Normalized defining polynomial
\( x^{12} - 6 x^{11} + 15 x^{10} - 14 x^{9} - 3 x^{8} + 12 x^{7} + 8 x^{6} - 12 x^{5} - 3 x^{4} + 14 x^{3} + \cdots + 1 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[0, 6]$ |
| |
| Discriminant: |
\(320979616137216\)
\(\medspace = 2^{26}\cdot 3^{14}\)
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| |
| Root discriminant: | \(16.18\) |
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| Galois root discriminant: | $2^{11/4}3^{7/6}\approx 24.23672593327708$ | ||
| Ramified primes: |
\(2\), \(3\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2^2$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-1}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{58}a^{11}+\frac{3}{29}a^{10}+\frac{15}{58}a^{8}-\frac{13}{29}a^{7}-\frac{5}{29}a^{6}+\frac{2}{29}a^{5}-\frac{11}{29}a^{4}+\frac{23}{58}a^{3}-\frac{7}{29}a-\frac{17}{58}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
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| Narrow class group: | $C_{2}$, which has order $2$ |
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Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( \frac{165}{58} a^{11} - \frac{1011}{58} a^{10} + \frac{89}{2} a^{9} - \frac{1242}{29} a^{8} - \frac{318}{29} a^{7} + \frac{1350}{29} a^{6} + \frac{330}{29} a^{5} - \frac{1119}{29} a^{4} - \frac{207}{58} a^{3} + \frac{93}{2} a^{2} + \frac{1953}{58} a + \frac{236}{29} \)
(order $4$)
|
| |
| Fundamental units: |
$\frac{90}{29}a^{11}-\frac{591}{29}a^{10}+58a^{9}-\frac{2188}{29}a^{8}+\frac{879}{29}a^{7}+\frac{753}{29}a^{6}+\frac{128}{29}a^{5}-\frac{1081}{29}a^{4}+\frac{330}{29}a^{3}+38a^{2}+\frac{712}{29}a+\frac{123}{29}$, $a$, $\frac{225}{58}a^{11}-\frac{775}{29}a^{10}+81a^{9}-\frac{6949}{58}a^{8}+\frac{2179}{29}a^{7}+\frac{267}{29}a^{6}+\frac{102}{29}a^{5}-\frac{1460}{29}a^{4}+\frac{1811}{58}a^{3}+39a^{2}+\frac{455}{29}a+\frac{3}{58}$, $\frac{357}{58}a^{11}-\frac{1191}{29}a^{10}+119a^{9}-\frac{9261}{58}a^{8}+\frac{2000}{29}a^{7}+\frac{1666}{29}a^{6}-\frac{243}{29}a^{5}-\frac{2100}{29}a^{4}+\frac{2005}{58}a^{3}+72a^{2}+\frac{1010}{29}a+\frac{311}{58}$, $\frac{18}{29}a^{11}-\frac{161}{58}a^{10}+\frac{5}{2}a^{9}+\frac{743}{58}a^{8}-\frac{1019}{29}a^{7}+\frac{806}{29}a^{6}+\frac{420}{29}a^{5}-\frac{570}{29}a^{4}-\frac{282}{29}a^{3}+\frac{41}{2}a^{2}+\frac{975}{58}a+\frac{287}{58}$
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| Regulator: | \( 402.569505236 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 402.569505236 \cdot 2}{4\cdot\sqrt{320979616137216}}\cr\approx \mathstrut & 0.691275813037 \end{aligned}\]
Galois group
$C_2^2\times S_4$ (as 12T48):
| A solvable group of order 96 |
| The 20 conjugacy class representatives for $C_2^2\times S_4$ |
| Character table for $C_2^2\times S_4$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 3.1.216.1, 6.2.2239488.2, 6.0.8957952.1, 6.0.746496.3 |
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 sibling: | data not computed |
| Minimal sibling: | 12.0.5015306502144.2 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{8}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.4a1.1 | $x^{4} + 2 x^{3} + 5 x^{2} + 4 x + 5$ | $2$ | $2$ | $4$ | $C_2^2$ | $$[2]^{2}$$ |
| 2.1.8.22d1.15 | $x^{8} + 4 x^{7} + 4 x^{6} + 10 x^{4} + 4 x^{2} + 8 x + 10$ | $8$ | $1$ | $22$ | $D_4$ | $$[2, 3, \frac{7}{2}]$$ | |
|
\(3\)
| 3.2.6.14a2.1 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2624 x^{6} + 3264 x^{5} + 3126 x^{4} + 2264 x^{3} + 1200 x^{2} + 432 x + 91$ | $6$ | $2$ | $14$ | $D_6$ | $$[\frac{3}{2}]_{2}^{2}$$ |