Normalized defining polynomial
\( x^{6} - x^{5} - 3x^{4} + 8x^{3} - 20x^{2} - 7x - 49 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[2, 2]$ |
| |
| Discriminant: |
\(313761125\)
\(\medspace = 5^{3}\cdot 61\cdot 41149\)
|
| |
| Root discriminant: | \(26.07\) |
| |
| Galois root discriminant: | $5^{1/2}61^{1/2}41149^{1/2}\approx 3542.660723241784$ | ||
| Ramified primes: |
\(5\), \(61\), \(41149\)
|
| |
| Discriminant root field: | $\Q(\sqrt{12550445}$) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{497}a^{5}+\frac{62}{497}a^{4}-\frac{73}{497}a^{3}-\frac{118}{497}a^{2}+\frac{1}{497}a+\frac{8}{71}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{3}$, which has order $3$ |
| |
| Narrow class group: | $C_{3}$, which has order $3$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{8}{497}a^{5}-\frac{1}{497}a^{4}-\frac{87}{497}a^{3}+\frac{50}{497}a^{2}+\frac{8}{497}a-\frac{78}{71}$, $\frac{31}{497}a^{5}-\frac{66}{497}a^{4}-\frac{275}{497}a^{3}+\frac{318}{497}a^{2}+\frac{528}{497}a+\frac{177}{71}$, $\frac{33}{497}a^{5}+\frac{58}{497}a^{4}+\frac{76}{497}a^{3}+\frac{579}{497}a^{2}+\frac{530}{497}a+\frac{122}{71}$
|
| |
| Regulator: | \( 21.5310161043 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 21.5310161043 \cdot 3}{2\cdot\sqrt{313761125}}\cr\approx \mathstrut & 0.287922726431 \end{aligned}\]
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
| \(\Q(\sqrt{5}) \) |
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 | ${\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.6.0.1}{6} }$ | R | ${\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(61\)
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 61.1.2.1a1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 61.2.1.0a1.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
|
\(41149\)
| $\Q_{41149}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{41149}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |