Normalized defining polynomial
\( x^{6} - x^{5} + 8x^{4} - 22x^{3} - 20x^{2} + 426x + 1611 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[0, 3]$ |
| |
| Discriminant: |
\(-6240321451\)
\(\medspace = -\,7^{5}\cdot 13^{5}\)
|
| |
| Root discriminant: | \(42.91\) |
| |
| Galois root discriminant: | $7^{5/6}13^{5/6}\approx 42.90762670326212$ | ||
| Ramified primes: |
\(7\), \(13\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-91}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_6$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(91=7\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{91}(1,·)$, $\chi_{91}(82,·)$, $\chi_{91}(81,·)$, $\chi_{91}(9,·)$, $\chi_{91}(10,·)$, $\chi_{91}(90,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-91}) \), 6.0.6240321451.2$^{3}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{141231}a^{5}-\frac{1437}{47077}a^{4}-\frac{4999}{47077}a^{3}+\frac{109}{47077}a^{2}+\frac{49997}{141231}a-\frac{116}{263}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{6}$, which has order $6$ |
| |
| Narrow class group: | $C_{6}$, which has order $6$ |
| |
| Relative class number: | $2$ |
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{22}{47077}a^{5}-\frac{688}{47077}a^{4}-\frac{395}{47077}a^{3}+\frac{7194}{47077}a^{2}+\frac{17163}{47077}a-\frac{29}{263}$, $\frac{273}{47077}a^{5}+\frac{22}{47077}a^{4}+\frac{1518}{47077}a^{3}-\frac{4883}{47077}a^{2}-\frac{50226}{47077}a+\frac{202}{263}$
|
| |
| Regulator: | \( 31.7983118742 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 31.7983118742 \cdot 6}{2\cdot\sqrt{6240321451}}\cr\approx \mathstrut & 0.299544564630 \end{aligned}\]
Galois group
| A cyclic group of order 6 |
| The 6 conjugacy class representatives for $C_6$ |
| Character table for $C_6$ |
Intermediate fields
| \(\Q(\sqrt{-91}) \), 3.3.8281.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 3.3.8281.2 $\times$ \(\Q(\sqrt{-91}) \) $\times$ \(\Q\) |
| 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 | ${\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.2.0.1}{2} }^{3}$ | ${\href{/padicField/5.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/11.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.1.0.1}{1} }^{6}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.1.6.5a1.3 | $x^{6} + 21$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ |
|
\(13\)
| 13.1.6.5a1.6 | $x^{6} + 104$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *6 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *6 | 1.91.2t1.a.a | $1$ | $ 7 \cdot 13 $ | \(\Q(\sqrt{-91}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *6 | 1.91.3t1.a.a | $1$ | $ 7 \cdot 13 $ | 3.3.8281.2 | $C_3$ (as 3T1) | $0$ | $1$ |
| *6 | 1.91.6t1.c.a | $1$ | $ 7 \cdot 13 $ | 6.0.6240321451.2 | $C_6$ (as 6T1) | $0$ | $-1$ |
| *6 | 1.91.3t1.a.b | $1$ | $ 7 \cdot 13 $ | 3.3.8281.2 | $C_3$ (as 3T1) | $0$ | $1$ |
| *6 | 1.91.6t1.c.b | $1$ | $ 7 \cdot 13 $ | 6.0.6240321451.2 | $C_6$ (as 6T1) | $0$ | $-1$ |
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 *.