Normalized defining polynomial
\( x^{12} - x^{9} + 2x^{6} + x^{3} + 1 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $(0, 6)$ |
| |
| Discriminant: |
\(6053445140625\)
\(\medspace = 3^{18}\cdot 5^{6}\)
|
| |
| Root discriminant: | \(11.62\) |
| |
| Galois root discriminant: | $3^{31/18}5^{1/2}\approx 14.831798946842026$ | ||
| Ramified primes: |
\(3\), \(5\)
|
| |
| 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: | \(\Q(\sqrt{-3}, \sqrt{5})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{2}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: |
\( \frac{1}{2} a^{9} - a^{6} + a^{3} + \frac{1}{2} \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{1}{2}a^{10}-a^{7}+a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}+a^{7}-a^{4}+\frac{3}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $a^{11}-\frac{1}{2}a^{10}-a^{9}-a^{8}+a^{7}+a^{6}+2a^{5}-a^{4}-2a^{3}+a^{2}+\frac{1}{2}a-1$, $\frac{1}{2}a^{11}+a^{10}+\frac{1}{2}a^{9}-a^{8}-a^{7}+2a^{5}+2a^{4}-\frac{3}{2}a^{2}+a+\frac{3}{2}$, $\frac{1}{2}a^{11}+\frac{1}{2}a^{10}-a^{7}+a^{4}-a^{3}+\frac{3}{2}a^{2}-\frac{1}{2}a$
|
| |
| Regulator: | \( 106.294858375 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 106.294858375 \cdot 1}{6\cdot\sqrt{6053445140625}}\cr\approx \mathstrut & 0.443035892439 \end{aligned}\]
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
| \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 6.0.2460375.2 |
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: | 18.6.402131117372361328125.1, 18.0.9651146816936671875.1 |
| 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 | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.2.0.1}{2} }^{3}$ | R | R | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.6.18a1.25 | $x^{12} + 15 x^{11} + 105 x^{10} + 460 x^{9} + 1416 x^{8} + 3240 x^{7} + 5672 x^{6} + 7680 x^{5} + 8016 x^{4} + 6320 x^{3} + 3600 x^{2} + 1344 x + 259$ | $6$ | $2$ | $18$ | $C_6\times S_3$ | $$[\frac{3}{2}, 2]_{2}^{2}$$ |
|
\(5\)
| 5.6.2.6a1.2 | $x^{12} + 2 x^{10} + 8 x^{9} + 3 x^{8} + 8 x^{7} + 22 x^{6} + 8 x^{5} + 5 x^{4} + 16 x^{3} + 4 x^{2} + 9$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $$[\ ]_{2}^{6}$$ |
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 | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *36 | 1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *36 | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.45.6t1.a.a | $1$ | $ 3^{2} \cdot 5 $ | 6.6.820125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.9.6t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.9.6t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.45.6t1.b.a | $1$ | $ 3^{2} \cdot 5 $ | 6.0.2460375.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.45.6t1.a.b | $1$ | $ 3^{2} \cdot 5 $ | 6.6.820125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.45.6t1.b.b | $1$ | $ 3^{2} \cdot 5 $ | 6.0.2460375.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 2.135.3t2.b.a | $2$ | $ 3^{3} \cdot 5 $ | 3.1.135.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.135.6t3.b.a | $2$ | $ 3^{3} \cdot 5 $ | 6.0.54675.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| *36 | 2.405.6t5.a.a | $2$ | $ 3^{4} \cdot 5 $ | 6.0.2460375.2 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| *36 | 2.405.6t5.a.b | $2$ | $ 3^{4} \cdot 5 $ | 6.0.2460375.2 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| *36 | 2.405.12t18.a.a | $2$ | $ 3^{4} \cdot 5 $ | 12.0.6053445140625.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| *36 | 2.405.12t18.a.b | $2$ | $ 3^{4} \cdot 5 $ | 12.0.6053445140625.2 | $C_6\times S_3$ (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 *.