Normalized defining polynomial
\( x^{12} - 2x^{11} + 6x^{10} - 7x^{9} + 12x^{8} - 7x^{7} + 10x^{6} - 7x^{5} + 12x^{4} - 7x^{3} + 6x^{2} - 2x + 1 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[0, 6]$ |
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| Discriminant: |
\(2071265703125\)
\(\medspace = 5^{7}\cdot 19^{2}\cdot 271^{2}\)
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| Root discriminant: | \(10.63\) |
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| Galois root discriminant: | $5^{3/4}19^{1/2}271^{1/2}\approx 239.93242815578742$ | ||
| Ramified primes: |
\(5\), \(19\), \(271\)
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{8}a^{10}-\frac{1}{4}a^{9}+\frac{3}{8}a^{8}-\frac{1}{8}a^{7}+\frac{3}{8}a^{6}-\frac{1}{2}a^{5}+\frac{1}{8}a^{4}-\frac{3}{8}a^{3}+\frac{1}{8}a^{2}+\frac{1}{4}a+\frac{3}{8}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{10}+\frac{1}{16}a^{9}+\frac{1}{8}a^{8}+\frac{1}{8}a^{7}-\frac{1}{16}a^{6}-\frac{3}{16}a^{5}+\frac{3}{8}a^{4}-\frac{1}{8}a^{3}+\frac{3}{16}a^{2}+\frac{5}{16}a-\frac{5}{16}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
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| Narrow class group: | Trivial group, which has order $1$ |
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Unit group
| Rank: | $5$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$a^{11}-2a^{10}+6a^{9}-7a^{8}+12a^{7}-7a^{6}+10a^{5}-7a^{4}+12a^{3}-7a^{2}+6a-2$, $\frac{7}{8}a^{11}-\frac{15}{8}a^{10}+\frac{43}{8}a^{9}-\frac{27}{4}a^{8}+\frac{43}{4}a^{7}-\frac{59}{8}a^{6}+\frac{67}{8}a^{5}-\frac{31}{4}a^{4}+\frac{39}{4}a^{3}-\frac{67}{8}a^{2}+\frac{31}{8}a-\frac{23}{8}$, $\frac{15}{8}a^{11}-\frac{23}{8}a^{10}+\frac{75}{8}a^{9}-\frac{31}{4}a^{8}+\frac{63}{4}a^{7}-\frac{19}{8}a^{6}+\frac{91}{8}a^{5}-\frac{19}{4}a^{4}+\frac{59}{4}a^{3}-\frac{27}{8}a^{2}+\frac{31}{8}a+\frac{1}{8}$, $\frac{3}{2}a^{11}-\frac{19}{8}a^{10}+\frac{31}{4}a^{9}-\frac{57}{8}a^{8}+\frac{111}{8}a^{7}-\frac{37}{8}a^{6}+11a^{5}-\frac{55}{8}a^{4}+\frac{97}{8}a^{3}-\frac{43}{8}a^{2}+\frac{17}{4}a-\frac{13}{8}$, $\frac{1}{8}a^{11}+a^{10}-\frac{13}{8}a^{9}+\frac{49}{8}a^{8}-\frac{47}{8}a^{7}+\frac{47}{4}a^{6}-\frac{31}{8}a^{5}+\frac{63}{8}a^{4}-\frac{33}{8}a^{3}+\frac{23}{2}a^{2}-\frac{37}{8}a+\frac{13}{4}$
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| Regulator: | \( 17.5482156786 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 17.5482156786 \cdot 1}{2\cdot\sqrt{2071265703125}}\cr\approx \mathstrut & 0.375114861844 \end{aligned}\]
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
| \(\Q(\sqrt{5}) \), 6.2.643625.1 |
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 | ${\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.2.0.1}{2} }^{2}$ | ${\href{/padicField/3.12.0.1}{12} }$ | R | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.12.0.1}{12} }$ | R | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.4.3a1.4 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 5.4.2.4a1.2 | $x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ | |
|
\(19\)
| 19.2.1.0a1.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 19.2.2.2a1.2 | $x^{4} + 36 x^{3} + 328 x^{2} + 72 x + 23$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 19.6.1.0a1.1 | $x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ | |
|
\(271\)
| $\Q_{271}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{271}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |