Normalized defining polynomial
\( x^{12} - 3x^{11} + x^{10} - 5x^{9} + 5x^{8} - 3x^{7} + 9x^{6} - 3x^{5} + 5x^{4} - 5x^{3} + x^{2} - 3x + 1 \)
Invariants
| Degree: | $12$ |
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| Signature: | $[4, 4]$ |
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| Discriminant: |
\(1027371533203125\)
\(\medspace = 3^{2}\cdot 5^{11}\cdot 11^{2}\cdot 139^{2}\)
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| Root discriminant: | \(17.82\) |
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| Galois root discriminant: | $3^{1/2}5^{11/12}11^{1/2}139^{1/2}\approx 296.1330461861123$ | ||
| Ramified primes: |
\(3\), \(5\), \(11\), \(139\)
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| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}$
| 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: |
$\frac{1}{3}a^{11}-\frac{2}{3}a^{10}-\frac{2}{3}a^{9}-\frac{5}{3}a^{8}+\frac{5}{3}a^{7}-\frac{5}{3}a^{6}+\frac{11}{3}a^{5}-\frac{2}{3}a^{4}+\frac{10}{3}a^{3}-\frac{5}{3}a^{2}+\frac{4}{3}a-2$, $a^{11}+3a^{10}-a^{9}+5a^{8}-5a^{7}+3a^{6}-9a^{5}+3a^{4}-5a^{3}+5a^{2}-a+3$, $a^{11}-\frac{7}{3}a^{10}-\frac{4}{3}a^{9}-\frac{10}{3}a^{8}+\frac{5}{3}a^{7}+\frac{4}{3}a^{6}+\frac{14}{3}a^{5}+\frac{7}{3}a^{4}+\frac{5}{3}a^{3}-\frac{4}{3}a^{2}-\frac{4}{3}a-\frac{4}{3}$, $\frac{2}{3}a^{11}+a^{10}+2a^{9}+4a^{8}-\frac{5}{3}a^{7}+\frac{8}{3}a^{6}-\frac{29}{3}a^{5}+\frac{2}{3}a^{4}-8a^{3}+5a^{2}-2a+\frac{14}{3}$, $\frac{2}{3}a^{11}+2a^{10}-a^{9}+4a^{8}-\frac{5}{3}a^{7}-\frac{1}{3}a^{6}-\frac{11}{3}a^{5}-\frac{4}{3}a^{4}-a^{3}+2a+\frac{2}{3}$, $\frac{1}{3}a^{11}-2a^{9}-3a^{8}-\frac{5}{3}a^{7}+\frac{2}{3}a^{6}+\frac{13}{3}a^{5}+\frac{11}{3}a^{4}+4a^{3}-a-\frac{10}{3}$, $\frac{1}{3}a^{11}-\frac{4}{3}a^{10}+\frac{5}{3}a^{9}-\frac{7}{3}a^{8}+2a^{7}-4a^{6}+a^{5}-3a^{4}+\frac{5}{3}a^{3}-\frac{1}{3}a^{2}+\frac{5}{3}a+\frac{1}{3}$
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| Regulator: | \( 738.9379603482661 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 738.9379603482661 \cdot 1}{2\cdot\sqrt{1027371533203125}}\cr\approx \mathstrut & 0.287444282059852 \end{aligned}\]
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
| \(\Q(\sqrt{5}) \), 6.6.4778125.1 |
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 | ${\href{/padicField/2.12.0.1}{12} }$ | R | R | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.12.0.1}{12} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.1.0a1.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |
| 3.4.1.0a1.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
| 3.2.2.2a1.1 | $x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ | |
|
\(5\)
| 5.1.12.11a1.4 | $x^{12} + 20$ | $12$ | $1$ | $11$ | $S_3 \times C_4$ | $$[\ ]_{12}^{2}$$ |
|
\(11\)
| $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 11.2.1.0a1.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 11.2.2.2a1.2 | $x^{4} + 14 x^{3} + 53 x^{2} + 28 x + 15$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 11.4.1.0a1.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
|
\(139\)
| 139.1.2.1a1.2 | $x^{2} + 278$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 139.1.2.1a1.2 | $x^{2} + 278$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 139.2.1.0a1.1 | $x^{2} + 138 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 139.6.1.0a1.1 | $x^{6} + 4 x^{4} + 46 x^{3} + 10 x^{2} + 118 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ |