Normalized defining polynomial
\( x^{12} - 12x^{10} - 4x^{9} + 54x^{8} + 36x^{7} - 104x^{6} - 108x^{5} + 57x^{4} + 96x^{3} + 36x^{2} + 36x + 33 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[0, 6]$ |
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| Discriminant: |
\(6499837226778624\)
\(\medspace = 2^{24}\cdot 3^{18}\)
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| Root discriminant: | \(20.78\) |
| |
| Galois root discriminant: | $2^{2}3^{31/18}\approx 26.531928538998848$ | ||
| Ramified primes: |
\(2\), \(3\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_6$ |
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| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{21}a^{9}-\frac{3}{7}a^{7}+\frac{1}{3}a^{6}+\frac{2}{7}a^{5}-\frac{3}{7}a^{3}+\frac{3}{7}a-\frac{1}{7}$, $\frac{1}{21}a^{10}-\frac{3}{7}a^{8}+\frac{1}{3}a^{7}+\frac{2}{7}a^{6}-\frac{3}{7}a^{4}+\frac{3}{7}a^{2}-\frac{1}{7}a$, $\frac{1}{693}a^{11}-\frac{10}{693}a^{10}-\frac{1}{63}a^{9}+\frac{139}{693}a^{8}+\frac{248}{693}a^{7}+\frac{31}{693}a^{6}-\frac{4}{33}a^{5}+\frac{79}{231}a^{4}-\frac{4}{77}a^{3}+\frac{53}{231}a^{2}+\frac{109}{231}a+\frac{4}{21}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
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: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{2}{21}a^{9}-\frac{6}{7}a^{7}-\frac{1}{3}a^{6}+\frac{18}{7}a^{5}+2a^{4}-\frac{13}{7}a^{3}-3a^{2}-\frac{15}{7}a-\frac{9}{7}$, $\frac{9}{77}a^{11}-\frac{50}{231}a^{10}-\frac{22}{21}a^{9}+\frac{129}{77}a^{8}+\frac{811}{231}a^{7}-\frac{923}{231}a^{6}-\frac{415}{77}a^{5}+\frac{164}{77}a^{4}+\frac{281}{77}a^{3}+\frac{12}{77}a^{2}+\frac{116}{77}a+\frac{5}{7}$, $\frac{23}{231}a^{11}+\frac{23}{231}a^{10}-\frac{6}{7}a^{9}-\frac{235}{231}a^{8}+\frac{512}{231}a^{7}+\frac{256}{77}a^{6}-\frac{72}{77}a^{5}-\frac{251}{77}a^{4}-\frac{19}{11}a^{3}-\frac{101}{77}a^{2}-\frac{122}{77}a-\frac{4}{7}$, $\frac{3}{77}a^{11}-\frac{13}{231}a^{10}-\frac{1}{3}a^{9}+\frac{32}{77}a^{8}+\frac{263}{231}a^{7}-\frac{260}{231}a^{6}-\frac{131}{77}a^{5}+\frac{95}{77}a^{4}+\frac{57}{77}a^{3}-\frac{62}{77}a^{2}+\frac{46}{77}a+\frac{6}{7}$, $\frac{10}{231}a^{11}-\frac{23}{231}a^{10}-\frac{8}{21}a^{9}+\frac{235}{231}a^{8}+\frac{40}{33}a^{7}-\frac{922}{231}a^{6}-\frac{159}{77}a^{5}+\frac{559}{77}a^{4}+\frac{276}{77}a^{3}-\frac{471}{77}a^{2}-\frac{307}{77}a+\frac{3}{7}$
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| Regulator: | \( 951.410419675 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 951.410419675 \cdot 2}{2\cdot\sqrt{6499837226778624}}\cr\approx \mathstrut & 0.726099057666 \end{aligned}\]
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
| \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-2}, \sqrt{3})\), 6.0.10077696.3 |
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: | 18.6.663221759162200073699328.1, 18.0.1768591357765866863198208.2 |
| 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 | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.4.24b1.9 | $x^{12} + 4 x^{10} + 8 x^{9} + 6 x^{8} + 24 x^{7} + 24 x^{6} + 24 x^{5} + 41 x^{4} + 32 x^{3} + 20 x^{2} + 24 x + 17$ | $4$ | $3$ | $24$ | $C_6\times C_2$ | $$[2, 3]^{3}$$ |
|
\(3\)
| 3.1.6.9a1.12 | $x^{6} + 3 x^{5} + 3 x^{4} + 24$ | $6$ | $1$ | $9$ | $S_3\times C_3$ | $$[\frac{3}{2}, 2]_{2}$$ |
| 3.1.6.9a1.12 | $x^{6} + 3 x^{5} + 3 x^{4} + 24$ | $6$ | $1$ | $9$ | $S_3\times C_3$ | $$[\frac{3}{2}, 2]_{2}$$ |
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 | 1.24.2t1.b.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{-6}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *36 | 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.72.6t1.a.a | $1$ | $ 2^{3} \cdot 3^{2}$ | 6.0.3359232.1 | $C_6$ (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) | $0$ | $1$ | |
| 1.72.6t1.c.a | $1$ | $ 2^{3} \cdot 3^{2}$ | 6.0.10077696.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.72.6t1.a.b | $1$ | $ 2^{3} \cdot 3^{2}$ | 6.0.3359232.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.72.6t1.c.b | $1$ | $ 2^{3} \cdot 3^{2}$ | 6.0.10077696.1 | $C_6$ (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) | $0$ | $1$ | |
| 2.216.3t2.b.a | $2$ | $ 2^{3} \cdot 3^{3}$ | 3.1.216.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.864.6t3.d.a | $2$ | $ 2^{5} \cdot 3^{3}$ | 6.2.2239488.3 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| *36 | 2.648.6t5.b.a | $2$ | $ 2^{3} \cdot 3^{4}$ | 6.0.10077696.3 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| *36 | 2.648.6t5.b.b | $2$ | $ 2^{3} \cdot 3^{4}$ | 6.0.10077696.3 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| *36 | 2.2592.12t18.b.a | $2$ | $ 2^{5} \cdot 3^{4}$ | 12.0.6499837226778624.37 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| *36 | 2.2592.12t18.b.b | $2$ | $ 2^{5} \cdot 3^{4}$ | 12.0.6499837226778624.37 | $C_6\times S_3$ (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 *.