Normalized defining polynomial
\( x^{6} - 3x^{5} + 5x^{4} - 3x^{3} + 4x^{2} + 2x + 2 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(0, 3)$ |
| |
| Discriminant: |
\(-679024\)
\(\medspace = -\,2^{4}\cdot 31\cdot 37^{2}\)
|
| |
| Root discriminant: | \(9.38\) |
| |
| Galois root discriminant: | $2^{2/3}31^{1/2}37^{1/2}\approx 53.76112804302788$ | ||
| Ramified primes: |
\(2\), \(31\), \(37\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-31}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | 8.0.20228803984.1$^{4}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}$
| 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 $1$ |
| |
| Relative class number: | $1$ |
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{2}a^{5}-\frac{3}{2}a^{4}+2a^{3}+1$, $\frac{1}{4}a^{5}-a^{4}+\frac{5}{4}a^{3}-\frac{1}{2}$
|
| |
| Regulator: | \( 6.64934608307 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 6.64934608307 \cdot 1}{2\cdot\sqrt{679024}}\cr\approx \mathstrut & 1.0007969765 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 6T11):
| A solvable group of order 48 |
| The 10 conjugacy class representatives for $S_4\times C_2$ |
| Character table for $S_4\times C_2$ |
Intermediate fields
| 3.3.148.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | \(\Q(\sqrt{-1147}) \) $\times$ 4.4.142228.1 |
| Degree 6 sibling: | 6.0.25123888.1 |
| Degree 8 siblings: | 8.0.20228803984.1, 8.0.27693232654096.1 |
| Degree 12 siblings: | deg 12, deg 12, deg 12, 12.0.631209748236544.1, deg 12, deg 12 |
| Degree 16 sibling: | deg 16 |
| Degree 24 siblings: | deg 24, deg 24, deg 24, deg 24 |
| 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 | ${\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | R | R | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.3.2a1.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ |
| 2.1.3.2a1.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
|
\(31\)
| 31.2.1.0a1.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 31.1.2.1a1.1 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 31.2.1.0a1.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
|
\(37\)
| 37.2.1.0a1.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 37.2.2.2a1.2 | $x^{4} + 66 x^{3} + 1093 x^{2} + 132 x + 41$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *48 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.1147.2t1.a.a | $1$ | $ 31 \cdot 37 $ | \(\Q(\sqrt{-1147}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.37.2t1.a.a | $1$ | $ 37 $ | \(\Q(\sqrt{37}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.31.2t1.a.a | $1$ | $ 31 $ | \(\Q(\sqrt{-31}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.142228.6t3.b.a | $2$ | $ 2^{2} \cdot 31^{2} \cdot 37 $ | 6.0.24144056368.3 | $D_{6}$ (as 6T3) | $1$ | $-2$ | |
| *48 | 2.148.3t2.a.a | $2$ | $ 2^{2} \cdot 37 $ | 3.3.148.1 | $S_3$ (as 3T2) | $1$ | $2$ |
| 3.142228.4t5.a.a | $3$ | $ 2^{2} \cdot 31^{2} \cdot 37 $ | 4.4.142228.1 | $S_4$ (as 4T5) | $1$ | $3$ | |
| *48 | 3.4588.6t11.b.a | $3$ | $ 2^{2} \cdot 31 \cdot 37 $ | 6.0.679024.1 | $S_4\times C_2$ (as 6T11) | $1$ | $-3$ |
| 3.169756.6t11.b.a | $3$ | $ 2^{2} \cdot 31 \cdot 37^{2}$ | 6.0.679024.1 | $S_4\times C_2$ (as 6T11) | $1$ | $-3$ | |
| 3.5262436.6t8.a.a | $3$ | $ 2^{2} \cdot 31^{2} \cdot 37^{2}$ | 4.4.142228.1 | $S_4$ (as 4T5) | $1$ | $3$ |
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 *.