Normalized defining polynomial
\( x^{12} - 4x^{10} + 52x^{8} + 148x^{6} + 469x^{4} + 1176x^{2} + 686 \)
Invariants
| Degree: | $12$ |
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| Signature: | $[0, 6]$ |
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| Discriminant: |
\(113186888059191296\)
\(\medspace = 2^{37}\cdot 7^{7}\)
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| Root discriminant: | \(26.37\) |
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| Galois root discriminant: | $2^{27/8}7^{5/6}\approx 52.5078942662214$ | ||
| Ramified primes: |
\(2\), \(7\)
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| Discriminant root field: | \(\Q(\sqrt{14}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | 4.0.14336.1 | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}$, $\frac{1}{28}a^{8}-\frac{1}{7}a^{6}-\frac{11}{28}a^{4}+\frac{2}{7}a^{2}-\frac{1}{2}$, $\frac{1}{56}a^{9}-\frac{1}{56}a^{8}+\frac{5}{28}a^{7}-\frac{5}{28}a^{6}+\frac{3}{56}a^{5}-\frac{3}{56}a^{4}-\frac{5}{14}a^{3}+\frac{5}{14}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{2836120}a^{10}+\frac{17139}{2836120}a^{8}+\frac{275509}{2836120}a^{6}+\frac{182227}{567224}a^{4}-\frac{24509}{202580}a^{2}-\frac{969}{28940}$, $\frac{1}{5672240}a^{11}-\frac{1}{5672240}a^{10}+\frac{17139}{5672240}a^{9}-\frac{17139}{5672240}a^{8}+\frac{275509}{5672240}a^{7}-\frac{275509}{5672240}a^{6}+\frac{182227}{1134448}a^{5}-\frac{182227}{1134448}a^{4}+\frac{178071}{405160}a^{3}-\frac{178071}{405160}a^{2}+\frac{27971}{57880}a-\frac{27971}{57880}$
| 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: | $5$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{5577}{2836120}a^{10}-\frac{33557}{2836120}a^{8}+\frac{349553}{2836120}a^{6}+\frac{17151}{567224}a^{4}+\frac{112687}{202580}a^{2}+\frac{7667}{28940}$, $\frac{69}{14470}a^{10}-\frac{663}{28940}a^{8}+\frac{1888}{7235}a^{6}+\frac{2807}{5788}a^{4}+\frac{13083}{7235}a^{2}+\frac{24237}{14470}$, $\frac{183}{567224}a^{10}-\frac{3553}{567224}a^{8}+\frac{16243}{567224}a^{6}-\frac{86925}{567224}a^{4}-\frac{5709}{5788}a^{2}-\frac{6581}{5788}$, $\frac{1623}{1418060}a^{11}-\frac{423}{709030}a^{10}-\frac{25661}{2836120}a^{9}+\frac{20397}{2836120}a^{8}+\frac{26923}{354515}a^{7}-\frac{62969}{1418060}a^{6}-\frac{34887}{567224}a^{5}+\frac{69579}{567224}a^{4}-\frac{10891}{50645}a^{3}+\frac{107629}{101290}a^{2}-\frac{41559}{28940}a-\frac{2797}{28940}$, $\frac{9353}{5672240}a^{11}-\frac{293}{115760}a^{10}-\frac{41003}{5672240}a^{9}-\frac{637}{115760}a^{8}+\frac{625817}{5672240}a^{7}-\frac{10477}{115760}a^{6}+\frac{83849}{1134448}a^{5}-\frac{12681}{23152}a^{4}+\frac{30003}{405160}a^{3}-\frac{87641}{57880}a^{2}+\frac{9633}{57880}a-\frac{109497}{57880}$
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| Regulator: | \( 4794.43933513 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 4794.43933513 \cdot 2}{2\cdot\sqrt{113186888059191296}}\cr\approx \mathstrut & 0.876837223311 \end{aligned}\]
Galois group
$C_6^2:C_4$ (as 12T82):
| A solvable group of order 144 |
| The 18 conjugacy class representatives for $C_6^2:C_4$ |
| Character table for $C_6^2:C_4$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.0.14336.1, 6.2.802816.1 |
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 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.4.1154968245501952.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{3}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.4.11a1.13 | $x^{4} + 4 x^{2} + 8 x + 2$ | $4$ | $1$ | $11$ | $D_{4}$ | $$[2, 3, 4]$$ |
| 2.1.8.26c1.11 | $x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + 8 x^{4} + 8 x^{3} + 2$ | $8$ | $1$ | $26$ | $C_2^2:C_4$ | $$[2, 3, \frac{7}{2}, 4]$$ | |
|
\(7\)
| 7.3.2.3a1.2 | $x^{6} + 12 x^{5} + 36 x^{4} + 8 x^{3} + 48 x^{2} + 23$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
| 7.2.3.4a1.1 | $x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 169 x + 27$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ |