Normalized defining polynomial
\( x^{6} - 2x^{5} - x^{4} + 5x^{3} - 2x^{2} - 3x + 45 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(0, 3)$ |
| |
| Discriminant: |
\(-33917755\)
\(\medspace = -\,5\cdot 19^{3}\cdot 23\cdot 43\)
|
| |
| Root discriminant: | \(17.99\) |
| |
| Galois root discriminant: | $5^{1/2}19^{1/2}23^{1/2}43^{1/2}\approx 306.52079864178876$ | ||
| Ramified primes: |
\(5\), \(19\), \(23\), \(43\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-93955}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-19}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{4}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{51}a^{5}-\frac{8}{51}a^{4}-\frac{4}{51}a^{3}-\frac{22}{51}a^{2}-\frac{23}{51}a-\frac{6}{17}$
| Monogenic: | Not computed | |
| 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: |
$\frac{5}{51}a^{5}-\frac{2}{17}a^{4}-\frac{20}{51}a^{3}+\frac{26}{51}a^{2}+\frac{55}{51}a-\frac{13}{17}$, $\frac{6}{17}a^{5}-\frac{8}{51}a^{4}-\frac{89}{51}a^{3}-\frac{13}{17}a^{2}+\frac{385}{51}a+\frac{62}{17}$
|
| |
| Regulator: | \( 35.1193746553 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 35.1193746553 \cdot 1}{2\cdot\sqrt{33917755}}\cr\approx \mathstrut & 0.747898796137 \end{aligned}\]
Galois group
$\SOPlus(4,2)$ (as 6T13):
| A solvable group of order 72 |
| The 9 conjugacy class representatives for $C_3^2:D_4$ |
| Character table for $C_3^2:D_4$ |
Intermediate fields
| \(\Q(\sqrt{-19}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | deg 6 |
| Degree 6 sibling: | deg 6 |
| Degree 9 sibling: | deg 9 |
| Degree 12 siblings: | deg 12, deg 12, deg 12, deg 12, deg 12, deg 12 |
| Degree 18 siblings: | deg 18, deg 18, deg 18 |
| Degree 24 siblings: | deg 24, deg 24 |
| Degree 36 siblings: | deg 36, deg 36, deg 36 |
| 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 | ${\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | R | R | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.6.0.1}{6} }$ | R | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 5.1.2.1a1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
|
\(19\)
| 19.3.2.3a1.2 | $x^{6} + 8 x^{4} + 34 x^{3} + 16 x^{2} + 136 x + 308$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
|
\(23\)
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 23.1.2.1a1.2 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(43\)
| $\Q_{43}$ | $x + 40$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{43}$ | $x + 40$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 43.2.1.0a1.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 43.1.2.1a1.2 | $x^{2} + 129$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |