Normalized defining polynomial
\( x^{12} - 3x^{11} + 9x^{10} - 2x^{9} + 3x^{8} - 3x^{7} - 6x^{6} - 3x^{5} + 3x^{4} - 2x^{3} + 9x^{2} - 3x + 1 \)
Invariants
| Degree: | $12$ |
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| Signature: | $[0, 6]$ |
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| Discriminant: |
\(144054149089536\)
\(\medspace = 2^{8}\cdot 3^{14}\cdot 7^{6}\)
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| Root discriminant: | \(15.13\) |
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| Galois root discriminant: | $2^{2/3}3^{7/6}7^{3/4}\approx 24.612269258377225$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2^2$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | 6.0.1714608.1 | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{9}a^{8}-\frac{1}{9}a^{7}+\frac{1}{9}a^{6}+\frac{2}{9}a^{5}+\frac{4}{9}a^{4}-\frac{1}{9}a^{3}-\frac{2}{9}a^{2}-\frac{1}{9}a+\frac{4}{9}$, $\frac{1}{18}a^{9}-\frac{1}{6}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{6}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{7}{18}$, $\frac{1}{18}a^{10}-\frac{1}{6}a^{7}-\frac{1}{3}a^{5}+\frac{1}{6}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{7}{18}a-\frac{1}{3}$, $\frac{1}{36}a^{11}-\frac{1}{36}a^{9}-\frac{1}{36}a^{8}+\frac{1}{9}a^{7}-\frac{1}{36}a^{6}-\frac{17}{36}a^{5}+\frac{1}{18}a^{4}+\frac{7}{36}a^{3}+\frac{5}{12}a^{2}-\frac{7}{18}a+\frac{1}{36}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
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| Narrow class group: | Trivial group, which has order $1$ |
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Unit group
| Rank: | $5$ |
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| Torsion generator: |
\( \frac{13}{36} a^{11} - \frac{17}{18} a^{10} + \frac{103}{36} a^{9} + \frac{5}{12} a^{8} + \frac{7}{6} a^{7} - \frac{11}{12} a^{6} - \frac{31}{12} a^{5} - a^{4} + \frac{3}{4} a^{3} - \frac{5}{36} a^{2} + \frac{26}{9} a + \frac{1}{36} \)
(order $6$)
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| Fundamental units: |
$\frac{1}{36}a^{11}-\frac{4}{9}a^{10}+\frac{43}{36}a^{9}-\frac{35}{12}a^{8}-\frac{1}{3}a^{7}-\frac{5}{4}a^{6}+\frac{3}{4}a^{5}+\frac{5}{2}a^{4}+\frac{13}{12}a^{3}-\frac{29}{36}a^{2}+\frac{7}{18}a-\frac{107}{36}$, $\frac{7}{36}a^{11}-\frac{5}{9}a^{10}+\frac{53}{36}a^{9}+\frac{5}{36}a^{8}-\frac{5}{9}a^{7}-\frac{79}{36}a^{6}-\frac{119}{36}a^{5}-\frac{23}{18}a^{4}+\frac{61}{36}a^{3}+\frac{19}{12}a^{2}+\frac{43}{18}a+\frac{7}{36}$, $\frac{1}{36}a^{11}+\frac{1}{18}a^{10}-\frac{1}{36}a^{9}+\frac{23}{36}a^{8}+\frac{23}{18}a^{7}-\frac{25}{36}a^{6}-\frac{53}{36}a^{5}-\frac{19}{9}a^{4}-\frac{53}{36}a^{3}-\frac{1}{4}a^{2}+\frac{1}{3}a+\frac{1}{36}$, $\frac{35}{36}a^{11}-\frac{17}{6}a^{10}+\frac{307}{36}a^{9}-\frac{43}{36}a^{8}+\frac{47}{18}a^{7}-\frac{61}{36}a^{6}-\frac{251}{36}a^{5}-\frac{40}{9}a^{4}+\frac{91}{36}a^{3}-\frac{107}{36}a^{2}+\frac{70}{9}a-\frac{17}{12}$, $\frac{11}{36}a^{11}-\frac{7}{9}a^{10}+\frac{9}{4}a^{9}+\frac{11}{12}a^{8}-\frac{1}{3}a^{7}+\frac{1}{4}a^{6}-\frac{15}{4}a^{5}-\frac{3}{2}a^{4}-\frac{3}{4}a^{3}-\frac{31}{36}a^{2}+\frac{37}{18}a-\frac{1}{4}$
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| Regulator: | \( 594.096723246 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 594.096723246 \cdot 1}{6\cdot\sqrt{144054149089536}}\cr\approx \mathstrut & 0.507600721880 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 12T24):
| A solvable group of order 48 |
| The 10 conjugacy class representatives for $C_2 \times S_4$ |
| Character table for $C_2 \times S_4$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.3.756.1, 6.4.4000752.1, 6.0.1714608.1, 6.2.12002256.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 6 siblings: | 6.4.4000752.1, 6.4.84015792.1 |
| Degree 8 siblings: | 8.0.12350321424.2, 8.0.1372257936.2 |
| Degree 12 siblings: | deg 12, deg 12, deg 12, deg 12, deg 12 |
| Degree 16 sibling: | deg 16 |
| Degree 24 siblings: | data not computed |
| Minimal sibling: | 6.4.4000752.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.3.4a1.2 | $x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ |
| 2.2.3.4a1.2 | $x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
|
\(3\)
| 3.2.6.14a2.1 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2624 x^{6} + 3264 x^{5} + 3126 x^{4} + 2264 x^{3} + 1200 x^{2} + 432 x + 91$ | $6$ | $2$ | $14$ | $D_6$ | $$[\frac{3}{2}]_{2}^{2}$$ |
|
\(7\)
| 7.2.1.0a1.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 7.2.1.0a1.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 7.1.4.3a1.1 | $x^{4} + 7$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ | |
| 7.1.4.3a1.1 | $x^{4} + 7$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |