Normalized defining polynomial
\( x^{12} - 2x^{9} + 2x^{6} - 4x^{3} + 4 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $(0, 6)$ |
| |
| Discriminant: |
\(25389989167104\)
\(\medspace = 2^{16}\cdot 3^{18}\)
|
| |
| Root discriminant: | \(13.09\) |
| |
| Galois root discriminant: | $2^{4/3}3^{31/18}\approx 16.714067630295588$ | ||
| Ramified primes: |
\(2\), \(3\)
|
| |
| 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: | \(\Q(\zeta_{12})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{2}a^{11}$
| 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: |
\( \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - 1 \)
(order $12$)
|
| |
| Fundamental units: |
$\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-2$, $\frac{1}{2}a^{6}+a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}+\frac{1}{2}a^{6}+a^{4}-2a$, $\frac{1}{2}a^{10}+\frac{1}{2}a^{6}-a$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}$
|
| |
| Regulator: | \( 357.058513593 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 357.058513593 \cdot 1}{12\cdot\sqrt{25389989167104}}\cr\approx \mathstrut & 0.363334049309 \end{aligned}\]
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
| \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{12})\), 6.0.314928.1 |
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: | 18.6.10362839986909376151552.1, 18.0.3454279995636458717184.2 |
| 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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.6.16a2.1 | $x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 128 x^{7} + 149 x^{6} + 144 x^{5} + 116 x^{4} + 76 x^{3} + 41 x^{2} + 18 x + 5$ | $6$ | $2$ | $16$ | $C_6\times S_3$ | $$[2]_{3}^{6}$$ |
|
\(3\)
| 3.2.6.18a1.22 | $x^{12} + 12 x^{11} + 75 x^{10} + 310 x^{9} + 936 x^{8} + 2160 x^{7} + 3896 x^{6} + 5520 x^{5} + 6096 x^{4} + 5120 x^{3} + 3120 x^{2} + 1248 x + 259$ | $6$ | $2$ | $18$ | $C_6\times S_3$ | $$[\frac{3}{2}, 2]_{2}^{2}$$ |
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 | 1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *36 | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *36 | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.36.6t1.b.a | $1$ | $ 2^{2} \cdot 3^{2}$ | 6.0.419904.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.36.6t1.a.a | $1$ | $ 2^{2} \cdot 3^{2}$ | \(\Q(\zeta_{36})^+\) | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.36.6t1.a.b | $1$ | $ 2^{2} \cdot 3^{2}$ | \(\Q(\zeta_{36})^+\) | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.9.6t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.36.6t1.b.b | $1$ | $ 2^{2} \cdot 3^{2}$ | 6.0.419904.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.9.6t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (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) | $0$ | $1$ | |
| 2.108.3t2.b.a | $2$ | $ 2^{2} \cdot 3^{3}$ | 3.1.108.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.432.6t3.a.a | $2$ | $ 2^{4} \cdot 3^{3}$ | 6.2.559872.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| *36 | 2.324.6t5.c.a | $2$ | $ 2^{2} \cdot 3^{4}$ | 6.0.314928.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| *36 | 2.324.6t5.c.b | $2$ | $ 2^{2} \cdot 3^{4}$ | 6.0.314928.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| *36 | 2.1296.12t18.a.a | $2$ | $ 2^{4} \cdot 3^{4}$ | 12.0.25389989167104.5 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| *36 | 2.1296.12t18.a.b | $2$ | $ 2^{4} \cdot 3^{4}$ | 12.0.25389989167104.5 | $C_6\times S_3$ (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 *.