Normalized defining polynomial
\( x^{6} - 5x^{4} - 17x^{2} + 93 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(0, 3)$ |
| |
| Discriminant: |
\(-5719872\)
\(\medspace = -\,2^{6}\cdot 3\cdot 31^{3}\)
|
| |
| Root discriminant: | \(13.37\) |
| |
| Galois root discriminant: | $2^{7/4}3^{1/2}31^{1/2}\approx 32.43724541951093$ | ||
| Ramified primes: |
\(2\), \(3\), \(31\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-93}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{12}a^{4}+\frac{1}{6}a^{2}-\frac{1}{4}$, $\frac{1}{12}a^{5}+\frac{1}{6}a^{3}-\frac{1}{4}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
| |
| Narrow class group: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{6}a^{4}-\frac{1}{6}a^{2}-3$, $\frac{5}{4}a^{5}-\frac{31}{12}a^{4}-a^{3}+\frac{7}{3}a^{2}-\frac{105}{4}a+\frac{221}{4}$
|
| |
| Regulator: | \( 5.8077394347 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 5.8077394347 \cdot 2}{2\cdot\sqrt{5719872}}\cr\approx \mathstrut & 0.6023564615 \end{aligned}\]
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
| 3.1.31.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 4.2.17856.2 $\times$ \(\Q(\sqrt{3}) \) |
| Degree 6 sibling: | 6.2.184512.1 |
| Degree 8 siblings: | 8.4.1275346944.1, 8.0.1225608413184.13 |
| Degree 12 siblings: | deg 12, deg 12, deg 12, deg 12, deg 12, deg 12 |
| Degree 16 sibling: | deg 16 |
| Degree 24 siblings: | deg 24, deg 24, deg 24, deg 24 |
| Minimal sibling: | 6.2.184512.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | R | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.6a2.1 | $x^{6} + 4 x^{4} + 2 x^{3} + 3 x^{2} + 4 x + 3$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $$[2, 2, 2]^{3}$$ |
|
\(3\)
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
|
\(31\)
| 31.1.2.1a1.2 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 31.1.2.1a1.1 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 31.1.2.1a1.1 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *48 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.31.2t1.a.a | $1$ | $ 31 $ | \(\Q(\sqrt{-31}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.372.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 31 $ | \(\Q(\sqrt{-93}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.4464.6t3.b.a | $2$ | $ 2^{4} \cdot 3^{2} \cdot 31 $ | 6.2.1660608.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| *48 | 2.31.3t2.b.a | $2$ | $ 31 $ | 3.1.31.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| 3.17856.4t5.b.a | $3$ | $ 2^{6} \cdot 3^{2} \cdot 31 $ | 4.2.17856.2 | $S_4$ (as 4T5) | $1$ | $1$ | |
| *48 | 3.184512.6t11.a.a | $3$ | $ 2^{6} \cdot 3 \cdot 31^{2}$ | 6.0.5719872.1 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ |
| 3.5952.6t11.a.a | $3$ | $ 2^{6} \cdot 3 \cdot 31 $ | 6.0.5719872.1 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
| 3.553536.6t8.b.a | $3$ | $ 2^{6} \cdot 3^{2} \cdot 31^{2}$ | 4.2.17856.2 | $S_4$ (as 4T5) | $1$ | $-1$ |
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 *.