Normalized defining polynomial
\( x^{16} + 8x^{14} + 52x^{12} + 136x^{10} + 566x^{8} + 184x^{6} - 300x^{4} - 72x^{2} + 81 \)
Invariants
| Degree: | $16$ |
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| Signature: | $[0, 8]$ |
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| Discriminant: |
\(484116351470433472610304\)
\(\medspace = 2^{66}\cdot 3^{8}\)
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| Root discriminant: | \(30.22\) |
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| Galois root discriminant: | $2^{555/128}3^{3/4}\approx 46.035004279893066$ | ||
| Ramified primes: |
\(2\), \(3\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2^2$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-2}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{4}-\frac{3}{8}$, $\frac{1}{16}a^{9}-\frac{1}{16}a^{8}-\frac{1}{8}a^{5}+\frac{1}{8}a^{4}-\frac{3}{16}a+\frac{3}{16}$, $\frac{1}{48}a^{10}+\frac{1}{48}a^{8}-\frac{1}{8}a^{6}+\frac{5}{24}a^{4}-\frac{23}{48}a^{2}+\frac{3}{16}$, $\frac{1}{48}a^{11}+\frac{1}{48}a^{9}-\frac{1}{8}a^{7}-\frac{1}{24}a^{5}-\frac{1}{4}a^{4}-\frac{23}{48}a^{3}-\frac{1}{16}a-\frac{1}{4}$, $\frac{1}{96}a^{12}-\frac{1}{96}a^{8}-\frac{1}{12}a^{6}+\frac{1}{32}a^{4}-\frac{5}{12}a^{2}-\frac{9}{32}$, $\frac{1}{576}a^{13}-\frac{1}{192}a^{12}-\frac{1}{144}a^{11}-\frac{17}{576}a^{9}-\frac{11}{192}a^{8}-\frac{7}{72}a^{7}-\frac{1}{12}a^{6}+\frac{59}{576}a^{5}-\frac{1}{64}a^{4}+\frac{67}{144}a^{3}-\frac{5}{12}a^{2}-\frac{17}{192}a-\frac{19}{64}$, $\frac{1}{72000}a^{14}-\frac{13}{2880}a^{12}-\frac{473}{72000}a^{10}+\frac{329}{14400}a^{8}-\frac{469}{72000}a^{6}+\frac{361}{72000}a^{4}-\frac{2921}{24000}a^{2}-\frac{3777}{8000}$, $\frac{1}{216000}a^{15}+\frac{1}{4320}a^{13}-\frac{1}{192}a^{12}+\frac{1027}{216000}a^{11}-\frac{173}{21600}a^{9}-\frac{11}{192}a^{8}-\frac{21469}{216000}a^{7}-\frac{1}{12}a^{6}-\frac{757}{108000}a^{5}-\frac{1}{64}a^{4}-\frac{10421}{72000}a^{3}-\frac{5}{12}a^{2}+\frac{1183}{4000}a-\frac{19}{64}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: | $7$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{287}{216000}a^{15}-\frac{229}{36000}a^{14}-\frac{109}{8640}a^{13}-\frac{83}{1440}a^{12}-\frac{17249}{216000}a^{11}-\frac{13933}{36000}a^{10}-\frac{10423}{43200}a^{9}-\frac{8891}{7200}a^{8}-\frac{159397}{216000}a^{7}-\frac{167099}{36000}a^{6}-\frac{121607}{216000}a^{5}-\frac{193669}{36000}a^{4}+\frac{25703}{8000}a^{3}-\frac{13841}{12000}a^{2}+\frac{53999}{24000}a-\frac{817}{4000}$, $\frac{287}{216000}a^{15}-\frac{229}{36000}a^{14}+\frac{109}{8640}a^{13}-\frac{83}{1440}a^{12}+\frac{17249}{216000}a^{11}-\frac{13933}{36000}a^{10}+\frac{10423}{43200}a^{9}-\frac{8891}{7200}a^{8}+\frac{159397}{216000}a^{7}-\frac{167099}{36000}a^{6}+\frac{121607}{216000}a^{5}-\frac{193669}{36000}a^{4}-\frac{25703}{8000}a^{3}-\frac{13841}{12000}a^{2}-\frac{53999}{24000}a-\frac{817