Normalized defining polynomial
\( x^{16} + 8x^{14} + 22x^{12} + 17x^{10} - 27x^{8} - 58x^{6} - 33x^{4} - 2x^{2} + 1 \)
Invariants
| Degree: | $16$ |
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| Signature: | $[4, 6]$ |
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| Discriminant: |
\(101609625625600000000\)
\(\medspace = 2^{16}\cdot 5^{8}\cdot 251^{4}\)
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| Root discriminant: | \(17.80\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(5\), \(251\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17}a^{14}-\frac{6}{17}a^{12}+\frac{4}{17}a^{10}-\frac{5}{17}a^{8}-\frac{8}{17}a^{6}+\frac{3}{17}a^{4}-\frac{7}{17}a^{2}-\frac{6}{17}$, $\frac{1}{17}a^{15}-\frac{6}{17}a^{13}+\frac{4}{17}a^{11}-\frac{5}{17}a^{9}-\frac{8}{17}a^{7}+\frac{3}{17}a^{5}-\frac{7}{17}a^{3}-\frac{6}{17}a$
| 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: | Trivial group, which has order $1$ |
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Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{13}{17}a^{15}+\frac{92}{17}a^{13}+\frac{205}{17}a^{11}+\frac{54}{17}a^{9}-\frac{376}{17}a^{7}-\frac{454}{17}a^{5}-\frac{91}{17}a^{3}+\frac{75}{17}a$, $\frac{2}{17}a^{14}+\frac{22}{17}a^{12}+\frac{76}{17}a^{10}+\frac{75}{17}a^{8}-\frac{84}{17}a^{6}-\frac{198}{17}a^{4}-\frac{116}{17}a^{2}-\frac{12}{17}$, $\frac{10}{17}a^{14}+\frac{59}{17}a^{12}+\frac{91}{17}a^{10}-\frac{50}{17}a^{8}-\frac{199}{17}a^{6}-\frac{106}{17}a^{4}-\frac{2}{17}a^{2}-\frac{9}{17}$, $\frac{9}{17}a^{14}+\frac{65}{17}a^{12}+\frac{155}{17}a^{10}+\frac{74}{17}a^{8}-\frac{242}{17}a^{6}-\frac{364}{17}a^{4}-\frac{131}{17}a^{2}-\frac{3}{17}$, $\frac{32}{17}a^{15}+\frac{233}{17}a^{13}+\frac{553}{17}a^{11}+\frac{248}{17}a^{9}-\frac{868}{17}a^{7}-\frac{1281}{17}a^{5}-\frac{496}{17}a^{3}+\frac{46}{17}a$, $\frac{1}{17}a^{15}+\frac{11}{17}a^{13}+\frac{38}{17}a^{11}+\frac{29}{17}a^{9}-\frac{76}{17}a^{7}-\frac{116}{17}a^{5}-\frac{7}{17}a^{3}+a^{2}+\frac{28}{17}a+1$, $\frac{4}{17}a^{15}+\frac{13}{17}a^{14}+\frac{27}{17}a^{13}+\frac{92}{17}a^{12}+\frac{50}{17}a^{11}+\frac{205}{17}a^{10}-\frac{20}{17}a^{9}+\frac{54}{17}a^{8}-\frac{134}{17}a^{7}-\frac{359}{17}a^{6}-\frac{90}{17}a^{5}-\frac{420}{17}a^{4}+\frac{40}{17}a^{3}-\frac{125}{17}a^{2}+\frac{61}{17}a+\frac{7}{17}$, $a^{15}+\frac{3}{17}a^{14}+7a^{13}+\frac{16}{17}a^{12}+15a^{11}+\frac{12}{17}a^{10}+2a^{9}-\frac{49}{17}a^{8}-29a^{7}-\frac{58}{17}a^{6}-29a^{5}+\frac{26}{17}a^{4}-4a^{3}+\frac{30}{17}a^{2}+3a+\frac{16}{17}$, $\frac{4}{17}a^{14}+a^{13}+\frac{27}{17}a^{12}+6a^{11}+\frac{50}{17}a^{10}+10a^{9}-\frac{20}{17}a^{8}-3a^{7}-\frac{134}{17}a^{6}-21a^{5}-\frac{73}{17}a^{4}-16a^{3}+\frac{57}{17}a^{2}-2a+\frac{27}{17}$
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| Regulator: | \( 4719.24941014 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 4719.24941014 \cdot 1}{2\cdot\sqrt{101609625625600000000}}\cr\approx \mathstrut & 0.230448933049 \end{aligned}\]
Galois group
$C_2^7.(C_2\times S_4)$ (as 16T1665):
| A solvable group of order 6144 |
| The 78 conjugacy class representatives for $C_2^7.(C_2\times S_4)$ |
| Character table for $C_2^7.(C_2\times S_4)$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.2.1255.1, 8.4.39375625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 16.4.1040482566406144000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.4a2.2 | $x^{4} + 4 x^{3} + 5 x^{2} + 4 x + 7$ | $2$ | $2$ | $4$ | $D_{4}$ | $$[2, 2]^{2}$$ |
| 2.6.2.12a13.2 | $x^{12} + 2 x^{11} + 4 x^{10} + 6 x^{9} + 5 x^{8} + 8 x^{7} + 9 x^{6} + 6 x^{5} + 10 x^{4} + 6 x^{3} + x^{2} + 4 x + 9$ | $2$ | $6$ | $12$ | 12T134 | $$[2, 2, 2, 2, 2, 2]^{6}$$ | |
|
\(5\)
| 5.4.2.4a1.2 | $x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |
| 5.4.2.4a1.2 | $x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ | |
|
\(251\)
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |