Normalized defining polynomial
\( x^{12} - 4x^{10} + 20x^{8} - 4x^{6} + 5x^{4} - 2 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[2, 5]$ |
| |
| Discriminant: |
\(-329990927286272\)
\(\medspace = -\,2^{37}\cdot 7^{4}\)
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| Root discriminant: | \(16.21\) |
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| Galois root discriminant: | $2^{27/8}7^{2/3}\approx 37.964259098544694$ | ||
| Ramified primes: |
\(2\), \(7\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}$, $\frac{1}{4}a^{8}+\frac{1}{4}a^{4}-\frac{1}{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{3}{8}a^{5}+\frac{1}{8}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{88}a^{10}+\frac{9}{88}a^{8}+\frac{5}{88}a^{6}-\frac{27}{88}a^{4}-\frac{19}{44}a^{2}+\frac{17}{44}$, $\frac{1}{176}a^{11}-\frac{1}{176}a^{10}+\frac{9}{176}a^{9}-\frac{9}{176}a^{8}+\frac{5}{176}a^{7}-\frac{5}{176}a^{6}-\frac{27}{176}a^{5}+\frac{27}{176}a^{4}-\frac{19}{88}a^{3}+\frac{19}{88}a^{2}+\frac{17}{88}a-\frac{17}{88}$
| 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: | Trivial group, which has order $1$ |
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Unit group
| Rank: | $6$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{19}{88}a^{10}-\frac{71}{88}a^{8}+\frac{359}{88}a^{6}+\frac{37}{88}a^{4}+\frac{35}{44}a^{2}+\frac{37}{44}$, $\frac{9}{44}a^{10}-\frac{10}{11}a^{8}+\frac{199}{44}a^{6}-\frac{61}{22}a^{4}+\frac{49}{22}a^{2}+\frac{16}{11}$, $\frac{5}{176}a^{11}-\frac{5}{176}a^{10}+\frac{1}{176}a^{9}-\frac{1}{176}a^{8}+\frac{25}{176}a^{7}-\frac{25}{176}a^{6}+\frac{349}{176}a^{5}-\frac{349}{176}a^{4}+\frac{81}{88}a^{3}-\frac{81}{88}a^{2}-\frac{47}{88}a+\frac{47}{88}$, $\frac{5}{176}a^{11}+\frac{5}{176}a^{10}+\frac{1}{176}a^{9}+\frac{1}{176}a^{8}+\frac{25}{176}a^{7}+\frac{25}{176}a^{6}+\frac{349}{176}a^{5}+\frac{349}{176}a^{4}+\frac{81}{88}a^{3}+\frac{81}{88}a^{2}-\frac{47}{88}a-\frac{47}{88}$, $\frac{7}{88}a^{11}-\frac{47}{88}a^{9}+\frac{211}{88}a^{7}-\frac{387}{88}a^{5}-\frac{1}{44}a^{3}-\frac{35}{44}a-1$, $\frac{51}{176}a^{11}+\frac{49}{176}a^{10}-\frac{179}{176}a^{9}-\frac{197}{176}a^{8}+\frac{915}{176}a^{7}+\frac{993}{176}a^{6}+\frac{317}{176}a^{5}-\frac{245}{176}a^{4}+\frac{43}{88}a^{3}+\frac{169}{88}a^{2}+\frac{97}{88}a+\frac{63}{88}$
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| Regulator: | \( 962.205672057 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 962.205672057 \cdot 1}{2\cdot\sqrt{329990927286272}}\cr\approx \mathstrut & 1.03740001591 \end{aligned}\]
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
| \(\Q(\sqrt{2}) \), 4.2.2048.1, 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: | 12.2.164995463643136.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{5}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.4.11a1.14 | $x^{4} + 4 x^{2} + 8 x + 18$ | $4$ | $1$ | $11$ | $D_{4}$ | $$[2, 3, 4]$$ |
| 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 | $x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ |