Normalized defining polynomial
\( x^{6} - 2x^{5} - x^{4} + 2x^{3} + 2x^{2} - 2x + 1 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[0, 3]$ |
| |
| Discriminant: |
\(-65600\)
\(\medspace = -\,2^{6}\cdot 5^{2}\cdot 41\)
|
| |
| Root discriminant: | \(6.35\) |
| |
| Galois root discriminant: | $2\cdot 5^{2/3}41^{1/2}\approx 37.44569770046876$ | ||
| Ramified primes: |
\(2\), \(5\), \(41\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-41}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-1}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$
| Monogenic: | Yes | |
| 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: |
\( -a^{5} + a^{4} + 2 a^{3} - a^{2} - 2 a + 1 \)
(order $4$)
|
| |
| Fundamental units: |
$a^{5}-a^{4}-a^{3}+a-1$, $a^{4}-2a^{2}-a+1$
|
| |
| Regulator: | \( 1.57718616336 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 1.57718616336 \cdot 1}{4\cdot\sqrt{65600}}\cr\approx \mathstrut & 0.381865701368 \end{aligned}\]
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
| \(\Q(\sqrt{-1}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 6.4.172302500.1 |
| Degree 6 sibling: | 6.4.172302500.1 |
| Degree 9 sibling: | 9.3.68921000000.1 |
| 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 | R | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.6a1.1 | $x^{6} + 2 x^{4} + 4 x^{3} + x^{2} + 4 x + 5$ | $2$ | $3$ | $6$ | $C_6$ | $$[2]^{3}$$ |
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 5.1.3.2a1.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
|
\(41\)
| $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 41.1.2.1a1.1 | $x^{2} + 41$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 41.3.1.0a1.1 | $x^{3} + x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *72 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.164.2t1.a.a | $1$ | $ 2^{2} \cdot 41 $ | \(\Q(\sqrt{-41}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *72 | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.41.2t1.a.a | $1$ | $ 41 $ | \(\Q(\sqrt{41}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 2.164.4t3.a.a | $2$ | $ 2^{2} \cdot 41 $ | 4.2.6724.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| *72 | 4.16400.6t13.d.a | $4$ | $ 2^{4} \cdot 5^{2} \cdot 41 $ | 6.0.65600.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ |
| 4.27568400.12t34.d.a | $4$ | $ 2^{4} \cdot 5^{2} \cdot 41^{3}$ | 6.0.65600.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| 4.4202500.6t13.b.a | $4$ | $ 2^{2} \cdot 5^{4} \cdot 41^{2}$ | 6.0.65600.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
| 4.67240000.12t34.b.a | $4$ | $ 2^{6} \cdot 5^{4} \cdot 41^{2}$ | 6.0.65600.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ |
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 *.