Normalized defining polynomial
\( x^{6} - x^{5} - 4x^{4} - 2x^{3} - 16x^{2} - x + 59 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[2, 2]$ |
| |
| Discriminant: |
\(38130625\)
\(\medspace = 5^{4}\cdot 13^{2}\cdot 19^{2}\)
|
| |
| Root discriminant: | \(18.35\) |
| |
| Galois root discriminant: | $5^{3/4}13^{1/2}19^{2/3}\approx 85.84234639792447$ | ||
| Ramified primes: |
\(5\), \(13\), \(19\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{942}a^{5}-\frac{7}{157}a^{4}-\frac{83}{471}a^{3}+\frac{35}{157}a^{2}-\frac{74}{471}a+\frac{415}{942}$
| 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: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{23}{942}a^{5}-\frac{4}{157}a^{4}-\frac{25}{471}a^{3}+\frac{20}{157}a^{2}-\frac{289}{471}a-\frac{817}{942}$, $\frac{13}{942}a^{5}+\frac{66}{157}a^{4}-\frac{137}{471}a^{3}-\frac{487}{157}a^{2}-\frac{20}{471}a+\frac{5395}{942}$, $\frac{527}{942}a^{5}-\frac{235}{157}a^{4}+\frac{62}{471}a^{3}-\frac{81}{157}a^{2}-\frac{4615}{471}a+\frac{17117}{942}$
|
| |
| Regulator: | \( 42.7931802738 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 42.7931802738 \cdot 1}{2\cdot\sqrt{38130625}}\cr\approx \mathstrut & 0.547176369706 \end{aligned}\]
Galois group
$C_3^2:C_4$ (as 6T10):
| A solvable group of order 36 |
| The 6 conjugacy class representatives for $C_3^2:C_4$ |
| Character table for $C_3^2:C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Galois closure: | deg 36 |
| Twin sextic algebra: | 6.2.13765155625.1 |
| Degree 6 sibling: | 6.2.13765155625.1 |
| Degree 9 sibling: | 9.1.20994959488140625.1 |
| Degree 12 siblings: | deg 12, deg 12 |
| Degree 18 sibling: | deg 18 |
| 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 | ${\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | R | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.1.4.3a1.4 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
|
\(13\)
| 13.2.1.0a1.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 13.2.2.2a1.1 | $x^{4} + 24 x^{3} + 148 x^{2} + 61 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ | |
|
\(19\)
| 19.1.3.2a1.3 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
| 19.3.1.0a1.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |