Normalized defining polynomial
\( x^{16} - 18 x^{14} + 192 x^{12} - 252 x^{11} - 135 x^{10} + 987 x^{9} - 2123 x^{8} - 4788 x^{7} + \cdots + 1681 \)
Invariants
| Degree: | $16$ |
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| Signature: | $[0, 8]$ |
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| Discriminant: |
\(66556997328875790787650649\)
\(\medspace = 7^{12}\cdot 37^{10}\)
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| Root discriminant: | \(41.11\) |
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| Galois root discriminant: | $7^{3/4}37^{3/4}\approx 64.56168002238364$ | ||
| Ramified primes: |
\(7\), \(37\)
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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: | \(\Q(\sqrt{-7}, \sqrt{37})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{18}a^{10}-\frac{1}{18}a^{9}-\frac{1}{18}a^{8}-\frac{1}{6}a^{6}+\frac{1}{6}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{7}{18}a^{2}-\frac{4}{9}a-\frac{5}{18}$, $\frac{1}{18}a^{11}-\frac{1}{9}a^{9}-\frac{1}{18}a^{8}-\frac{1}{6}a^{7}-\frac{1}{6}a^{5}+\frac{1}{3}a^{4}+\frac{5}{18}a^{3}+\frac{1}{6}a^{2}+\frac{5}{18}a-\frac{5}{18}$, $\frac{1}{36}a^{12}-\frac{1}{36}a^{10}-\frac{1}{18}a^{9}-\frac{1}{9}a^{8}+\frac{1}{3}a^{6}+\frac{1}{4}a^{5}+\frac{17}{36}a^{4}-\frac{1}{12}a^{3}-\frac{1}{18}a^{2}-\frac{13}{36}a-\frac{5}{36}$, $\frac{1}{36}a^{13}-\frac{1}{36}a^{11}-\frac{1}{6}a^{9}-\frac{1}{18}a^{8}+\frac{1}{3}a^{7}+\frac{1}{12}a^{6}-\frac{13}{36}a^{5}-\frac{5}{12}a^{4}-\frac{7}{18}a^{3}+\frac{1}{4}a^{2}+\frac{5}{12}a-\frac{5}{18}$, $\frac{1}{576}a^{14}+\frac{5}{576}a^{13}+\frac{1}{144}a^{12}+\frac{5}{576}a^{11}+\frac{13}{576}a^{10}-\frac{37}{288}a^{9}+\frac{1}{18}a^{8}-\frac{13}{192}a^{7}-\frac{23}{288}a^{6}-\frac{101}{576}a^{5}+\frac{17}{36}a^{4}-\frac{85}{288}a^{3}+\frac{5}{144}a^{2}+\frac{13}{288}a+\frac{283}{576}$, $\frac{1}{70\cdots 68}a^{15}+\frac{42\cdots 79}{14\cdots 04}a^{14}-\frac{61\cdots 89}{70\cdots 68}a^{13}+\frac{14\cdots 45}{17\cdots 48}a^{12}-\frac{17\cdots 21}{88\cdots 96}a^{11}+\frac{18\cdots 53}{70\cdots 68}a^{10}-\frac{11\cdots 13}{11\cdots 28}a^{9}-\frac{10\cdots 43}{70\cdots 68}a^{8}+\frac{11\cdots 05}{70\cdots 68}a^{7}+\frac{11\cdots 71}{23\cdots 56}a^{6}+\frac{18\cdots 25}{70\cdots 68}a^{5}-\frac{10\cdots 97}{35\cdots 84}a^{4}-\frac{13\cdots 17}{35\cdots 84}a^{3}-\frac{69\cdots 09}{35\cdots 84}a^{2}-\frac{10\cdots 25}{23\cdots 56}a-\frac{44\cdots 79}{17\cdots 48}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$ |
