Normalized defining polynomial
\( x^{12} + 6x^{10} + 51x^{8} + 132x^{6} + 441x^{4} - 54x^{2} - 441 \)
Invariants
| Degree: | $12$ |
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| Signature: | $[2, 5]$ |
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| Discriminant: |
\(-1314732507698036736\)
\(\medspace = -\,2^{38}\cdot 3^{14}\)
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| Root discriminant: | \(32.35\) |
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| Galois root discriminant: | $2^{137/32}3^{25/18}\approx 89.42379121066101$ | ||
| Ramified primes: |
\(2\), \(3\)
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| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{6}a^{8}-\frac{1}{2}$, $\frac{1}{6}a^{9}-\frac{1}{2}a$, $\frac{1}{62472}a^{10}+\frac{279}{20824}a^{8}+\frac{1401}{10412}a^{6}-\frac{1891}{10412}a^{4}+\frac{91}{1096}a^{2}-\frac{75}{20824}$, $\frac{1}{874608}a^{11}-\frac{1}{124944}a^{10}+\frac{10691}{291536}a^{9}+\frac{9575}{124944}a^{8}-\frac{9011}{145768}a^{7}+\frac{6209}{62472}a^{6}-\frac{43539}{145768}a^{5}+\frac{1891}{20824}a^{4}+\frac{13}{2192}a^{3}-\frac{91}{2192}a^{2}+\frac{51985}{291536}a-\frac{10337}{41648}$
| 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: | $C_{2}$, which has order $2$ |
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Unit group
| Rank: | $6$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{1}{456}a^{10}+\frac{1}{456}a^{8}+\frac{23}{228}a^{6}+\frac{9}{76}a^{4}+\frac{11}{8}a^{2}+\frac{153}{152}$, $\frac{7}{15618}a^{10}+\frac{653}{15618}a^{8}+\frac{788}{7809}a^{6}+\frac{2381}{2603}a^{4}+\frac{89}{274}a^{2}-\frac{5731}{5206}$, $\frac{8473}{874608}a^{11}-\frac{7}{6576}a^{10}+\frac{42977}{874608}a^{9}+\frac{169}{6576}a^{8}+\frac{242455}{437304}a^{7}+\frac{719}{3288}a^{6}+\frac{178629}{145768}a^{5}+\frac{1181}{1096}a^{4}+\frac{13701}{2192}a^{3}+\frac{9817}{2192}a^{2}-\frac{41991}{291536}a+\frac{3265}{2192}$, $\frac{147}{20824}a^{10}+\frac{4697}{62472}a^{8}+\frac{3533}{10412}a^{6}+\frac{9441}{10412}a^{4}-\frac{421}{1096}a^{2}-\frac{22663}{20824}$, $\frac{1047}{145768}a^{11}-\frac{47}{7809}a^{10}+\frac{15605}{437304}a^{9}-\frac{98}{2603}a^{8}+\frac{72091}{218652}a^{7}-\frac{2029}{7809}a^{6}+\frac{47269}{72884}a^{5}-\frac{1853}{2603}a^{4}+\frac{2473}{1096}a^{3}-\frac{304}{137}a^{2}-\frac{412579}{145768}a+\frac{6128}{2603}$, $\frac{1955}{437304}a^{11}-\frac{22}{7809}a^{10}+\frac{22475}{437304}a^{9}-\frac{193}{7809}a^{8}+\frac{21423}{72884}a^{7}-\frac{119}{7809}a^{6}+\frac{82651}{72884}a^{5}-\frac{92}{2603}a^{4}+\frac{2399}{1096}a^{3}+\frac{327}{137}a^{2}+\frac{176147}{145768}a+\frac{6856}{2603}$
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| Regulator: | \( 89363.76853 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 89363.76853 \cdot 1}{2\cdot\sqrt{1314732507698036736}}\cr\approx \mathstrut & 1.526413196 \end{aligned}\]
Galois group
$(C_2^2\times A_4^2):C_4$ (as 12T241):
| A solvable group of order 2304 |
| The 40 conjugacy class representatives for $(C_2^2\times A_4^2):C_4$ |
| Character table for $(C_2^2\times A_4^2):C_4$ |
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 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.2.328683126924509184.1 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.4.8b1.6 | $x^{4} + 4 x^{3} + 2 x^{2} + 4 x + 14$ | $4$ | $1$ | $8$ | $C_2^2$ | $$[2, 3]$$ |
| 2.1.8.30a1.70 | $x^{8} + 8 x^{7} + 16 x^{6} + 4 x^{4} + 18$ | $8$ | $1$ | $30$ | $((C_8 : C_2):C_2):C_2$ | $$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]$$ | |
|
\(3\)
| 3.2.6.14a1.2 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2630 x^{6} + 3303 x^{5} + 3240 x^{4} + 2462 x^{3} + 1410 x^{2} + 567 x + 124$ | $6$ | $2$ | $14$ | $(C_3\times C_3):C_4$ | $$[\frac{3}{2}, \frac{3}{2}]_{2}^{2}$$ |