Normalized defining polynomial
\( x^{12} - 10x^{10} + 32x^{8} - 24x^{6} - 48x^{4} + 64x^{2} + 64 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $(0, 6)$ |
| |
| Discriminant: |
\(24904730935296\)
\(\medspace = 2^{18}\cdot 3^{6}\cdot 19^{4}\)
|
| |
| Root discriminant: | \(13.07\) |
| |
| Galois root discriminant: | $2^{3/2}3^{1/2}19^{2/3}\approx 34.88253362095716$ | ||
| Ramified primes: |
\(2\), \(3\), \(19\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_6$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{2}, \sqrt{-3})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( \frac{1}{16} a^{8} - \frac{1}{2} a^{6} + \frac{5}{4} a^{4} - \frac{1}{2} a^{2} - 1 \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{1}{32}a^{10}-\frac{1}{4}a^{8}+\frac{5}{8}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{32}a^{10}-\frac{5}{16}a^{8}+a^{6}-a^{4}-\frac{1}{2}a^{2}+2$, $\frac{1}{16}a^{8}-\frac{1}{8}a^{7}-\frac{1}{2}a^{6}+\frac{3}{4}a^{5}+\frac{5}{4}a^{4}-a^{3}-\frac{1}{2}a^{2}-2$, $\frac{1}{2}a^{2}-a$, $\frac{1}{32}a^{11}-\frac{5}{16}a^{9}+a^{7}+\frac{1}{8}a^{6}-a^{5}-\frac{3}{4}a^{4}+\frac{1}{2}a^{2}+2$
|
| |
| Regulator: | \( 74.34391887043572 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 74.34391887043572 \cdot 1}{6\cdot\sqrt{24904730935296}}\cr\approx \mathstrut & 0.152768031638102 \end{aligned}\]
Galois group
$C_6\times S_3$ (as 12T18):
| A solvable group of order 36 |
| The 18 conjugacy class representatives for $C_6\times S_3$ |
| Character table for $C_6\times S_3$ |
Intermediate fields
| \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), 6.0.9747.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 36 |
| Degree 18 siblings: | 18.6.216561186743676164093509632.1, 18.0.5847152042079256430524760064.1 |
| 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 | R | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.2.18a1.17 | $x^{12} + 2 x^{10} + 2 x^{9} + x^{8} + 4 x^{7} + 7 x^{6} + 2 x^{5} + 8 x^{4} + 6 x^{3} + x^{2} + 6 x + 7$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $$[3]^{6}$$ |
|
\(3\)
| 3.6.2.6a1.2 | $x^{12} + 4 x^{10} + 6 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} + 9 x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $$[\ ]_{2}^{6}$$ |
|
\(19\)
| 19.2.1.0a1.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 19.2.1.0a1.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 19.2.1.0a1.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 19.2.3.4a1.2 | $x^{6} + 54 x^{5} + 978 x^{4} + 6048 x^{3} + 1956 x^{2} + 216 x + 27$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *36 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *36 | 1.24.2t1.b.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{-6}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *36 | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *36 | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.57.6t1.a.a | $1$ | $ 3 \cdot 19 $ | 6.0.3518667.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.152.6t1.b.a | $1$ | $ 2^{3} \cdot 19 $ | 6.6.66724352.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.456.6t1.c.a | $1$ | $ 2^{3} \cdot 3 \cdot 19 $ | 6.0.1801557504.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.57.6t1.a.b | $1$ | $ 3 \cdot 19 $ | 6.0.3518667.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.456.6t1.c.b | $1$ | $ 2^{3} \cdot 3 \cdot 19 $ | 6.0.1801557504.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.152.6t1.b.b | $1$ | $ 2^{3} \cdot 19 $ | 6.6.66724352.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 2.1083.3t2.b.a | $2$ | $ 3 \cdot 19^{2}$ | 3.1.1083.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.69312.6t3.f.a | $2$ | $ 2^{6} \cdot 3 \cdot 19^{2}$ | 6.0.1801557504.3 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| *36 | 2.57.6t5.a.a | $2$ | $ 3 \cdot 19 $ | 6.0.9747.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| *36 | 2.3648.12t18.b.a | $2$ | $ 2^{6} \cdot 3 \cdot 19 $ | 12.0.24904730935296.4 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| *36 | 2.57.6t5.a.b | $2$ | $ 3 \cdot 19 $ | 6.0.9747.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| *36 | 2.3648.12t18.b.b | $2$ | $ 2^{6} \cdot 3 \cdot 19 $ | 12.0.24904730935296.4 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
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 *.