}{4000}$, $\frac{1031}{108000}a^{15}+\frac{143}{72000}a^{14}-\frac{367}{4320}a^{13}+\frac{31}{2880}a^{12}-\frac{60587}{108000}a^{11}+\frac{4361}{72000}a^{10}-\frac{37099}{21600}a^{9}-\frac{203}{14400}a^{8}-\frac{690961}{108000}a^{7}+\frac{22933}{72000}a^{6}-\frac{677441}{108000}a^{5}-\frac{202627}{72000}a^{4}+\frac{122801}{36000}a^{3}-\frac{51703}{24000}a^{2}+\frac{34087}{12000}a+\frac{7139}{8000}$, $\frac{1039}{43200}a^{15}+\frac{3}{320}a^{14}-\frac{347}{1728}a^{13}+\frac{5}{64}a^{12}-\frac{56653}{43200}a^{11}+\frac{161}{320}a^{10}-\frac{31541}{8640}a^{9}+\frac{87}{64}a^{8}-\frac{620909}{43200}a^{7}+\frac{1673}{320}a^{6}-\frac{346729}{43200}a^{5}+\frac{683}{320}a^{4}+\frac{48073}{4800}a^{3}-\frac{2389}{320}a^{2}+\frac{39253}{4800}a-\frac{1519}{320}$, $\frac{299}{24000}a^{15}+\frac{127}{24000}a^{14}+\frac{269}{2880}a^{13}+\frac{29}{960}a^{12}+\frac{43219}{72000}a^{11}+\frac{4429}{24000}a^{10}+\frac{19963}{14400}a^{9}+\frac{683}{4800}a^{8}+\frac{454307}{72000}a^{7}+\frac{39437}{24000}a^{6}-\frac{70433}{72000}a^{5}-\frac{125153}{24000}a^{4}-\frac{284911}{72000}a^{3}-\frac{7467}{8000}a^{2}+\frac{25343}{24000}a+\frac{10463}{8000}$, $\frac{511}{108000}a^{15}+\frac{193}{72000}a^{14}-\frac{91}{2160}a^{13}+\frac{71}{2880}a^{12}-\frac{30547}{108000}a^{11}+\frac{12211}{72000}a^{10}-\frac{4811}{5400}a^{9}+\frac{8297}{14400}a^{8}-\frac{370841}{108000}a^{7}+\frac{158483}{72000}a^{6}-\frac{206423}{54000}a^{5}+\frac{200173}{72000}a^{4}-\frac{13373}{12000}a^{3}+\frac{20747}{24000}a^{2}+\frac{209}{1500}a-\frac{1961}{8000}$, $\frac{4061}{108000}a^{15}-\frac{383}{24000}a^{14}+\frac{1207}{4320}a^{13}-\frac{37}{320}a^{12}+\frac{193397}{108000}a^{11}-\frac{5947}{8000}a^{10}+\frac{87619}{21600}a^{9}-\frac{7757}{4800}a^{8}+\frac{2023891}{108000}a^{7}-\frac{62791}{8000}a^{6}-\frac{449479}{108000}a^{5}+\frac{23329}{8000}a^{4}-\frac{405431}{36000}a^{3}+\frac{51229}{24000}a^{2}+\frac{60353}{12000}a-\frac{8977}{8000}$
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| Regulator: | \( 363419.3955430768 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 363419.3955430768 \cdot 1}{2\cdot\sqrt{484116351470433472610304}}\cr\approx \mathstrut & 0.634369328885186 \end{aligned}\]
Galois group
$C_2^6:D_4$ (as 16T969):
| A solvable group of order 512 |
| The 44 conjugacy class representatives for $C_2^6:D_4$ |
| Character table for $C_2^6:D_4$ |
Intermediate fields
| \(\Q(\sqrt{-2}) \), 4.0.6144.1, 8.0.28991029248.6, 8.0.28991029248.2, 8.0.21743271936.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.16.66j1.9 | $x^{16} + 8 x^{10} + 16 x^{5} + 16 x^{3} + 2$ | $16$ | $1$ | $66$ | 16T969 | $$[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$$ |
|
\(3\)
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 3.2.4.6a1.2 | $x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 64 x + 19$ | $4$ | $2$ | $6$ | $D_4$ | $$[\ ]_{4}^{2}$$ |