Normalized defining polynomial
\( x^{16} - 2 x^{15} - 18 x^{14} + 18 x^{13} + 163 x^{12} - 172 x^{11} - 552 x^{10} + 1347 x^{9} + \cdots + 16381 \)
Invariants
| Degree: | $16$ |
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| Signature: | $[0, 8]$ |
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| Discriminant: |
\(483823969655762431707489\)
\(\medspace = 3^{8}\cdot 97^{10}\)
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| Root discriminant: | \(30.22\) |
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| Galois root discriminant: | $3^{1/2}97^{3/4}\approx 53.535199825186666$ | ||
| Ramified primes: |
\(3\), \(97\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2\times C_4$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-3}, \sqrt{97})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{14}a^{14}+\frac{1}{7}a^{13}-\frac{1}{14}a^{12}-\frac{3}{14}a^{11}+\frac{1}{7}a^{10}+\frac{5}{14}a^{9}-\frac{3}{14}a^{8}-\frac{3}{7}a^{7}-\frac{3}{14}a^{6}+\frac{1}{14}a^{5}+\frac{2}{7}a^{4}+\frac{1}{14}a^{3}-\frac{5}{14}a^{2}-\frac{3}{7}a+\frac{3}{14}$, $\frac{1}{65\cdots 98}a^{15}+\frac{20\cdots 59}{32\cdots 49}a^{14}+\frac{25\cdots 64}{32\cdots 49}a^{13}+\frac{77\cdots 91}{32\cdots 49}a^{12}-\frac{14\cdots 02}{32\cdots 49}a^{11}-\frac{79\cdots 91}{32\cdots 49}a^{10}+\frac{30\cdots 41}{32\cdots 49}a^{9}-\frac{11\cdots 36}{32\cdots 49}a^{8}-\frac{19\cdots 33}{32\cdots 49}a^{7}+\frac{57\cdots 82}{32\cdots 49}a^{6}+\frac{13\cdots 84}{32\cdots 49}a^{5}-\frac{40\cdots 95}{32\cdots 49}a^{4}-\frac{15\cdots 80}{32\cdots 49}a^{3}+\frac{15\cdots 90}{32\cdots 49}a^{2}+\frac{14\cdots 02}{32\cdots 49}a-\frac{93\cdots 51}{94\cdots 14}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
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| Narrow class group: | Trivial group, which has order $1$ (assuming GRH) |
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Unit group
| Rank: | $7$ |
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| Torsion generator: |
\( \frac{1635044887083493}{25925487728830265582} a^{15} - \frac{2266218916099537}{25925487728830265582} a^{14} - \frac{28982100472406337}{25925487728830265582} a^{13} + \frac{3380228533810795}{12962743864415132791} a^{12} + \frac{243174892866235377}{25925487728830265582} a^{11} - \frac{83579489680520449}{25925487728830265582} a^{10} - \frac{368938399675886743}{12962743864415132791} a^{9} + \frac{1377627882526655127}{25925487728830265582} a^{8} + \frac{2402718408110796251}{25925487728830265582} a^{7} - \frac{3853528833812264616}{12962743864415132791} a^{6} + \frac{248620025616991945}{25925487728830265582} a^{5} + \frac{13011015059914101035}{25925487728830265582} a^{4} - \frac{3673169606597379325}{12962743864415132791} a^{3} - \frac{14729991011466854863}{25925487728830265582} a^{2} + \frac{45273357898236017427}{25925487728830265582} a - \frac{12615760837496252921}{25925487728830265582} \)
