Normalized defining polynomial
\( x^{6} + 10x^{4} - 20x^{3} + 15x^{2} - 80x + 90 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[2, 2]$ |
| |
| Discriminant: |
\(193600000\)
\(\medspace = 2^{9}\cdot 5^{5}\cdot 11^{2}\)
|
| |
| Root discriminant: | \(24.05\) |
| |
| Galois root discriminant: | $2^{3/2}5^{5/6}11^{1/2}\approx 35.86875806065302$ | ||
| Ramified primes: |
\(2\), \(5\), \(11\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{10}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{6}a^{4}-\frac{1}{3}a^{3}+\frac{1}{6}a^{2}-\frac{1}{3}a$, $\frac{1}{36}a^{5}-\frac{1}{36}a^{4}+\frac{11}{36}a^{3}+\frac{5}{36}a^{2}+\frac{5}{18}a-\frac{1}{2}$
| Monogenic: | No | |
| Index: | $8$ | |
| Inessential primes: | $2$ |
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: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{11}{4}a^{3}-\frac{5}{4}a^{2}+\frac{5}{2}a-\frac{19}{2}$, $\frac{11}{12}a^{5}+\frac{9}{4}a^{4}+\frac{47}{4}a^{3}+\frac{15}{4}a^{2}+\frac{11}{6}a-\frac{163}{2}$, $\frac{79}{36}a^{5}-\frac{133}{36}a^{4}+\frac{653}{36}a^{3}-\frac{2107}{36}a^{2}+\frac{917}{18}a-\frac{9}{2}$
|
| |
| Regulator: | \( 379.233450856 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 379.233450856 \cdot 2}{2\cdot\sqrt{193600000}}\cr\approx \mathstrut & 4.30401414059 \end{aligned}\]
Galois group
| A solvable group of order 12 |
| The 6 conjugacy class representatives for $D_{6}$ |
| Character table for $D_{6}$ |
Intermediate fields
| \(\Q(\sqrt{10}) \), 3.1.2200.1 |
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: | 3.1.2200.1 $\times$ \(\Q(\sqrt{-55}) \) $\times$ \(\Q\) |
| Degree 6 sibling: | 6.0.266200000.1 |
| 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 | ${\href{/padicField/3.2.0.1}{2} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.3a1.4 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ |
| 2.1.2.3a1.4 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ | |
| 2.1.2.3a1.4 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ | |
|
\(5\)
| 5.1.6.5a1.2 | $x^{6} + 10$ | $6$ | $1$ | $5$ | $D_{6}$ | $$[\ ]_{6}^{2}$$ |
|
\(11\)
| 11.2.1.0a1.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 11.2.2.2a1.2 | $x^{4} + 14 x^{3} + 53 x^{2} + 28 x + 15$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *12 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.88.2t1.b.a | $1$ | $ 2^{3} \cdot 11 $ | \(\Q(\sqrt{-22}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.55.2t1.a.a | $1$ | $ 5 \cdot 11 $ | \(\Q(\sqrt{-55}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *12 | 1.40.2t1.a.a | $1$ | $ 2^{3} \cdot 5 $ | \(\Q(\sqrt{10}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *12 | 2.2200.3t2.a.a | $2$ | $ 2^{3} \cdot 5^{2} \cdot 11 $ | 3.1.2200.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| *12 | 2.2200.6t3.a.a | $2$ | $ 2^{3} \cdot 5^{2} \cdot 11 $ | 6.2.193600000.1 | $D_{6}$ (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 *.