Normalized defining polynomial
\( x^{6} - 2x^{5} + 11x^{4} - 12x^{3} + 25x^{2} - 14x - 1 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[2, 2]$ |
| |
| Discriminant: |
\(24920064\)
\(\medspace = 2^{14}\cdot 3^{2}\cdot 13^{2}\)
|
| |
| Root discriminant: | \(17.09\) |
| |
| Galois root discriminant: | $2^{11/4}3^{1/2}13^{1/2}\approx 42.011171440958684$ | ||
| Ramified primes: |
\(2\), \(3\), \(13\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{8}a^{5}+\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{3}{8}a+\frac{3}{8}$
| 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: |
$\frac{1}{8}a^{5}-\frac{3}{8}a^{4}+\frac{7}{4}a^{3}-\frac{9}{4}a^{2}+\frac{35}{8}a-\frac{17}{8}$, $\frac{1}{8}a^{5}-\frac{3}{8}a^{4}+\frac{3}{4}a^{3}-\frac{5}{4}a^{2}+\frac{3}{8}a-\frac{1}{8}$, $\frac{3}{8}a^{5}-\frac{1}{8}a^{4}+\frac{9}{4}a^{3}+\frac{5}{4}a^{2}+\frac{9}{8}a-\frac{19}{8}$
|
| |
| Regulator: | \( 22.7867388452 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 22.7867388452 \cdot 3}{2\cdot\sqrt{24920064}}\cr\approx \mathstrut & 1.08123124035 \end{aligned}\]
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
| \(\Q(\sqrt{2}) \) |
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: | 6.2.24920064.2 |
| Degree 6 sibling: | 6.2.24920064.2 |
| Degree 9 sibling: | 9.1.9703274840064.1 |
| 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 | R | R | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | ${\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.3a1.3 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ |
| 2.1.4.11a1.11 | $x^{4} + 8 x^{3} + 4 x^{2} + 2$ | $4$ | $1$ | $11$ | $C_4$ | $$[3, 4]$$ | |
|
\(3\)
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 3.2.2.2a1.1 | $x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ | |
|
\(13\)
| 13.2.1.0a1.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 13.2.2.2a1.1 | $x^{4} + 24 x^{3} + 148 x^{2} + 61 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.624.4t1.b.a | $1$ | $ 2^{4} \cdot 3 \cdot 13 $ | 4.0.3115008.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
| 1.624.4t1.b.b | $1$ | $ 2^{4} \cdot 3 \cdot 13 $ | 4.0.3115008.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
| * | 4.3115008.6t10.a.a | $4$ | $ 2^{11} \cdot 3^{2} \cdot 13^{2}$ | 6.2.24920064.1 | $C_3^2:C_4$ (as 6T10) | $1$ | $0$ |
| 4.3115008.6t10.b.a | $4$ | $ 2^{11} \cdot 3^{2} \cdot 13^{2}$ | 6.2.24920064.1 | $C_3^2:C_4$ (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 *.