Normalized defining polynomial
\( x^{6} - x^{5} + 56x^{4} - 179x^{3} - 56x^{2} - 9217x + 76193 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(0, 3)$ |
| |
| Discriminant: |
\(-50325149888\)
\(\medspace = -\,2^{6}\cdot 13^{3}\cdot 71^{3}\)
|
| |
| Root discriminant: | \(60.76\) |
| |
| Galois root discriminant: | $2^{3/2}13^{1/2}71^{1/2}\approx 85.93020423576334$ | ||
| Ramified primes: |
\(2\), \(13\), \(71\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-923}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-923}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{926425762}a^{5}-\frac{15618038}{463212881}a^{4}-\frac{208168523}{926425762}a^{3}+\frac{52182915}{463212881}a^{2}-\frac{123995151}{926425762}a-\frac{121658361}{463212881}$
| Monogenic: | No | |
| Index: | $2$ | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{30}$, which has order $30$ |
| |
| Narrow class group: | $C_{30}$, which has order $30$ |
|
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{32165}{463212881}a^{5}+\frac{354349}{463212881}a^{4}+\frac{1652560}{463212881}a^{3}+\frac{23173343}{463212881}a^{2}-\frac{41126505}{463212881}a-\frac{300738635}{463212881}$, $\frac{68185}{926425762}a^{5}+\frac{10492389}{463212881}a^{4}-\frac{201641153}{926425762}a^{3}+\frac{1080346076}{463212881}a^{2}-\frac{14870679115}{926425762}a+\frac{21711933570}{463212881}$
|
| |
| Regulator: | \( 35.8874405812 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 35.8874405812 \cdot 30}{2\cdot\sqrt{50325149888}}\cr\approx \mathstrut & 0.595224518048 \end{aligned}\]
Galois group
| A solvable group of order 12 |
| The 6 conjugacy class representatives for $D_{6}$ |
| Character table for $D_{6}$ |
Intermediate fields
| \(\Q(\sqrt{-923}) \), 3.1.104.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Galois closure: | deg 12 |
| Twin sextic algebra: | \(\Q\) $\times$ \(\Q(\sqrt{142}) \) $\times$ 3.1.104.1 |
| Degree 6 sibling: | 6.2.30969323008.1 |
| Minimal sibling: | 6.2.30969323008.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.1.0a1.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 2.2.2.6a1.1 | $x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 3$ | $2$ | $2$ | $6$ | $C_2^2$ | $$[3]^{2}$$ | |
|
\(13\)
| 13.1.2.1a1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 13.1.2.1a1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 13.1.2.1a1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(71\)
| 71.3.2.3a1.1 | $x^{6} + 8 x^{4} + 128 x^{3} + 16 x^{2} + 583 x + 4096$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *12 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.104.2t1.b.a | $1$ | $ 2^{3} \cdot 13 $ | \(\Q(\sqrt{-26}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *12 | 1.923.2t1.a.a | $1$ | $ 13 \cdot 71 $ | \(\Q(\sqrt{-923}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.568.2t1.a.a | $1$ | $ 2^{3} \cdot 71 $ | \(\Q(\sqrt{142}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *12 | 2.104.3t2.b.a | $2$ | $ 2^{3} \cdot 13 $ | 3.1.104.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| *12 | 2.524264.6t3.a.a | $2$ | $ 2^{3} \cdot 13 \cdot 71^{2}$ | 6.0.50325149888.5 | $D_{6}$ (as 6T3) | $1$ | $0$ |
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 *.