Normalized defining polynomial
\( x^{12} - 12x^{10} + 108x^{8} - 428x^{6} + 1272x^{4} - 72x^{2} + 4 \)
Invariants
| Degree: | $12$ |
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| Signature: | $[0, 6]$ |
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| Discriminant: |
\(1015599566684160000\)
\(\medspace = 2^{22}\cdot 3^{18}\cdot 5^{4}\)
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| Root discriminant: | \(31.66\) |
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| Galois root discriminant: | $2^{11/6}3^{3/2}5^{1/2}\approx 41.40523078218084$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
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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 a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \), 6.0.7873200.1$^{3}$, 6.0.335923200.1$^{3}$, 12.0.1015599566684160000.1$^{12}$, 12.0.25389989167104000000.1$^{6}$, deg 12$^{6}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{12}a^{6}+\frac{1}{6}$, $\frac{1}{12}a^{7}+\frac{1}{6}a$, $\frac{1}{12}a^{8}+\frac{1}{6}a^{2}$, $\frac{1}{12}a^{9}+\frac{1}{6}a^{3}$, $\frac{1}{5688}a^{10}+\frac{35}{2844}a^{8}+\frac{20}{711}a^{6}+\frac{421}{2844}a^{4}+\frac{515}{1422}a^{2}-\frac{224}{711}$, $\frac{1}{5688}a^{11}+\frac{35}{2844}a^{9}+\frac{20}{711}a^{7}+\frac{421}{2844}a^{5}+\frac{515}{1422}a^{3}-\frac{224}{711}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{3}$, which has order $3$ |
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| Narrow class group: | $C_{3}$, which has order $3$ |
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| Relative class number: | $3$ |
Unit group
| Rank: | $5$ |
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| Torsion generator: |
\( -\frac{9}{632} a^{10} + \frac{161}{948} a^{8} - \frac{483}{316} a^{6} + \frac{1899}{316} a^{4} - \frac{8533}{474} a^{2} + \frac{161}{158} \)
(order $6$)
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| Fundamental units: |
$\frac{27}{316}a^{11}-\frac{9}{316}a^{10}-\frac{161}{158}a^{9}+\frac{161}{474}a^{8}+\frac{1449}{158}a^{7}-\frac{483}{158}a^{6}-\frac{5697}{158}a^{5}+\frac{1899}{158}a^{4}+\frac{8454}{79}a^{3}-\frac{8533}{237}a^{2}-\frac{9}{79}a+\frac{82}{79}$, $\frac{63}{632}a^{11}+\frac{17}{2844}a^{10}-\frac{1127}{948}a^{9}-\frac{58}{711}a^{8}+\frac{3381}{316}a^{7}+\frac{2009}{2844}a^{6}-\frac{13293}{316}a^{5}-\frac{4219}{1422}a^{4}+\frac{59257}{474}a^{3}+\frac{5200}{711}a^{2}-\frac{21}{158}a-\frac{301}{1422}$, $\frac{63}{632}a^{11}-\frac{17}{2844}a^{10}-\frac{1127}{948}a^{9}+\frac{58}{711}a^{8}+\frac{3381}{316}a^{7}-\frac{2009}{2844}a^{6}-\frac{13293}{316}a^{5}+\frac{4219}{1422}a^{4}+\frac{59257}{474}a^{3}-\frac{5200}{711}a^{2}-\frac{21}{158}a+\frac{301}{1422}$, $\frac{903}{632}a^{11}+\frac{1145}{2844}a^{10}-\frac{4045}{237}a^{9}-\frac{6851}{1422}a^{8}+\frac{48461}{316}a^{7}+\frac{123239}{2844}a^{6}-\frac{190533}{316}a^{5}-\frac{243175}{1422}a^{4}+\frac{422608}{237}a^{3}+\frac{360733}{711}a^{2}-\frac{301}{158}a-\frac{20803}{1422}$, $\frac{53}{948}a^{11}+\frac{59}{5688}a^{10}-\frac{635}{948}a^{9}-\frac{305}{2844}a^{8}+\frac{1905}{316}a^{7}+\frac{2587}{2844}a^{6}-\frac{11341}{474}a^{5}-\frac{9289}{2844}a^{4}+\frac{33655}{474}a^{3}+\frac{13795}{1422}a^{2}-\frac{635}{158}a-\frac{1547}{1422}$
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| Regulator: | \( 19168.46386576396 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 19168.46386576396 \cdot 3}{6\cdot\sqrt{1015599566684160000}}\cr\approx \mathstrut & 0.585160862736377 \end{aligned}\]
Galois group
$C_2\times D_6$ (as 12T10):
| A solvable group of order 24 |
| The 12 conjugacy class representatives for $S_3 \times C_2^2$ |
| Character table for $S_3 \times C_2^2$ |
Intermediate fields
| \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \), 3.3.1620.1, \(\Q(\sqrt{-2}, \sqrt{-3})\), 6.0.7873200.1, 6.6.1007769600.2, 6.0.335923200.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 24 |
| Degree 12 siblings: | deg 12, 12.0.25389989167104000000.1, deg 12 |
| 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 | R | ${\href{/padicField/7.2.0.1}{2} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.6.22a1.1 | $x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 126 x^{7} + 141 x^{6} + 126 x^{5} + 90 x^{4} + 50 x^{3} + 21 x^{2} + 6 x + 3$ | $6$ | $2$ | $22$ | $D_6$ | $$[3]_{3}^{2}$$ |
|
\(3\)
| 3.1.6.9a2.1 | $x^{6} + 3 x^{4} + 3$ | $6$ | $1$ | $9$ | $D_{6}$ | $$[2]_{2}^{2}$$ |
| 3.1.6.9a2.1 | $x^{6} + 3 x^{4} + 3$ | $6$ | $1$ | $9$ | $D_{6}$ | $$[2]_{2}^{2}$$ | |
|
\(5\)
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |