Normalized defining polynomial
\( x^{16} - 3 x^{15} - 15 x^{14} - 9 x^{13} + 67 x^{12} + 28 x^{11} + 318 x^{10} - 337 x^{9} - 1061 x^{8} + \cdots + 9 \)
Invariants
| Degree: | $16$ |
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| Signature: | $[8, 4]$ |
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| Discriminant: |
\(27466182620559501230159449\)
\(\medspace = 13^{14}\cdot 17^{8}\)
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| Root discriminant: | \(38.90\) |
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| Galois root discriminant: | $13^{7/8}17^{1/2}\approx 38.897787010151276$ | ||
| Ramified primes: |
\(13\), \(17\)
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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. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{9}a^{14}+\frac{1}{9}a^{13}-\frac{1}{9}a^{11}-\frac{1}{9}a^{9}-\frac{1}{9}a^{8}+\frac{2}{9}a^{7}-\frac{2}{9}a^{6}+\frac{4}{9}a^{5}-\frac{4}{9}a^{4}+\frac{1}{3}a^{3}-\frac{4}{9}a^{2}$, $\frac{1}{11\cdots 33}a^{15}+\frac{32\cdots 07}{10\cdots 03}a^{14}-\frac{24\cdots 59}{11\cdots 33}a^{13}+\frac{16\cdots 17}{11\cdots 33}a^{12}-\frac{73\cdots 35}{11\cdots 33}a^{11}-\frac{21\cdots 39}{11\cdots 33}a^{10}-\frac{29\cdots 92}{11\cdots 33}a^{9}+\frac{28\cdots 48}{19\cdots 53}a^{8}-\frac{13\cdots 69}{35\cdots 01}a^{7}+\frac{31\cdots 71}{11\cdots 33}a^{6}-\frac{54\cdots 22}{13\cdots 37}a^{5}+\frac{65\cdots 40}{11\cdots 33}a^{4}+\frac{29\cdots 17}{20\cdots 57}a^{3}+\frac{26\cdots 23}{11\cdots 33}a^{2}+\frac{32\cdots 83}{39\cdots 11}a-\frac{12\cdots 16}{13\cdots 37}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
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| Narrow class group: | $C_{2}$, which has order $2$ (assuming GRH) |
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Unit group
| Rank: | $11$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{28\cdots 37}{43\cdots 79}a^{15}-\frac{63\cdots 22}{35\cdots 01}a^{14}-\frac{40\cdots 90}{39\cdots 11}a^{13}-\frac{37\cdots 47}{43\cdots 79}a^{12}+\frac{15\cdots 81}{39\cdots 11}a^{11}+\frac{36\cdots 16}{13\cdots 37}a^{10}+\frac{84\cdots 94}{39\cdots 11}a^{9}-\frac{10\cdots 91}{64\cdots 51}a^{8}-\frac{25\cdots 99}{35\cdots 01}a^{7}+\frac{16\cdots 16}{39\cdots 11}a^{6}-\frac{77\cdots 48}{39\cdots 11}a^{5}+\frac{31\cdots 87}{39\cdots 11}a^{4}-\frac{18\cdots 57}{76\cdots 91}a^{3}-\frac{79\cdots 79}{39\cdots 11}a^{2}+\frac{59\cdots 36}{13\cdots 37}a+\frac{83\cdots 92}{43\cdots 79}$, $\frac{26\cdots 98}{11\cdots 33}a^{15}-\frac{51\cdots 32}{10\cdots 03}a^{14}-\frac{44\cdots 90}{11\cdots 33}a^{13}-\frac{65\cdots 06}{11\cdots 33}a^{12}+\frac{10\cdots 75}{11\cdots 33}a^{11}+\frac{14\cdots 27}{11\cdots 33}a^{10}+\frac{10\cdots 07}{11\cdots 33}a^{9}+\frac{11\cdots 86}{19\cdots 53}a^{8}-\frac{72\cdots 48}{35\cdots 01}a^{7}+\frac{23\cdots 84}{11\cdots 33}a^{6}-\frac{74\cdots 58}{39\cdots 11}a^{5}+\frac{20\cdots 69}{11\cdots 33}a^{4}-\frac{91\cdots 07}{20\cdots 57}a^{3}+\frac{18\cdots 09}{11\cdots 33}a^{2}+\frac{18\cdots 52}{39\cdots 11}a-\frac{34\cdots 62}{13\cdots 37}$, $\frac{55\cdots 39}{11\cdots 33}a^{15}-\frac{16\cdots 98}{10\cdots 03}a^{14}-\frac{89\cdots 85}{11\cdots 33}a^{13}-\frac{95\cdots 90}{11\cdots 33}a^{12}+\frac{56\cdots 89}{11\cdots 33}a^{11}+\frac{29\cdots 