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: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{82\cdots 25}{11\cdots 84}a^{15}+\frac{75\cdots 13}{13\cdots 12}a^{14}-\frac{82\cdots 39}{12\cdots 76}a^{13}-\frac{25\cdots 43}{27\cdots 24}a^{12}+\frac{16\cdots 39}{56\cdots 92}a^{11}+\frac{86\cdots 03}{11\cdots 84}a^{10}-\frac{22\cdots 05}{62\cdots 88}a^{9}-\frac{68\cdots 91}{11\cdots 84}a^{8}+\frac{30\cdots 71}{11\cdots 84}a^{7}-\frac{12\cdots 67}{11\cdots 84}a^{6}-\frac{88\cdots 71}{37\cdots 28}a^{5}+\frac{33\cdots 95}{56\cdots 92}a^{4}+\frac{70\cdots 57}{56\cdots 92}a^{3}+\frac{70\cdots 71}{56\cdots 92}a^{2}-\frac{23\cdots 61}{37\cdots 28}a+\frac{47\cdots 51}{27\cdots 24}$, $\frac{47\cdots 97}{88\cdots 96}a^{15}+\frac{92\cdots 17}{10\cdots 28}a^{14}-\frac{88\cdots 11}{98\cdots 44}a^{13}-\frac{31\cdots 33}{21\cdots 56}a^{12}+\frac{40\cdots 37}{44\cdots 48}a^{11}+\frac{41\cdots 71}{29\cdots 32}a^{10}-\frac{70\cdots 81}{44\cdots 48}a^{9}+\frac{33\cdots 37}{88\cdots 96}a^{8}-\frac{46\cdots 61}{88\cdots 96}a^{7}-\frac{33\cdots 03}{88\cdots 96}a^{6}+\frac{10\cdots 63}{29\cdots 32}a^{5}+\frac{18\cdots 26}{11\cdots 87}a^{4}+\frac{71\cdots 11}{11\cdots 87}a^{3}-\frac{38\cdots 79}{49\cdots 72}a^{2}-\frac{57\cdots 05}{88\cdots 96}a+\frac{14\cdots 05}{21\cdots 56}$, $\frac{11\cdots 45}{29\cdots 24}a^{15}-\frac{179524842118333}{14\cdots 28}a^{14}-\frac{40\cdots 67}{58\cdots 48}a^{13}+\frac{14\cdots 79}{71\cdots 64}a^{12}+\frac{42\cdots 51}{58\cdots 48}a^{11}-\frac{69\cdots 95}{58\cdots 48}a^{10}+\frac{66\cdots 43}{32\cdots 36}a^{9}+\frac{88\cdots 25}{29\cdots 24}a^{8}-\frac{55\cdots 47}{58\cdots 48}a^{7}-\frac{96\cdots 21}{73\cdots 56}a^{6}+\frac{34\cdots 55}{58\cdots 48}a^{5}+\frac{57\cdots 77}{14\cdots 12}a^{4}-\frac{19\cdots 93}{29\cdots 24}a^{3}-\frac{72\cdots 89}{73\cdots 56}a^{2}+\frac{30\cdots 91}{81\cdots 84}a-\frac{15\cdots 05}{14\cdots 28}$, $\frac{43\cdots 91}{58\cdots 64}a^{15}+\frac{17\cdots 15}{21\cdots 56}a^{14}-\frac{76\cdots 81}{58\cdots 64}a^{13}-\frac{22\cdots 01}{14\cdots 04}a^{12}+\frac{12\cdots 41}{88\cdots 96}a^{11}-\frac{28\cdots 73}{17\cdots 92}a^{10}-\frac{24\cdots 77}{88\cdots 96}a^{9}+\frac{24\cdots 15}{58\cdots 64}a^{8}-\frac{13\cdots 95}{58\cdots 64}a^{7}-\frac{10\cdots 31}{17\cdots 92}a^{6}+\frac{15\cdots 63}{19\cdots 88}a^{5}+\frac{23\cdots 39}{98\cdots 44}a^{4}-\frac{20\cdots 07}{88\cdots 96}a^{3}-\frac{33\cdots 09}{88\cdots 96}a^{2}+\frac{39\cdots 37}{17\cdots 92}a-\frac{57\cdots 03}{14\cdots 04}$, $\frac{18\cdots 01}{35\cdots 84}a^{15}-\frac{75\cdots 11}{28\cdots 08}a^{14}-\frac{13\cdots 49}{17\cdots 92}a^{13}+\frac{12\cdots 23}{28\cdots 08}a^{12}+\frac{25\cdots 15}{35\cdots 84}a^{11}-\frac{23\cdots 25}{44\cdots 48}a^{10}+\frac{81\cdots 47}{88\cdots 96}a^{9}-\frac{18\cdots 71}{35\cdots 84}a^{8}-\frac{22\cdots 39}{88\cdots 96}a^{7}+\frac{20\cdots 11}{39\cdots 76}a^{6}+\frac{16\cdots 59}{17\cdots 92}a^{5}-\frac{30\cdots 41}{19\cdots 88}a^{4}-\frac{67\cdots 11}{44\cdots 48}a^{3}+\frac{60\cdots 29}{17\cdots 92}a^{2}-\frac{68\cdots 81}{35\cdots 84}a+\frac{16\cdots 87}{43\cdots 12}$, $\frac{30\cdots 53}{35\cdots 84}a^{15}-\frac{15\cdots 75}{43\cdots 12}a^{14}-\frac{69\cdots 63}{35\cdots 84}a^{13}-\frac{86\cdots 55}{86\cdots 24}a^{12}+\frac{13\cdots 33}{58\cdots 64}a^{11}-\frac{52\cdots 97}{39\cdots 76}a^{10}-\frac{10\cdots 29}{17\cdots 92}a^{9}+\frac{21\cdots 37}{35\cdots 84}a^{8}-\frac{48\cdots 05}{35\cdots 84}a^{7}-\frac{18\cdots 03}{35\cdots 84}a^{6}+\frac{68\cdots 83}{35\cdots 84}a^{5}+\frac{57\cdots 91}{17\cdots 92}a^{4}-\frac{39\cdots 11}{19\cdots 88}a^{3}-\frac{75\cdots 29}{19\cdots 88}a^{2}+\frac{11\cdots 01}{35\cdots 84}a-\frac{36\cdots 05}{86\cdots 24}$, $\frac{13\cdots 97}{23\cdots 56}a^{15}-\frac{11\cdots 85}{17\cdots 48}a^{14}-\frac{83\cdots 79}{88\cdots 96}a^{13}+\frac{58\cdots 39}{17\cdots 48}a^{12}+\frac{66\cdots 43}{70\cdots 68}a^{11}-\frac{50\cdots 93}{35\cdots 84}a^{10}+\frac{28\cdots 15}{29\cdots 32}a^{9}+\frac{13\cdots 79}{70\cdots 68}a^{8}-\frac{39\cdots 71}{39\cdots 76}a^{7}-\frac{14\cdots 47}{70\cdots 68}a^{6}+\frac{11\cdots 97}{17\cdots 92}a^{5}+\frac{21\cdots 81}{35\cdots 84}a^{4}-\frac{73\cdots 07}{17\cdots 92}a^{3}-\frac{20\cdots 21}{35\cdots 84}a^{2}-\frac{12\cdots 59}{78\cdots 52}a+\frac{13\cdots 13}{43\cdots 12}$
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| Regulator: | \( 9547939.47561 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 9547939.47561 \cdot 1}{2\cdot\sqrt{66556997328875790787650649}}\cr\approx \mathstrut & 1.42141779492 \end{aligned}\] (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
| \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{-259}) \), 4.0.1813.1 x2, 4.2.9583.1 x2, \(\Q(\sqrt{-7}, \sqrt{37})\), 8.0.8158247197093.1, 8.0.5959274797.1, 8.0.4499860561.1 |
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: | 8.0.8158247197093.1, 8.0.5959274797.1 |
| Degree 16 sibling: | 16.4.91116529343230957588293738481.1 |
| Minimal sibling: | 8.0.5959274797.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{4}$ | ${\href{/padicField/3.2.0.1}{2} }^{8}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.2.4.6a1.3 | $x^{8} + 24 x^{7} + 228 x^{6} + 1080 x^{5} + 2646 x^{4} + 3240 x^{3} + 2052 x^{2} + 655 x + 109$ | $4$ | $2$ | $6$ | $Q_8$ | $$[\ ]_{4}^{2}$$ |
| 7.2.4.6a1.3 | $x^{8} + 24 x^{7} + 228 x^{6} + 1080 x^{5} + 2646 x^{4} + 3240 x^{3} + 2052 x^{2} + 655 x + 109$ | $4$ | $2$ | $6$ | $Q_8$ | $$[\ ]_{4}^{2}$$ | |
|
\(37\)
| 37.2.4.6a1.4 | $x^{8} + 132 x^{7} + 6542 x^{6} + 144540 x^{5} + 1212081 x^{4} + 289080 x^{3} + 26168 x^{2} + 1204 x + 1311$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ |
| 37.4.2.4a1.2 | $x^{8} + 12 x^{6} + 48 x^{5} + 40 x^{4} + 288 x^{3} + 600 x^{2} + 96 x + 41$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |