Normalized defining polynomial
\( x^{12} - 24x^{9} - 12x^{8} + 72x^{7} + 188x^{6} - 312x^{5} - 123x^{4} + 264x^{3} - 204x^{2} + 192x - 14 \)
Invariants
| Degree: | $12$ |
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| Signature: | $[4, 4]$ |
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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: | $2^{3}3^{25/18}\approx 36.79242356556043$ | ||
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{3}{8}a^{4}-\frac{1}{4}$, $\frac{1}{8}a^{9}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{3}{8}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a$, $\frac{1}{232}a^{10}-\frac{13}{232}a^{9}+\frac{7}{232}a^{8}-\frac{5}{116}a^{7}+\frac{115}{232}a^{6}+\frac{23}{232}a^{5}+\frac{109}{232}a^{4}-\frac{4}{29}a^{3}+\frac{3}{116}a^{2}+\frac{17}{116}a+\frac{3}{116}$, $\frac{1}{601922608}a^{11}-\frac{353643}{601922608}a^{10}+\frac{6166155}{601922608}a^{9}+\frac{23520383}{601922608}a^{8}-\frac{130490681}{601922608}a^{7}+\frac{44590195}{601922608}a^{6}-\frac{135023635}{601922608}a^{5}-\frac{159959103}{601922608}a^{4}+\frac{80524135}{300961304}a^{3}-\frac{14514785}{300961304}a^{2}-\frac{16759165}{300961304}a+\frac{3404329}{42994472}$
| 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: | $7$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{6198}{185321}a^{11}+\frac{9627}{370642}a^{10}+\frac{5465}{185321}a^{9}-\frac{565473}{741284}a^{8}-\frac{182181}{185321}a^{7}+\frac{265347}{185321}a^{6}+\frac{1271058}{185321}a^{5}-\frac{3655797}{741284}a^{4}-\frac{1016592}{185321}a^{3}+\frac{1030830}{185321}a^{2}-\frac{1027056}{185321}a+\frac{1045075}{370642}$, $\frac{3165697}{150480652}a^{11}+\frac{2313013}{37620163}a^{10}+\frac{428161}{5188988}a^{9}-\frac{119203593}{300961304}a^{8}-\frac{239059419}{150480652}a^{7}-\frac{154116777}{150480652}a^{6}+\frac{755263277}{150480652}a^{5}+\frac{1981658787}{300961304}a^{4}-\frac{298983891}{75240326}a^{3}-\frac{170063121}{37620163}a^{2}-\frac{196156697}{75240326}a-\frac{116812025}{21497236}$, $\frac{3865143}{150480652}a^{11}-\frac{1580213}{300961304}a^{10}-\frac{5533209}{150480652}a^{9}-\frac{102858291}{150480652}a^{8}-\frac{46402389}{150480652}a^{7}+\frac{765393751}{300961304}a^{6}+\frac{909745715}{150480652}a^{5}-\frac{648194641}{75240326}a^{4}-\frac{530774813}{75240326}a^{3}+\frac{1339361129}{150480652}a^{2}-\frac{159304851}{75240326}a-\frac{2660244}{5374309}$, $\frac{68041923}{601922608}a^{11}+\frac{79638523}{601922608}a^{10}+\frac{88000643}{601922608}a^{9}-\frac{1538163987}{601922608}a^{8}-\frac{2638076807}{601922608}a^{7}+\frac{1907122653}{601922608}a^{6}+\frac{15232477697}{601922608}a^{5}-\frac{3253415901}{601922608}a^{4}-\frac{6450201427}{300961304}a^{3}+\frac{1382138593}{300961304}a^{2}-\frac{5577624585}{300961304}a+\frac{40238403}{42994472}$, $\frac{730739}{601922608}a^{11}+\frac{1248199}{601922608}a^{10}+\frac{2586721}{601922608}a^{9}-\frac{11050855}{601922608}a^{8}-\frac{21999339}{601922608}a^{7}+\frac{21745001}{601922608}a^{6}+\frac{138643047}{601922608}a^{5}-\frac{126780625}{601922608}a^{4}-\frac{167796155}{300961304}a^{3}-\frac{51578523}{300961304}a^{2}-\frac{117539967}{300961304}a-\frac{246917}{1482568}$, $\frac{4838985}{42994472}a^{11}+\frac{3550807}{42994472}a^{10}+\frac{4454271}{42994472}a^{9}-\frac{110506819}{42994472}a^{8}-\frac{136484889}{42994472}a^{7}+\frac{205697113}{42994472}a^{6}+\frac{983187417}{42994472}a^{5}-\frac{751081997}{42994472}a^{4}-\frac{357278029}{21497236}a^{3}+\frac{372132169}{21497236}a^{2}-\frac{379563393}{21497236}a+\frac{220005089}{21497236}$, $\frac{11664155}{601922608}a^{11}+\frac{21082477}{601922608}a^{10}+\frac{33592305}{601922608}a^{9}-\frac{226872791}{601922608}a^{8}-\frac{562636635}{601922608}a^{7}-\frac{76167153}{601922608}a^{6}+\frac{2219665327}{601922608}a^{5}+\frac{444086499}{601922608}a^{4}-\frac{843216555}{300961304}a^{3}+\frac{46570039}{300961304}a^{2}-\frac{416732103}{300961304}a+\frac{9062435}{42994472}$
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| Regulator: | \( 42465.0056166 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 42465.0056166 \cdot 1}{2\cdot\sqrt{164341563462254592}}\cr\approx \mathstrut & 1.30607145905 \end{aligned}\]
Galois group
$C_2\times C_3^2:C_4$ (as 12T41):
| A solvable group of order 72 |
| The 12 conjugacy class representatives for $C_2\times C_3^2:C_4$ |
| Character table for $C_2\times C_3^2:C_4$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.4.18432.1, 6.2.11943936.2 |
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 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.2282521714753536.28 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.4.11a1.10 | $x^{4} + 4 x^{2} + 18$ | $4$ | $1$ | $11$ | $C_4$ | $$[3, 4]$$ |
| 2.1.8.24c1.2 | $x^{8} + 2 x^{4} + 4 x^{2} + 8 x + 18$ | $8$ | $1$ | $24$ | $C_4\times C_2$ | $$[2, 3, 4]$$ | |
|
\(3\)
| 3.2.6.14a1.2 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2630 x^{6} + 3303 x^{5} + 3240 x^{4} + 2462 x^{3} + 1410 x^{2} + 567 x + 124$ | $6$ | $2$ | $14$ | $(C_3\times C_3):C_4$ | $$[\frac{3}{2}, \frac{3}{2}]_{2}^{2}$$ |