39}{11\cdots 33}a^{10}+\frac{11\cdots 62}{11\cdots 33}a^{9}-\frac{51\cdots 97}{19\cdots 53}a^{8}-\frac{86\cdots 24}{11\cdots 67}a^{7}+\frac{62\cdots 57}{11\cdots 33}a^{6}+\frac{29\cdots 66}{39\cdots 11}a^{5}+\frac{79\cdots 59}{11\cdots 33}a^{4}-\frac{21\cdots 37}{20\cdots 57}a^{3}+\frac{13\cdots 05}{11\cdots 33}a^{2}-\frac{34\cdots 33}{39\cdots 11}a-\frac{86\cdots 11}{13\cdots 37}$, $\frac{10\cdots 22}{39\cdots 11}a^{15}-\frac{31\cdots 10}{35\cdots 01}a^{14}-\frac{17\cdots 45}{43\cdots 79}a^{13}-\frac{48\cdots 06}{39\cdots 11}a^{12}+\frac{27\cdots 93}{13\cdots 37}a^{11}+\frac{26\cdots 00}{39\cdots 11}a^{10}+\frac{31\cdots 10}{39\cdots 11}a^{9}-\frac{77\cdots 54}{64\cdots 51}a^{8}-\frac{11\cdots 32}{35\cdots 01}a^{7}+\frac{11\cdots 78}{39\cdots 11}a^{6}-\frac{21\cdots 64}{39\cdots 11}a^{5}+\frac{18\cdots 85}{43\cdots 79}a^{4}-\frac{13\cdots 66}{69\cdots 19}a^{3}-\frac{52\cdots 63}{43\cdots 79}a^{2}-\frac{78\cdots 75}{13\cdots 37}a-\frac{55\cdots 48}{43\cdots 79}$, $\frac{11\cdots 84}{39\cdots 11}a^{15}-\frac{11\cdots 67}{11\cdots 67}a^{14}-\frac{16\cdots 15}{39\cdots 11}a^{13}-\frac{19\cdots 37}{39\cdots 11}a^{12}+\frac{93\cdots 41}{39\cdots 11}a^{11}+\frac{22\cdots 51}{39\cdots 11}a^{10}+\frac{10\cdots 08}{13\cdots 37}a^{9}-\frac{32\cdots 75}{21\cdots 17}a^{8}-\frac{12\cdots 19}{35\cdots 01}a^{7}+\frac{47\cdots 54}{13\cdots 37}a^{6}-\frac{14\cdots 65}{39\cdots 11}a^{5}+\frac{20\cdots 80}{39\cdots 11}a^{4}-\frac{15\cdots 70}{69\cdots 19}a^{3}-\frac{50\cdots 42}{39\cdots 11}a^{2}-\frac{40\cdots 33}{43\cdots 79}a-\frac{86\cdots 36}{43\cdots 79}$, $\frac{21164790259}{5663045130237}a^{15}+\frac{3564137023}{514822284567}a^{14}-\frac{566119371278}{5663045130237}a^{13}-\frac{1834440313489}{5663045130237}a^{12}-\frac{543658015726}{5663045130237}a^{11}+\frac{5652136786022}{5663045130237}a^{10}+\frac{11577580746034}{5663045130237}a^{9}+\frac{30240279273430}{5663045130237}a^{8}-\frac{329793847151}{57202476063}a^{7}-\frac{80874356709907}{5663045130237}a^{6}+\frac{8128484247547}{1887681710079}a^{5}-\frac{41658926721961}{5663045130237}a^{4}+\frac{87975668049044}{5663045130237}a^{3}-\frac{27802968150811}{5663045130237}a^{2}+\frac{2651437946587}{1887681710079}a+\frac{3514699948}{629227236693}$, $\frac{36\cdots 97}{11\cdots 33}a^{15}-\frac{86\cdots 23}{10\cdots 03}a^{14}-\frac{58\cdots 17}{11\cdots 33}a^{13}-\frac{52\cdots 19}{11\cdots 33}a^{12}+\frac{23\cdots 03}{11\cdots 33}a^{11}+\frac{19\cdots 98}{11\cdots 33}a^{10}+\frac{11\cdots 36}{11\cdots 33}a^{9}-\frac{12\cdots 92}{19\cdots 53}a^{8}-\frac{13\cdots 09}{35\cdots 01}a^{7}+\frac{18\cdots 71}{11\cdots 33}a^{6}-\frac{16\cdots 47}{13\cdots 37}a^{5}+\frac{42\cdots 46}{11\cdots 33}a^{4}-\frac{91\cdots 14}{20\cdots 57}a^{3}-\frac{19\cdots 00}{11\cdots 33}a^{2}+\frac{31\cdots 58}{39\cdots 11}a+\frac{97\cdots 62}{13\cdots 37}$, $\frac{38\cdots 15}{39\cdots 11}a^{15}-\frac{10\cdots 42}{35\cdots 01}a^{14}-\frac{56\cdots 22}{39\cdots 11}a^{13}-\frac{29\cdots 94}{39\cdots 11}a^{12}+\frac{25\cdots 22}{39\cdots 11}a^{11}+\frac{85\cdots 70}{39\cdots 11}a^{10}+\frac{12\cdots 99}{39\cdots 11}a^{9}-\frac{23\cdots 97}{64\cdots 51}a^{8}-\frac{11\cdots 28}{11\cdots 67}a^{7}+\frac{37\cdots 74}{39\cdots 11}a^{6}-\frac{26\cdots 94}{43\cdots 79}a^{5}+\frac{60\cdots 05}{39\cdots 11}a^{4}-\frac{51\cdots 20}{69\cdots 