Normalized defining polynomial
\( x^{12} - 6x^{10} - 15x^{8} + 96x^{6} + 144x^{4} - 576x^{2} + 288 \)
Invariants
| Degree: | $12$ |
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| Signature: | $[4, 4]$ |
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| Discriminant: |
\(657366253849018368\)
\(\medspace = 2^{37}\cdot 3^{14}\)
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| Root discriminant: | \(30.54\) |
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| Galois root discriminant: | $2^{67/16}3^{25/18}\approx 83.79758760322899$ | ||
| Ramified primes: |
\(2\), \(3\)
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| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{12}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{12}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{12}a^{8}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{24}a^{9}+\frac{1}{8}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2424}a^{10}+\frac{43}{1212}a^{8}+\frac{19}{2424}a^{6}-\frac{63}{404}a^{4}-\frac{1}{2}a^{3}+\frac{43}{202}a^{2}+\frac{35}{101}$, $\frac{1}{4848}a^{11}+\frac{43}{2424}a^{9}-\frac{61}{1616}a^{7}+\frac{19}{404}a^{5}-\frac{29}{202}a^{3}-\frac{33}{101}a$
| 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: | $C_{2}$, which has order $2$ |
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Unit group
| Rank: | $7$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{5}{606}a^{10}-\frac{49}{1212}a^{8}-\frac{107}{606}a^{6}+\frac{255}{404}a^{4}+\frac{228}{101}a^{2}-\frac{209}{101}$, $\frac{109}{4848}a^{11}+\frac{1}{101}a^{10}-\frac{131}{1212}a^{9}-\frac{79}{1212}a^{8}-\frac{2171}{4848}a^{7}-\frac{25}{404}a^{6}+\frac{1213}{808}a^{5}+\frac{153}{202}a^{4}+\frac{1081}{202}a^{3}+\frac{123}{202}a^{2}-\frac{769}{101}a-\frac{271}{101}$, $\frac{109}{4848}a^{11}-\frac{1}{101}a^{10}-\frac{131}{1212}a^{9}+\frac{79}{1212}a^{8}-\frac{2171}{4848}a^{7}+\frac{25}{404}a^{6}+\frac{1213}{808}a^{5}-\frac{153}{202}a^{4}+\frac{1081}{202}a^{3}-\frac{123}{202}a^{2}-\frac{769}{101}a+\frac{271}{101}$, $\frac{23}{4848}a^{11}-\frac{31}{2424}a^{10}-\frac{7}{808}a^{9}+\frac{27}{404}a^{8}-\frac{977}{4848}a^{7}+\frac{421}{2424}a^{6}+\frac{67}{202}a^{5}-\frac{42}{101}a^{4}+\frac{222}{101}a^{3}-\frac{727}{202}a^{2}-\frac{254}{101}a+\frac{329}{101}$, $\frac{29}{1212}a^{11}-\frac{14}{303}a^{10}-\frac{11}{101}a^{9}+\frac{39}{202}a^{8}-\frac{661}{1212}a^{7}+\frac{193}{202}a^{6}+\frac{147}{101}a^{5}-\frac{256}{101}a^{4}+\frac{641}{101}a^{3}-\frac{994}{101}a^{2}-\frac{394}{101}a+\frac{827}{101}$, $\frac{23}{4848}a^{11}+\frac{31}{2424}a^{10}-\frac{7}{808}a^{9}-\frac{27}{404}a^{8}-\frac{977}{4848}a^{7}-\frac{421}{2424}a^{6}+\frac{67}{202}a^{5}+\frac{42}{101}a^{4}+\frac{222}{101}a^{3}+\frac{727}{202}a^{2}-\frac{254}{101}a-\frac{329}{101}$, $\frac{7}{404}a^{11}-\frac{11}{808}a^{10}-\frac{113}{1212}a^{9}+\frac{8}{101}a^{8}-\frac{409}{1212}a^{7}+\frac{787}{2424}a^{6}+\frac{485}{404}a^{5}-\frac{345}{404}a^{4}+\frac{695}{202}a^{3}-\frac{356}{101}a^{2}-\frac{550}{101}a+\frac{461}{101}$
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| Regulator: | \( 163329.982884 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 163329.982884 \cdot 1}{2\cdot\sqrt{657366253849018368}}\cr\approx \mathstrut & 2.51172260493 \end{aligned}\]
Galois group
$C_2\times A_4^2:C_4$ (as 12T198):
| A solvable group of order 1152 |
| The 26 conjugacy class representatives for $C_2\times A_4^2:C_4$ |
| Character table for $C_2\times A_4^2:C_4$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 6.2.11943936.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 sibling: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| 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 | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.3a1.3 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ |
| 2.1.2.3a1.3 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ | |
| 2.1.8.31a1.184 | $x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 50$ | $8$ | $1$ | $31$ | $(C_8:C_2):C_2$ | $$[2, 3, \frac{7}{2}, 4, 5]$$ | |
|
\(3\)
| 3.2.6.14a1.2 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2630 x^{6} + 3303 x^{5} + 3240 x^{4} + 2462 x^{3} + 1410 x^{2} + 567 x + 124$ | $6$ | $2$ | $14$ | $(C_3\times C_3):C_4$ | $$[\frac{3}{2}, \frac{3}{2}]_{2}^{2}$$ |