Normalized defining polynomial
\( x^{12} + 6x^{10} - 21x^{8} - 276x^{6} - 846x^{4} - 324x^{2} + 18 \)
Invariants
| Degree: | $12$ |
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| Signature: | $[4, 4]$ |
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| Discriminant: |
\(2629465015396073472\)
\(\medspace = 2^{39}\cdot 3^{14}\)
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| Root discriminant: | \(34.28\) |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{3}a^{8}$, $\frac{1}{3}a^{9}$, $\frac{1}{1139883}a^{10}+\frac{21708}{379961}a^{8}-\frac{110}{1139883}a^{6}-\frac{107986}{379961}a^{4}+\frac{105597}{379961}a^{2}+\frac{111121}{379961}$, $\frac{1}{1139883}a^{11}+\frac{21708}{379961}a^{9}-\frac{110}{1139883}a^{7}-\frac{107986}{379961}a^{5}+\frac{105597}{379961}a^{3}+\frac{111121}{379961}a$
| 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: | $C_{2}\times C_{4}$, which has order $8$ |
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Unit group
| Rank: | $7$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{7}{929}a^{10}+\frac{116}{2787}a^{8}-\frac{452}{2787}a^{6}-\frac{1875}{929}a^{4}-\frac{5560}{929}a^{2}-\frac{107}{929}$, $\frac{26158}{1139883}a^{10}+\frac{148429}{1139883}a^{8}-\frac{597614}{1139883}a^{6}-\frac{2347480}{379961}a^{4}-\frac{6569481}{379961}a^{2}-\frac{378493}{379961}$, $\frac{4802}{1139883}a^{10}+\frac{17545}{1139883}a^{8}-\frac{148259}{1139883}a^{6}-\frac{281968}{379961}a^{4}-\frac{551102}{379961}a^{2}-\frac{242163}{379961}$, $\frac{30155}{1139883}a^{11}-\frac{107}{379961}a^{10}+\frac{175772}{1139883}a^{9}-\frac{6949}{1139883}a^{8}-\frac{657323}{1139883}a^{7}+\frac{11770}{379961}a^{6}-\frac{2711787}{379961}a^{5}+\frac{87055}{379961}a^{4}-\frac{8154787}{379961}a^{3}-\frac{80108}{379961}a^{2}-\frac{2682031}{379961}a-\frac{713429}{379961}$, $\frac{3962}{1139883}a^{11}+\frac{8624}{1139883}a^{10}+\frac{27769}{1139883}a^{9}+\frac{47018}{1139883}a^{8}-\frac{55859}{1139883}a^{7}-\frac{62906}{379961}a^{6}-\frac{384407}{379961}a^{5}-\frac{746775}{379961}a^{4}-\frac{1481591}{379961}a^{3}-\frac{1997794}{379961}a^{2}-\frac{1633241}{379961}a-\frac{334099}{379961}$, $\frac{8210}{1139883}a^{10}+\frac{20971}{379961}a^{8}-\frac{47726}{379961}a^{6}-\frac{875969}{379961}a^{4}-\frac{3159320}{379961}a^{2}-\frac{1502795}{379961}$, $\frac{118940}{1139883}a^{11}-\frac{8414}{379961}a^{10}+\frac{723536}{1139883}a^{9}-\frac{49574}{379961}a^{8}-\frac{2444492}{1139883}a^{7}+\frac{165618}{379961}a^{6}-\frac{11052026}{379961}a^{5}+\frac{2222164}{379961}a^{4}-\frac{34480126}{379961}a^{3}+\frac{7166200}{379961}a^{2}-\frac{15030124}{379961}a+\frac{3375469}{379961}$
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| Regulator: | \( 67646.0496888 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 67646.0496888 \cdot 2}{2\cdot\sqrt{2629465015396073472}}\cr\approx \mathstrut & 1.04027508690 \end{aligned}\]
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 | ${\href{/padicField/5.12.0.1}{12} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.4.10a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 2$ | $4$ | $1$ | $10$ | $D_{4}$ | $$[2, 3, \frac{7}{2}]$$ |
| 2.1.8.29a1.26 | $x^{8} + 8 x^{7} + 20 x^{6} + 8 x^{4} + 2$ | $8$ | $1$ | $29$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]$$ | |
|
\(3\)
| 3.2.6.14a1.2 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2630 x^{6} + 3303 x^{5} + 3240 x^{4} + 2462 x^{3} + 1410 x^{2} + 567 x + 124$ | $6$ | $2$ | $14$ | $(C_3\times C_3):C_4$ | $$[\frac{3}{2}, \frac{3}{2}]_{2}^{2}$$ |