19}a^{3}-\frac{47\cdots 23}{39\cdots 11}a^{2}-\frac{49\cdots 69}{13\cdots 37}a-\frac{78\cdots 36}{43\cdots 79}$, $\frac{33\cdots 90}{39\cdots 11}a^{15}-\frac{30\cdots 97}{35\cdots 01}a^{14}-\frac{69\cdots 29}{39\cdots 11}a^{13}-\frac{13\cdots 97}{39\cdots 11}a^{12}+\frac{15\cdots 85}{39\cdots 11}a^{11}+\frac{49\cdots 79}{39\cdots 11}a^{10}+\frac{12\cdots 56}{39\cdots 11}a^{9}+\frac{17\cdots 91}{64\cdots 51}a^{8}-\frac{15\cdots 83}{11\cdots 67}a^{7}-\frac{35\cdots 02}{39\cdots 11}a^{6}+\frac{51\cdots 21}{43\cdots 79}a^{5}-\frac{25\cdots 30}{39\cdots 11}a^{4}+\frac{35\cdots 49}{69\cdots 19}a^{3}-\frac{14\cdots 98}{39\cdots 11}a^{2}+\frac{22\cdots 50}{43\cdots 79}a+\frac{58\cdots 65}{43\cdots 79}$, $\frac{15\cdots 78}{11\cdots 33}a^{15}-\frac{14\cdots 22}{10\cdots 03}a^{14}-\frac{28\cdots 16}{11\cdots 33}a^{13}-\frac{62\cdots 43}{11\cdots 33}a^{12}+\frac{15\cdots 58}{11\cdots 33}a^{11}+\frac{11\cdots 21}{11\cdots 33}a^{10}+\frac{57\cdots 13}{11\cdots 33}a^{9}+\frac{87\cdots 02}{19\cdots 53}a^{8}-\frac{39\cdots 01}{35\cdots 01}a^{7}-\frac{88\cdots 02}{11\cdots 33}a^{6}+\frac{54\cdots 94}{13\cdots 37}a^{5}+\frac{77\cdots 21}{11\cdots 33}a^{4}+\frac{15\cdots 92}{20\cdots 57}a^{3}-\frac{39\cdots 09}{11\cdots 33}a^{2}-\frac{49\cdots 32}{39\cdots 11}a+\frac{15\cdots 83}{13\cdots 37}$, $\frac{10\cdots 04}{11\cdots 33}a^{15}-\frac{24\cdots 22}{10\cdots 03}a^{14}-\frac{16\cdots 31}{11\cdots 33}a^{13}-\frac{15\cdots 71}{11\cdots 33}a^{12}+\frac{66\cdots 56}{11\cdots 33}a^{11}+\frac{55\cdots 82}{11\cdots 33}a^{10}+\frac{34\cdots 87}{11\cdots 33}a^{9}-\frac{37\cdots 34}{19\cdots 53}a^{8}-\frac{37\cdots 72}{35\cdots 01}a^{7}+\frac{51\cdots 37}{11\cdots 33}a^{6}-\frac{36\cdots 51}{39\cdots 11}a^{5}+\frac{13\cdots 80}{11\cdots 33}a^{4}-\frac{31\cdots 13}{20\cdots 57}a^{3}-\frac{46\cdots 25}{11\cdots 33}a^{2}-\frac{12\cdots 26}{39\cdots 11}a+\frac{13\cdots 30}{13\cdots 37}$
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| Regulator: | \( 13224148.4074 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 13224148.4074 \cdot 1}{2\cdot\sqrt{27466182620559501230159449}}\cr\approx \mathstrut & 0.503382375460 \end{aligned}\] (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
| \(\Q(\sqrt{13}) \), \(\Q(\sqrt{221}) \), \(\Q(\sqrt{17}) \), 4.4.37349.1 x2, 4.4.634933.1 x2, \(\Q(\sqrt{13}, \sqrt{17})\), 8.4.5240818888357.2, 8.4.5240818888357.1, 8.8.403139914489.1 |
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: | 8.4.5240818888357.2, 8.4.5240818888357.1 |
| Degree 16 sibling: | 16.0.95038694188787201488441.2 |
| Minimal sibling: | 8.4.5240818888357.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.2.0.1}{2} }^{8}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | R | R | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.1.8.7a1.3 | $x^{8} + 52$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $$[\ ]_{8}^{2}$$ |
| 13.1.8.7a1.3 | $x^{8} + 52$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $$[\ ]_{8}^{2}$$ | |
|
\(17\)
| 17.4.2.4a1.2 | $x^{8} + 14 x^{6} + 20 x^{5} + 55 x^{4} + 140 x^{3} + 142 x^{2} + 60 x + 26$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |
| 17.4.2.4a1.2 | $x^{8} + 14 x^{6} + 20 x^{5} + 55 x^{4} + 140 x^{3} + 142 x^{2} + 60 x + 26$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |