Normalized defining polynomial
\( x^{12} + 30x^{8} + 12x^{6} + 153x^{4} + 36x^{2} - 36 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[2, 5]$ |
| |
| Discriminant: |
\(-328683126924509184\)
\(\medspace = -\,2^{36}\cdot 3^{14}\)
|
| |
| Root discriminant: | \(28.82\) |
| |
| Galois root discriminant: | $2^{27/8}3^{25/18}\approx 47.713870191292706$ | ||
| Ramified primes: |
\(2\), \(3\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{12}a^{7}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{24}a^{8}-\frac{1}{12}a^{6}+\frac{1}{8}a^{4}-\frac{1}{2}$, $\frac{1}{24}a^{9}+\frac{1}{8}a^{5}+\frac{1}{4}a^{3}$, $\frac{1}{1680}a^{10}+\frac{29}{1680}a^{8}-\frac{13}{560}a^{6}-\frac{1}{2}a^{5}+\frac{47}{560}a^{4}-\frac{1}{2}a^{3}-\frac{1}{10}a^{2}+\frac{17}{140}$, $\frac{1}{1680}a^{11}+\frac{29}{1680}a^{9}-\frac{13}{560}a^{7}+\frac{47}{560}a^{5}-\frac{1}{2}a^{4}-\frac{1}{10}a^{3}-\frac{1}{2}a^{2}+\frac{17}{140}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{3}$, which has order $3$ |
| |
| Narrow class group: | $C_{6}$, which has order $6$ |
|
Unit group
| Rank: | $6$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{60}a^{10}-\frac{1}{60}a^{8}+\frac{31}{60}a^{6}-\frac{3}{20}a^{4}+\frac{27}{10}a^{2}-\frac{3}{5}$, $\frac{9}{560}a^{11}+\frac{17}{1680}a^{10}+\frac{13}{1680}a^{9}+\frac{1}{560}a^{8}+\frac{767}{1680}a^{7}+\frac{199}{560}a^{6}+\frac{219}{560}a^{5}+\frac{29}{560}a^{4}+\frac{23}{10}a^{3}+\frac{23}{10}a^{2}+\frac{109}{140}a+\frac{79}{140}$, $\frac{1}{60}a^{11}+\frac{1}{210}a^{10}-\frac{1}{60}a^{9}+\frac{11}{840}a^{8}+\frac{31}{60}a^{7}+\frac{9}{140}a^{6}-\frac{3}{20}a^{5}+\frac{83}{280}a^{4}+\frac{27}{10}a^{3}-\frac{4}{5}a^{2}+\frac{7}{5}a-\frac{107}{70}$, $\frac{1}{120}a^{11}-\frac{41}{1680}a^{10}-\frac{1}{120}a^{9}+\frac{1}{1680}a^{8}+\frac{31}{120}a^{7}-\frac{1061}{1680}a^{6}-\frac{3}{40}a^{5}-\frac{177}{560}a^{4}+\frac{27}{20}a^{3}-\frac{7}{5}a^{2}+\frac{6}{5}a-\frac{67}{140}$, $\frac{1}{120}a^{11}+\frac{41}{1680}a^{10}-\frac{1}{120}a^{9}-\frac{1}{1680}a^{8}+\frac{31}{120}a^{7}+\frac{1061}{1680}a^{6}-\frac{3}{40}a^{5}+\frac{177}{560}a^{4}+\frac{27}{20}a^{3}+\frac{7}{5}a^{2}+\frac{6}{5}a+\frac{67}{140}$, $\frac{1}{280}a^{10}-\frac{3}{140}a^{8}+\frac{31}{280}a^{6}+\frac{9}{70}a^{4}+\frac{2}{5}a^{2}+\frac{8}{35}$
|
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| Regulator: | \( 14842.120621 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 14842.120621 \cdot 3}{2\cdot\sqrt{328683126924509184}}\cr\approx \mathstrut & 1.5211002718 \end{aligned}\]
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
| \(\Q(\sqrt{2}) \), 4.2.9216.1, 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 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.2.73040694872113152.1 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{6}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.4.10a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 2$ | $4$ | $1$ | $10$ | $D_{4}$ | $$[2, 3, \frac{7}{2}]$$ |
| 2.1.8.26c1.10 | $x^{8} + 4 x^{6} + 8 x^{5} + 8 x^{4} + 8 x^{3} + 18$ | $8$ | $1$ | $26$ | $C_2^2:C_4$ | $$[2, 3, \frac{7}{2}, 4]$$ | |
|
\(3\)
| 3.2.6.14a1.2 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2630 x^{6} + 3303 x^{5} + 3240 x^{4} + 2462 x^{3} + 1410 x^{2} + 567 x + 124$ | $6$ | $2$ | $14$ | $(C_3\times C_3):C_4$ | $$[\frac{3}{2}, \frac{3}{2}]_{2}^{2}$$ |