t1.a.a | $1$ | $ 19 $ | \(\Q(\sqrt{-19}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.8303.6t3.b.a | $2$ | $ 19^{2} \cdot 23 $ | 6.2.83453453.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| *48 | 2.23.3t2.b.a | $2$ | $ 23 $ | 3.1.23.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| 3.8303.4t5.b.a | $3$ | $ 19^{2} \cdot 23 $ | 4.2.8303.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
| *48 | 3.437.6t11.b.a | $3$ | $ 19 \cdot 23 $ | 6.0.10051.1 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ |
| 3.10051.6t11.b.a | $3$ | $ 19 \cdot 23^{2}$ | 6.0.10051.1 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
| 3.190969.6t8.b.a | $3$ | $ 19^{2} \cdot 23^{2}$ | 4.2.8303.1 | $S_4$ (as 4T5) | $1$ | $-1$ |
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 *.