(order $6$)
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| Fundamental units: |
$\frac{34\cdots 33}{94\cdots 14}a^{15}-\frac{53\cdots 47}{65\cdots 98}a^{14}-\frac{37\cdots 61}{65\cdots 98}a^{13}+\frac{28\cdots 76}{32\cdots 49}a^{12}+\frac{29\cdots 71}{65\cdots 98}a^{11}-\frac{67\cdots 13}{65\cdots 98}a^{10}-\frac{36\cdots 43}{32\cdots 49}a^{9}+\frac{44\cdots 51}{65\cdots 98}a^{8}+\frac{20\cdots 51}{65\cdots 98}a^{7}-\frac{74\cdots 48}{32\cdots 49}a^{6}+\frac{16\cdots 95}{65\cdots 98}a^{5}+\frac{24\cdots 69}{65\cdots 98}a^{4}-\frac{11\cdots 12}{32\cdots 49}a^{3}-\frac{41\cdots 21}{65\cdots 98}a^{2}+\frac{83\cdots 71}{65\cdots 98}a-\frac{67\cdots 57}{65\cdots 98}$, $\frac{32\cdots 71}{94\cdots 14}a^{15}-\frac{66\cdots 85}{32\cdots 49}a^{14}-\frac{34\cdots 77}{65\cdots 98}a^{13}+\frac{10\cdots 65}{32\cdots 49}a^{12}+\frac{20\cdots 11}{32\cdots 49}a^{11}-\frac{19\cdots 05}{65\cdots 98}a^{10}-\frac{74\cdots 06}{32\cdots 49}a^{9}+\frac{46\cdots 02}{32\cdots 49}a^{8}-\frac{29\cdots 15}{65\cdots 98}a^{7}-\frac{19\cdots 03}{32\cdots 49}a^{6}+\frac{21\cdots 04}{32\cdots 49}a^{5}+\frac{70\cdots 79}{65\cdots 98}a^{4}-\frac{59\cdots 63}{32\cdots 49}a^{3}-\frac{22\cdots 78}{32\cdots 49}a^{2}+\frac{20\cdots 09}{65\cdots 98}a-\frac{12\cdots 83}{65\cdots 98}$, $\frac{36\cdots 90}{47\cdots 07}a^{15}-\frac{42\cdots 50}{47\cdots 07}a^{14}-\frac{15\cdots 59}{94\cdots 14}a^{13}+\frac{16\cdots 81}{47\cdots 07}a^{12}+\frac{74\cdots 82}{47\cdots 07}a^{11}-\frac{29\cdots 99}{94\cdots 14}a^{10}-\frac{32\cdots 85}{47\cdots 07}a^{9}+\frac{36\cdots 23}{47\cdots 07}a^{8}+\frac{21\cdots 87}{94\cdots 14}a^{7}-\frac{26\cdots 42}{47\cdots 07}a^{6}-\frac{18\cdots 86}{47\cdots 07}a^{5}+\frac{13\cdots 85}{94\cdots 14}a^{4}-\frac{22\cdots 57}{47\cdots 07}a^{3}-\frac{11\cdots 29}{47\cdots 07}a^{2}+\frac{21\cdots 73}{94\cdots 14}a-\frac{70\cdots 70}{47\cdots 07}$, $\frac{15\cdots 47}{94\cdots 14}a^{15}-\frac{20\cdots 52}{47\cdots 07}a^{14}-\frac{26\cdots 99}{94\cdots 14}a^{13}+\frac{22\cdots 18}{47\cdots 07}a^{12}+\frac{12\cdots 12}{47\cdots 07}a^{11}-\frac{41\cdots 79}{94\cdots 14}a^{10}-\frac{44\cdots 86}{47\cdots 07}a^{9}+\frac{12\cdots 70}{47\cdots 07}a^{8}+\frac{17\cdots 47}{94\cdots 14}a^{7}-\frac{62\cdots 55}{47\cdots 07}a^{6}+\frac{31\cdots 97}{47\cdots 07}a^{5}+\frac{22\cdots 93}{94\cdots 14}a^{4}-\frac{10\cdots 50}{47\cdots 07}a^{3}-\frac{13\cdots 30}{47\cdots 07}a^{2}+\frac{59\cdots 25}{94\cdots 14}a-\frac{28\cdots 09}{94\cdots 14}$, $\frac{13\cdots 37}{47\cdots 07}a^{15}-\frac{12\cdots 98}{32\cdots 49}a^{14}+\frac{49\cdots 04}{32\cdots 49}a^{13}+\frac{16\cdots 30}{32\cdots 49}a^{12}+\frac{12\cdots 40}{32\cdots 49}a^{11}-\frac{12\cdots 05}{32\cdots 49}a^{10}-\frac{81\cdots 70}{32\cdots 49}a^{9}+\frac{30\cdots 89}{32\cdots 49}a^{8}+\frac{17\cdots 35}{32\cdots 49}a^{7}-\frac{19\cdots 18}{32\cdots 49}a^{6}+\frac{11\cdots 42}{32\cdots 49}a^{5}+\frac{59\cdots 27}{32\cdots 49}a^{4}-\frac{18\cdots 94}{32\cdots 49}a^{3}-\frac{12\cdots 97}{32\cdots 49}a^{2}+\frac{12\cdots 20}{32\cdots 49}a-\frac{12\cdots 15}{32\cdots 49}$, $\frac{53\cdots 72}{32\cdots 49}a^{15}+\frac{51\cdots 03}{65\cdots 98}a^{14}-\frac{92\cdots 19}{32\cdots 49}a^{13}-\frac{10\cdots 25}{32\cdots 49}a^{12}+\frac{11\cdots 27}{65\cdots 98}a^{11}+\frac{52\cdots 18}{32\cdots 49}a^{10}-\frac{12\cdots 38}{32\cdots 49}a^{9}+\frac{55\cdots 11}{65\cdots 98}a^{8}+\frac{10\cdots 83}{32\cdots 49}a^{7}-\frac{51\cdots 17}{32\cdots 49}a^{6}-\frac{18\cdots 97}{65\cdots 98}a^{5}+\frac{20\cdots 74}{32\cdots 49}a^{4}+\frac{39\cdots 01}{32\cdots 49}a^{3}-\frac{71\cdots 07}{65\cdots 98}a^{2}+\frac{30\cdots 00}{32\cdots 49}a-\frac{22\cdots 56}{32\cdots 49}$, $\frac{24\cdots 41}{65\cdots 98}a^{15}-\frac{40\cdots 36}{47\cdots 07}a^{14}-\frac{35\cdots 43}{65\cdots 98}a^{13}+\frac{28\cdots 37}{32\cdots 49}a^{12}+\frac{13\cdots 00}{32\cdots 49}a^{11}-\frac{53\cdots 61}{65\cdots 98}a^{10}-\frac{19\cdots 66}{32\cdots 49}a^{9}+\frac{17\cdots 13}{47\cdots 07}a^{8}-\frac{13\cdots 27}{65\cdots 98}a^{7}-\frac{55\cdots 94}{47\cdots 07}a^{6}+\frac{89\cdots 16}{32\cdots 49}a^{5}-\frac{19\cdots 53}{94\cdots 14}a^{4}+\frac{85\cdots 70}{47\cdots 07}a^{3}+\frac{46\cdots 07}{32\cdots 49}a^{2}-\frac{78\cdots 93}{65\cdots 98}a+\frac{31\cdots 59}{65\cdots 98}$
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| Regulator: | \( 329160.9111 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 329160.9111 \cdot 1}{6\cdot\sqrt{483823969655762431707489}}\cr\approx \mathstrut & 0.1915809233 \end{aligned}\] (assuming GRH)
Galois group
$C_4\wr C_2$ (as 16T42):
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $C_4\wr C_2$ |
| Character table for $C_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{97}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-291}) \), 4.2.28227.1 x2, 4.0.873.1 x2, \(\Q(\sqrt{-3}, \sqrt{97})\), 8.0.73926513.1, 8.0.695574560817.1, 8.0.7170871761.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 32 |
| Degree 8 siblings: | 8.0.73926513.1, 8.0.695574560817.1 |
| Degree 16 sibling: | 16.4.505811081165674302215084889.1 |
| Minimal sibling: | 8.0.73926513.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.2.4a1.2 | $x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |
| 3.4.2.4a1.2 | $x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ | |
|
\(97\)
| 97.2.4.6a1.3 | $x^{8} + 384 x^{7} + 55316 x^{6} + 3544704 x^{5} + 85487766 x^{4} + 17723520 x^{3} + 1382900 x^{2} + 48097 x + 9549$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ |
| 97.4.2.4a1.2 | $x^{8} + 12 x^{6} + 160 x^{5} + 46 x^{4} + 960 x^{3} + 6460 x^{2} + 800 x + 122$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |