Normalized defining polynomial
\( x^{12} + 6x^{10} - 15x^{8} - 156x^{6} - 198x^{4} + 180x^{2} + 18 \)
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: | $2^{137/32}3^{25/18}\approx 89.42379121066101$ | ||
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{3}a^{8}$, $\frac{1}{3}a^{9}$, $\frac{1}{17652}a^{10}+\frac{2297}{17652}a^{8}+\frac{529}{4413}a^{6}-\frac{68}{1471}a^{4}+\frac{243}{2942}a^{2}+\frac{705}{2942}$, $\frac{1}{35304}a^{11}-\frac{1}{35304}a^{10}-\frac{3587}{35304}a^{9}+\frac{3587}{35304}a^{8}+\frac{529}{8826}a^{7}-\frac{529}{8826}a^{6}-\frac{34}{1471}a^{5}+\frac{34}{1471}a^{4}+\frac{243}{5884}a^{3}-\frac{243}{5884}a^{2}-\frac{2237}{5884}a+\frac{2237}{5884}$
| 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 $2$ |
|
Unit group
| Rank: | $7$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{77}{17652}a^{10}+\frac{349}{17652}a^{8}-\frac{455}{4413}a^{6}-\frac{823}{1471}a^{4}+\frac{1059}{2942}a^{2}+\frac{1329}{2942}$, $\frac{12}{1471}a^{10}+\frac{316}{4413}a^{8}-\frac{316}{4413}a^{6}-\frac{2437}{1471}a^{4}-\frac{4569}{1471}a^{2}-\frac{725}{1471}$, $\frac{65}{5884}a^{10}+\frac{731}{17652}a^{8}-\frac{1286}{4413}a^{6}-\frac{1492}{1471}a^{4}+\frac{313}{2942}a^{2}+\frac{2143}{2942}$, $\frac{2771}{35304}a^{11}-\frac{1225}{35304}a^{10}+\frac{16151}{35304}a^{9}-\frac{7157}{35304}a^{8}-\frac{3675}{2942}a^{7}+\frac{5099}{8826}a^{6}-\frac{17722}{1471}a^{5}+\frac{7817}{1471}a^{4}-\frac{79799}{5884}a^{3}+\frac{31829}{5884}a^{2}+\frac{97153}{5884}a-\frac{42807}{5884}$, $\frac{2771}{35304}a^{11}+\frac{1225}{35304}a^{10}+\frac{16151}{35304}a^{9}+\frac{7157}{35304}a^{8}-\frac{3675}{2942}a^{7}-\frac{5099}{8826}a^{6}-\frac{17722}{1471}a^{5}-\frac{7817}{1471}a^{4}-\frac{79799}{5884}a^{3}-\frac{31829}{5884}a^{2}+\frac{97153}{5884}a+\frac{42807}{5884}$, $\frac{4913}{35304}a^{11}-\frac{197}{35304}a^{10}+\frac{29069}{35304}a^{9}+\frac{559}{35304}a^{8}-\frac{19403}{8826}a^{7}+\frac{1699}{8826}a^{6}-\frac{31710}{1471}a^{5}-\frac{657}{1471}a^{4}-\frac{147693}{5884}a^{3}-\frac{12567}{5884}a^{2}+\frac{165683}{5884}a+\frac{11157}{5884}$, $\frac{423}{5884}a^{11}-\frac{673}{8826}a^{10}+\frac{8197}{17652}a^{9}-\frac{4273}{8826}a^{8}-\frac{1296}{1471}a^{7}+\frac{4343}{4413}a^{6}-\frac{17155}{1471}a^{5}+\frac{17978}{1471}a^{4}-\frac{56441}{2942}a^{3}+\frac{27691}{1471}a^{2}+\frac{14987}{2942}a-\frac{11100}{1471}$
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| Regulator: | \( 48271.1258774 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 48271.1258774 \cdot 1}{2\cdot\sqrt{657366253849018368}}\cr\approx \mathstrut & 0.742323460095 \end{aligned}\]
Galois group
$S_4^2:C_4$ (as 12T237):
| A solvable group of order 2304 |
| The 40 conjugacy class representatives for $S_4^2:C_4$ |
| Character table for $S_4^2:C_4$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 6.4.5971968.1 |
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 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: | 12.4.164341563462254592.5 |
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 | ${\href{/padicField/5.12.0.1}{12} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.4.8b1.1 | $x^{4} + 2 x^{2} + 4 x + 2$ | $4$ | $1$ | $8$ | $C_2^2$ | $$[2, 3]$$ |
| 2.1.8.29a1.22 | $x^{8} + 20 x^{6} + 8 x^{4} + 16 x^{3} + 2$ | $8$ | $1$ | $29$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]$$ | |
|
\(3\)
| 3.2.6.14a1.2 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2630 x^{6} + 3303 x^{5} + 3240 x^{4} + 2462 x^{3} + 1410 x^{2} + 567 x + 124$ | $6$ | $2$ | $14$ | $(C_3\times C_3):C_4$ | $$[\frac{3}{2}, \frac{3}{2}]_{2}^{2}$$ |