Normalized defining polynomial
\( x^{6} - x^{5} - 4x^{4} - 7x^{3} - 16x^{2} - 21x - 11 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[2, 2]$ |
| |
| Discriminant: |
\(11560000\)
\(\medspace = 2^{6}\cdot 5^{4}\cdot 17^{2}\)
|
| |
| Root discriminant: | \(15.04\) |
| |
| Galois root discriminant: | $2^{3/2}5^{3/4}17^{1/2}\approx 38.99392548461692$ | ||
| Ramified primes: |
\(2\), \(5\), \(17\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{14}a^{5}+\frac{1}{7}a^{4}-\frac{5}{14}a^{3}+\frac{3}{7}a^{2}-\frac{5}{14}a+\frac{3}{7}$
| Monogenic: | No | |
| Index: | $2$ | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{3}$, which has order $3$ |
| |
| Narrow class group: | $C_{3}$, which has order $3$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{3}{14}a^{5}-\frac{4}{7}a^{4}-\frac{1}{14}a^{3}-\frac{5}{7}a^{2}-\frac{29}{14}a-\frac{5}{7}$, $\frac{1}{14}a^{5}-\frac{5}{14}a^{4}+\frac{9}{14}a^{3}-\frac{15}{14}a^{2}+\frac{9}{14}a-\frac{1}{14}$, $\frac{5}{14}a^{5}-\frac{11}{14}a^{4}-\frac{11}{14}a^{3}-\frac{19}{14}a^{2}-\frac{39}{14}a-\frac{5}{14}$
|
| |
| Regulator: | \( 9.6076343801 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 9.6076343801 \cdot 3}{2\cdot\sqrt{11560000}}\cr\approx \mathstrut & 0.66934270985 \end{aligned}\]
Galois group
$C_3^2:C_4$ (as 6T10):
| A solvable group of order 36 |
| The 6 conjugacy class representatives for $C_3^2:C_4$ |
| Character table for $C_3^2:C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Galois closure: | deg 36 |
| Twin sextic algebra: | 6.2.11560000.3 |
| Degree 6 sibling: | 6.2.11560000.3 |
| Degree 9 sibling: | 9.1.5345344000000.1 |
| Degree 12 siblings: | deg 12, deg 12 |
| Degree 18 sibling: | deg 18 |
| 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 | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | R | ${\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.1.0a1.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 2.2.2.6a1.6 | $x^{4} + 2 x^{3} + 7 x^{2} + 14 x + 7$ | $2$ | $2$ | $6$ | $C_4$ | $$[3]^{2}$$ | |
|
\(5\)
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.1.4.3a1.1 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
|
\(17\)
| 17.2.1.0a1.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 17.2.2.2a1.1 | $x^{4} + 32 x^{3} + 262 x^{2} + 113 x + 9$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.680.4t1.h.a | $1$ | $ 2^{3} \cdot 5 \cdot 17 $ | 4.0.2312000.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
| 1.680.4t1.h.b | $1$ | $ 2^{3} \cdot 5 \cdot 17 $ | 4.0.2312000.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
| * | 4.2312000.6t10.a.a | $4$ | $ 2^{6} \cdot 5^{3} \cdot 17^{2}$ | 6.2.11560000.1 | $C_3^2:C_4$ (as 6T10) | $1$ | $0$ |
| 4.2312000.6t10.b.a | $4$ | $ 2^{6} \cdot 5^{3} \cdot 17^{2}$ | 6.2.11560000.1 | $C_3^2:C_4$ (as 6T10) | $1$ | $0$ |
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 *.