Normalized defining polynomial
\( x^{12} - 3x^{11} - 9x^{10} + 28x^{9} + 24x^{8} - 87x^{7} - 19x^{6} + 107x^{5} + 2x^{4} - 52x^{3} + 8x + 1 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[12, 0]$ |
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| Discriminant: |
\(2811266206890625\)
\(\medspace = 5^{6}\cdot 11^{2}\cdot 38561^{2}\)
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| Root discriminant: | \(19.38\) |
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| Galois root discriminant: | $5^{1/2}11^{1/2}38561^{1/2}\approx 1456.3155564643262$ | ||
| Ramified primes: |
\(5\), \(11\), \(38561\)
|
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}-\frac{1}{2}$
| 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: | $11$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$3a^{11}-\frac{13}{2}a^{10}-\frac{65}{2}a^{9}+57a^{8}+121a^{7}-\frac{323}{2}a^{6}-\frac{401}{2}a^{5}+161a^{4}+\frac{321}{2}a^{3}-\frac{73}{2}a^{2}-\frac{91}{2}a-6$, $a^{10}-\frac{1}{2}a^{9}-\frac{25}{2}a^{8}+48a^{6}+\frac{25}{2}a^{5}-\frac{133}{2}a^{4}-28a^{3}+\frac{47}{2}a^{2}+\frac{23}{2}a+\frac{3}{2}$, $a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-6a^{9}+2a^{8}+\frac{49}{2}a^{7}+\frac{9}{2}a^{6}-45a^{5}-\frac{51}{2}a^{4}+\frac{79}{2}a^{3}+\frac{51}{2}a^{2}-12a-6$, $\frac{3}{2}a^{11}-\frac{7}{2}a^{10}-15a^{9}+30a^{8}+\frac{97}{2}a^{7}-\frac{165}{2}a^{6}-64a^{5}+\frac{163}{2}a^{4}+\frac{83}{2}a^{3}-\frac{53}{2}a^{2}-11a+1$, $a^{11}-\frac{3}{2}a^{10}-12a^{9}+\frac{25}{2}a^{8}+48a^{7}-\frac{71}{2}a^{6}-79a^{5}+\frac{77}{2}a^{4}+\frac{103}{2}a^{3}-12a^{2}-9a-\frac{3}{2}$, $2a^{11}-\frac{5}{2}a^{10}-\frac{49}{2}a^{9}+18a^{8}+101a^{7}-\frac{79}{2}a^{6}-\frac{353}{2}a^{5}+23a^{4}+\frac{261}{2}a^{3}+\frac{15}{2}a^{2}-\frac{65}{2}a-7$, $\frac{5}{2}a^{10}-\frac{5}{2}a^{9}-29a^{8}+13a^{7}+\frac{207}{2}a^{6}-\frac{23}{2}a^{5}-132a^{4}-\frac{35}{2}a^{3}+\frac{91}{2}a^{2}+\frac{23}{2}a$, $2a^{11}-\frac{7}{2}a^{10}-24a^{9}+\frac{61}{2}a^{8}+101a^{7}-\frac{175}{2}a^{6}-189a^{5}+\frac{179}{2}a^{4}+\frac{317}{2}a^{3}-16a^{2}-44a-\frac{17}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-7a^{9}+4a^{8}+\frac{69}{2}a^{7}-\frac{21}{2}a^{6}-76a^{5}+\frac{13}{2}a^{4}+\frac{145}{2}a^{3}+\frac{15}{2}a^{2}-21a-4$, $6a^{11}-\frac{27}{2}a^{10}-\frac{127}{2}a^{9}+119a^{8}+227a^{7}-\frac{681}{2}a^{6}-\frac{699}{2}a^{5}+351a^{4}+\frac{503}{2}a^{3}-\frac{195}{2}a^{2}-\frac{127}{2}a-6$
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| Regulator: | \( 4466.4963181 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 4466.4963181 \cdot 1}{2\cdot\sqrt{2811266206890625}}\cr\approx \mathstrut & 0.17252258093 \end{aligned}\]
Galois group
$S_4\wr C_2$ (as 12T203):
| A solvable group of order 1152 |
| The 20 conjugacy class representatives for $S_4\wr C_2$ |
| Character table for $S_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 6.6.4820125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 sibling: | data not computed |
| Degree 12 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 8.8.2916175625.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | R | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | R | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.3.2.3a1.2 | $x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
| 5.3.2.3a1.2 | $x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
|
\(11\)
| 11.1.2.1a1.2 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 11.1.2.1a1.1 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 11.2.1.0a1.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 11.2.1.0a1.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 11.4.1.0a1.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
|
\(38561\)
| $\Q_{38561}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{38561}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{38561}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{38561}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |