Normalized defining polynomial
\( x^{12} + 2x^{8} - 4x^{6} + 9x^{4} - 20x^{2} - 4 \)
Invariants
| Degree: | $12$ |
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| Signature: | $[2, 5]$ |
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| Discriminant: |
\(-164995463643136\)
\(\medspace = -\,2^{36}\cdot 7^{4}\)
|
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| Root discriminant: | \(15.30\) |
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| Galois root discriminant: | $2^{27/8}7^{2/3}\approx 37.964259098544694$ | ||
| Ramified primes: |
\(2\), \(7\)
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{4}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{6}+\frac{1}{8}a^{5}-\frac{1}{8}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{40}a^{10}+\frac{3}{40}a^{8}-\frac{9}{40}a^{6}+\frac{9}{40}a^{4}+\frac{2}{5}a^{2}-\frac{3}{10}$, $\frac{1}{80}a^{11}-\frac{1}{80}a^{10}+\frac{3}{80}a^{9}-\frac{3}{80}a^{8}-\frac{9}{80}a^{7}-\frac{11}{80}a^{6}-\frac{11}{80}a^{5}+\frac{11}{80}a^{4}+\frac{9}{20}a^{3}+\frac{3}{10}a^{2}-\frac{3}{20}a-\frac{7}{20}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
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: | $6$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{3}{40}a^{11}-\frac{1}{40}a^{9}+\frac{3}{40}a^{7}-\frac{23}{40}a^{5}+\frac{9}{20}a^{3}-\frac{7}{5}a$, $\frac{1}{16}a^{11}-\frac{1}{80}a^{10}-\frac{1}{16}a^{9}-\frac{3}{80}a^{8}+\frac{3}{16}a^{7}-\frac{11}{80}a^{6}-\frac{7}{16}a^{5}+\frac{11}{80}a^{4}+a^{3}-\frac{7}{10}a^{2}-\frac{5}{4}a+\frac{13}{20}$, $\frac{1}{20}a^{11}-\frac{1}{40}a^{10}+\frac{1}{40}a^{9}+\frac{1}{20}a^{8}+\frac{1}{20}a^{7}-\frac{1}{40}a^{6}-\frac{7}{40}a^{5}-\frac{1}{10}a^{4}+\frac{11}{20}a^{3}-\frac{2}{5}a^{2}-\frac{3}{5}a-\frac{1}{5}$, $\frac{1}{20}a^{11}+\frac{1}{40}a^{10}+\frac{1}{40}a^{9}-\frac{1}{20}a^{8}+\frac{1}{20}a^{7}+\frac{1}{40}a^{6}-\frac{7}{40}a^{5}+\frac{1}{10}a^{4}+\frac{11}{20}a^{3}+\frac{2}{5}a^{2}-\frac{3}{5}a+\frac{1}{5}$, $\frac{1}{20}a^{10}-\frac{1}{10}a^{8}+\frac{1}{20}a^{6}-\frac{3}{10}a^{4}+\frac{13}{10}a^{2}-\frac{8}{5}$, $\frac{3}{40}a^{11}-\frac{1}{10}a^{10}-\frac{1}{40}a^{9}-\frac{1}{20}a^{8}+\frac{13}{40}a^{7}-\frac{1}{10}a^{6}-\frac{3}{40}a^{5}+\frac{7}{20}a^{4}+\frac{6}{5}a^{3}-\frac{11}{10}a^{2}-\frac{19}{10}a+\frac{1}{5}$
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| Regulator: | \( 600.705809198 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 600.705809198 \cdot 1}{2\cdot\sqrt{164995463643136}}\cr\approx \mathstrut & 0.915914991129 \end{aligned}\]
Galois group
$C_6^2:C_4$ (as 12T82):
| A solvable group of order 144 |
| The 18 conjugacy class representatives for $C_6^2:C_4$ |
| Character table for $C_6^2:C_4$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.2.1024.1, 6.2.802816.1 |
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 24 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 | ${\href{/padicField/3.4.0.1}{4} }^{3}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{5}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.4.10a1.1 | $x^{4} + 4 x^{3} + 2$ | $4$ | $1$ | $10$ | $D_{4}$ | $$[2, 3, \frac{7}{2}]$$ |
| 2.1.8.26c1.3 | $x^{8} + 4 x^{6} + 8 x^{5} + 8 x^{3} + 2$ | $8$ | $1$ | $26$ | $C_2^2:C_4$ | $$[2, 3, \frac{7}{2}, 4]$$ | |
|
\(7\)
| 7.1.3.2a1.2 | $x^{3} + 14$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
| 7.1.3.2a1.2 | $x^{3} + 14$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 7.6.1.0a1.1 | $x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ |