Normalized defining polynomial
\( x^{12} - 18x^{10} + 105x^{8} - 228x^{6} + 234x^{4} - 108x^{2} + 18 \)
Invariants
| Degree: | $12$ |
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| Signature: | $[8, 2]$ |
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| Discriminant: |
\(164341563462254592\)
\(\medspace = 2^{35}\cdot 3^{14}\)
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| Root discriminant: | \(27.20\) |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{3}a^{8}$, $\frac{1}{3}a^{9}$, $\frac{1}{12}a^{10}-\frac{1}{6}a^{9}-\frac{1}{12}a^{8}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{12}a^{11}-\frac{1}{12}a^{9}-\frac{1}{6}a^{8}-\frac{1}{2}a^{3}-\frac{1}{2}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
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| Narrow class group: | $C_{2}$, which has order $2$ |
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Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{1}{6}a^{10}-\frac{17}{6}a^{8}+\frac{44}{3}a^{6}-23a^{4}+13a^{2}-2$, $\frac{31}{12}a^{11}+a^{10}-\frac{181}{4}a^{9}-\frac{35}{2}a^{8}+\frac{748}{3}a^{7}+\frac{289}{3}a^{6}-468a^{5}-181a^{4}+\frac{753}{2}a^{3}+148a^{2}-\frac{193}{2}a-38$, $a^{11}-\frac{1}{12}a^{10}-\frac{35}{2}a^{9}+\frac{17}{12}a^{8}+\frac{289}{3}a^{7}-\frac{22}{3}a^{6}-181a^{5}+12a^{4}+148a^{3}-\frac{21}{2}a^{2}-38a+\frac{5}{2}$, $\frac{1}{4}a^{11}-\frac{7}{3}a^{10}-\frac{17}{4}a^{9}+\frac{245}{6}a^{8}+22a^{7}-\frac{674}{3}a^{6}-35a^{5}+421a^{4}+\frac{47}{2}a^{3}-342a^{2}-\frac{7}{2}a+90$, $\frac{31}{12}a^{11}-a^{10}-\frac{181}{4}a^{9}+\frac{35}{2}a^{8}+\frac{748}{3}a^{7}-\frac{289}{3}a^{6}-468a^{5}+181a^{4}+\frac{753}{2}a^{3}-148a^{2}-\frac{191}{2}a+38$, $\frac{31}{12}a^{11}+a^{10}-\frac{181}{4}a^{9}-\frac{35}{2}a^{8}+\frac{748}{3}a^{7}+\frac{289}{3}a^{6}-468a^{5}-181a^{4}+\frac{753}{2}a^{3}+148a^{2}-\frac{191}{2}a-38$, $\frac{43}{12}a^{11}+\frac{43}{12}a^{10}-\frac{251}{4}a^{9}-\frac{251}{4}a^{8}+\frac{1037}{3}a^{7}+\frac{1037}{3}a^{6}-649a^{5}-649a^{4}+\frac{1049}{2}a^{3}+\frac{1049}{2}a^{2}-\frac{269}{2}a-\frac{269}{2}$, $\frac{3}{4}a^{11}+\frac{3}{4}a^{10}-\frac{53}{4}a^{9}-\frac{53}{4}a^{8}+\frac{223}{3}a^{7}+\frac{223}{3}a^{6}-146a^{5}-146a^{4}+\frac{249}{2}a^{3}+\frac{249}{2}a^{2}-\frac{67}{2}a-\frac{67}{2}$, $\frac{15}{2}a^{11}-\frac{19}{4}a^{10}-\frac{395}{3}a^{9}+\frac{1001}{12}a^{8}+729a^{7}-462a^{6}-1386a^{5}+879a^{4}+1138a^{3}-\frac{1449}{2}a^{2}-303a+\frac{389}{2}$
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| Regulator: | \( 43838.1438943 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{2}\cdot 43838.1438943 \cdot 1}{2\cdot\sqrt{164341563462254592}}\cr\approx \mathstrut & 0.546447139708 \end{aligned}\]
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: | This field is its own minimal sibling |
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 | ${\href{/padicField/5.12.0.1}{12} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.6a1.5 | $x^{4} + 2 x^{3} + 7 x^{2} + 6 x + 7$ | $2$ | $2$ | $6$ | $C_2^2$ | $$[3]^{2}$$ |
| 2.1.8.29a1.20 | $x^{8} + 20 x^{6} + 16 x^{5} + 8 x^{4} + 2$ | $8$ | $1$ | $29$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]$$ | |
|
\(3\)
| 3.2.6.14a1.2 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2630 x^{6} + 3303 x^{5} + 3240 x^{4} + 2462 x^{3} + 1410 x^{2} + 567 x + 124$ | $6$ | $2$ | $14$ | $(C_3\times C_3):C_4$ | $$[\frac{3}{2}, \frac{3}{2}]_{2}^{2}$$ |