Normalized defining polynomial
\( x^{6} - x^{5} - x^{4} - 12x^{3} - 24x^{2} - 20x - 4 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[2, 2]$ |
| |
| Discriminant: |
\(14440000\)
\(\medspace = 2^{6}\cdot 5^{4}\cdot 19^{2}\)
|
| |
| Root discriminant: | \(15.60\) |
| |
| Galois root discriminant: | $2^{3/2}5^{3/4}19^{2/3}\approx 67.34027682701665$ | ||
| Ramified primes: |
\(2\), \(5\), \(19\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{8}a^{4}+\frac{3}{8}a^{3}+\frac{1}{8}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{8}a^{5}-\frac{1}{8}a^{2}-\frac{1}{4}$
| Monogenic: | No | |
| Index: | $2$ | |
| Inessential primes: | $2$ |
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: |
$\frac{1}{4}a^{5}-\frac{3}{8}a^{4}-\frac{1}{8}a^{3}-\frac{21}{8}a^{2}-\frac{19}{4}a-\frac{11}{4}$, $\frac{1}{8}a^{5}-\frac{9}{8}a^{2}-2a-\frac{5}{4}$, $\frac{3}{8}a^{5}-2a^{3}-\frac{43}{8}a^{2}-4a-\frac{3}{4}$
|
| |
| Regulator: | \( 32.9681500507 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 32.9681500507 \cdot 1}{2\cdot\sqrt{14440000}}\cr\approx \mathstrut & 0.685015997550 \end{aligned}\]
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: | 6.2.5212840000.4 |
| Degree 6 sibling: | 6.2.5212840000.4 |
| Degree 9 sibling: | 9.1.3010936384000000.3 |
| 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 | R | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | R | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.1.0a1.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 2.2.2.6a1.6 | $x^{4} + 2 x^{3} + 7 x^{2} + 14 x + 7$ | $2$ | $2$ | $6$ | $C_4$ | $$[3]^{2}$$ | |
|
\(5\)
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.1.4.3a1.4 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
|
\(19\)
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 19.1.3.2a1.2 | $x^{3} + 38$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.40.4t1.b.a | $1$ | $ 2^{3} \cdot 5 $ | 4.0.8000.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
| 1.40.4t1.b.b | $1$ | $ 2^{3} \cdot 5 $ | 4.0.8000.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
| * | 4.2888000.6t10.b.a | $4$ | $ 2^{6} \cdot 5^{3} \cdot 19^{2}$ | 6.2.14440000.2 | $C_3^2:C_4$ (as 6T10) | $1$ | $0$ |
| 4.1042568000.6t10.b.a | $4$ | $ 2^{6} \cdot 5^{3} \cdot 19^{4}$ | 6.2.14440000.2 | $C_3^2:C_4$ (as 6T10) | $1$ | $0$ |
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 *.