Normalized defining polynomial
\( x^{12} - 4x^{10} - 36x^{6} + 77x^{4} - 56x^{2} + 14 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[4, 4]$ |
| |
| Discriminant: |
\(2309936491003904\)
\(\medspace = 2^{37}\cdot 7^{5}\)
|
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| Root discriminant: | \(19.07\) |
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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: | \(\Q(\sqrt{14}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{16}a^{8}-\frac{1}{2}a^{7}-\frac{1}{8}a^{6}-\frac{1}{2}a^{5}-\frac{3}{16}a^{4}+\frac{1}{4}a^{2}+\frac{1}{8}$, $\frac{1}{16}a^{9}-\frac{1}{8}a^{7}-\frac{1}{2}a^{6}-\frac{3}{16}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}+\frac{1}{8}a$, $\frac{1}{208}a^{10}-\frac{1}{208}a^{8}+\frac{75}{208}a^{6}+\frac{33}{208}a^{4}-\frac{29}{104}a^{2}+\frac{41}{104}$, $\frac{1}{208}a^{11}-\frac{1}{208}a^{9}+\frac{75}{208}a^{7}+\frac{33}{208}a^{5}-\frac{29}{104}a^{3}+\frac{41}{104}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $7$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{33}{52}a^{10}-\frac{111}{52}a^{8}-\frac{73}{52}a^{6}-\frac{1225}{52}a^{4}+\frac{889}{26}a^{2}-\frac{311}{26}$, $\frac{95}{208}a^{11}-\frac{95}{208}a^{10}-\frac{145}{104}a^{9}+\frac{145}{104}a^{8}-\frac{285}{208}a^{7}+\frac{285}{208}a^{6}-\frac{229}{13}a^{5}+\frac{229}{13}a^{4}+\frac{1951}{104}a^{3}-\frac{1951}{104}a^{2}-\frac{141}{26}a+\frac{141}{26}$, $\frac{5}{208}a^{10}-\frac{5}{208}a^{8}-\frac{41}{208}a^{6}-\frac{251}{208}a^{4}-\frac{41}{104}a^{2}+\frac{101}{104}$, $\frac{69}{104}a^{11}+\frac{43}{208}a^{10}-\frac{53}{26}a^{9}-\frac{67}{104}a^{8}-\frac{207}{104}a^{7}-\frac{129}{208}a^{6}-\frac{1325}{52}a^{5}-\frac{409}{52}a^{4}+\frac{1457}{52}a^{3}+\frac{963}{104}a^{2}-\frac{191}{26}a-\frac{25}{13}$, $\frac{69}{104}a^{11}-\frac{43}{208}a^{10}-\frac{53}{26}a^{9}+\frac{67}{104}a^{8}-\frac{207}{104}a^{7}+\frac{129}{208}a^{6}-\frac{1325}{52}a^{5}+\frac{409}{52}a^{4}+\frac{1457}{52}a^{3}-\frac{963}{104}a^{2}-\frac{191}{26}a+\frac{25}{13}$, $\frac{7}{26}a^{10}-\frac{93}{104}a^{8}-\frac{29}{52}a^{6}-\frac{1065}{104}a^{4}+\frac{361}{26}a^{2}-a-\frac{321}{52}$, $\frac{137}{208}a^{11}+\frac{23}{104}a^{10}-\frac{449}{208}a^{9}-\frac{11}{13}a^{8}-\frac{333}{208}a^{7}-\frac{17}{104}a^{6}-\frac{5151}{208}a^{5}-\frac{407}{52}a^{4}+\frac{3411}{104}a^{3}+\frac{815}{52}a^{2}-\frac{1247}{104}a-\frac{211}{26}$
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| Regulator: | \( 3050.6748939 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 3050.6748939 \cdot 1}{2\cdot\sqrt{2309936491003904}}\cr\approx \mathstrut & 0.79141698850 \end{aligned}\]
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
| \(\Q(\sqrt{2}) \), 4.4.14336.1, 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: | 12.0.1154968245501952.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{3}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 | $x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + 8 x^{4} + 8 x^{3} + 2$ | $8$ | $1$ | $26$ | $C_2^2:C_4$ | $$[2, 3, \frac{7}{2}, 4]$$ | |
|
\(7\)
| 7.3.1.0a1.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 7.3.1.0a1.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 7.1.6.5a1.5 | $x^{6} + 35$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ |