Normalized defining polynomial
\( x^{16} - 42 x^{14} - 16 x^{13} + 582 x^{12} + 484 x^{11} - 2960 x^{10} - 4472 x^{9} + 2964 x^{8} + \cdots - 121 \)
Invariants
| Degree: | $16$ |
| |
| Signature: | $[12, 2]$ |
| |
| Discriminant: |
\(1272272789919032934400000000\)
\(\medspace = 2^{24}\cdot 5^{8}\cdot 97^{2}\cdot 379^{4}\)
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| |
| Root discriminant: | \(49.44\) |
| |
| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(5\), \(97\), \(379\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{5}$, $\frac{1}{10}a^{14}+\frac{1}{5}a^{13}+\frac{1}{10}a^{12}-\frac{1}{10}a^{11}-\frac{1}{10}a^{10}-\frac{1}{5}a^{9}+\frac{1}{5}a^{8}+\frac{2}{5}a^{7}-\frac{1}{2}a^{6}-\frac{1}{10}a^{4}-\frac{1}{2}a^{3}+\frac{1}{10}a^{2}+\frac{2}{5}$, $\frac{1}{13\cdots 50}a^{15}+\frac{18\cdots 93}{68\cdots 25}a^{14}-\frac{12\cdots 01}{13\cdots 50}a^{13}+\frac{96\cdots 19}{68\cdots 25}a^{12}-\frac{29\cdots 23}{13\cdots 05}a^{11}+\frac{63\cdots 09}{13\cdots 50}a^{10}+\frac{51\cdots 97}{68\cdots 25}a^{9}-\frac{47\cdots 53}{13\cdots 50}a^{8}+\frac{62\cdots 71}{13\cdots 50}a^{7}-\frac{10\cdots 95}{27\cdots 61}a^{6}+\frac{23\cdots 59}{13\cdots 50}a^{5}-\frac{21\cdots 37}{68\cdots 25}a^{4}-\frac{28\cdots 77}{68\cdots 25}a^{3}+\frac{51\cdots 79}{13\cdots 50}a^{2}+\frac{63\cdots 22}{68\cdots 25}a+\frac{46\cdots 81}{13\cdots 50}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
| |
| Narrow class group: | $C_{2}\times C_{2}\times C_{2}$, which has order $8$ (assuming GRH) |
|
Unit group
| Rank: | $13$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{23\cdots 78}{25\cdots 45}a^{15}-\frac{22\cdots 62}{25\cdots 45}a^{14}-\frac{68\cdots 23}{25\cdots 45}a^{13}+\frac{17\cdots 73}{51\cdots 90}a^{12}+\frac{90\cdots 97}{51\cdots 29}a^{11}-\frac{23\cdots 41}{51\cdots 90}a^{10}-\frac{65\cdots 78}{25\cdots 45}a^{9}+\frac{11\cdots 57}{51\cdots 90}a^{8}+\frac{24\cdots 08}{25\cdots 45}a^{7}-\frac{21\cdots 80}{51\cdots 29}a^{6}-\frac{27\cdots 88}{25\cdots 45}a^{5}+\frac{92\cdots 21}{51\cdots 90}a^{4}+\frac{82\cdots 83}{25\cdots 45}a^{3}-\frac{30\cdots 91}{51\cdots 90}a^{2}-\frac{50\cdots 93}{25\cdots 45}a+\frac{82\cdots 31}{51\cdots 90}$, $\frac{48\cdots 83}{68\cdots 25}a^{15}+\frac{25\cdots 11}{13\cdots 50}a^{14}-\frac{40\cdots 71}{13\cdots 50}a^{13}-\frac{22\cdots 57}{13\cdots 50}a^{12}+\frac{53\cdots 16}{13\cdots 05}a^{11}+\frac{23\cdots 92}{68\cdots 25}a^{10}-\frac{24\cdots 41}{13\cdots 50}a^{9}-\frac{33\cdots 53}{13\cdots 50}a^{8}+\frac{80\cdots 38}{68\cdots 25}a^{7}+\frac{44\cdots 27}{54\cdots 22}a^{6}+\frac{69\cdots 69}{13\cdots 50}a^{5}-\frac{25\cdots 69}{13\cdots 50}a^{4}+\frac{39\cdots 93}{68\cdots 25}a^{3}+\frac{10\cdots 12}{68\cdots 25}a^{2}-\frac{20\cdots 21}{13\cdots 50}a+\frac{36\cdots 61}{13\cdots 50}$, $\frac{56\cdots 24}{68\cdots 25}a^{15}+\frac{19\cdots 63}{13\cdots 50}a^{14}-\frac{49\cdots 03}{13\cdots 50}a^{13}-\frac{46\cdots 83}{68\cdots 25}a^{12}+\frac{13\cdots 64}{27\cdots 61}a^{11}+\frac{13\cdots 97}{13\cdots 50}a^{10}-\frac{31\cdots 83}{13\cdots 50}a^{9}-\frac{78\cdots 99}{13\cdots 50}a^{8}-\frac{26\cdots 51}{68\cdots 25}a^{7}+\frac{75\cdots 05}{54\cdots 22}a^{6}+\frac{29\cdots 07}{13\cdots 50}a^{5}-\frac{16\cdots 56}{68\cdots 25}a^{4}-\frac{22\cdots 21}{68\cdots 25}a^{3}+\frac{41\cdots 77}{13\cdots 50}a^{2}+\frac{26\cdots 87}{13\cdots 50}a-\frac{22\cdots 47}{13\cdots 50}$, $\frac{99\cdots 23}{68\cdots 25}a^{15}+\frac{46\cdots 41}{13\cdots 50}a^{14}-\frac{84\cdots 01}{13\cdots 50}a^{13}-\frac{26\cdots 71}{68\cdots 25}a^{12}+\frac{11\cdots 81}{13\cdots 05}a^{11}+\frac{12\cdots 79}{13\cdots 50}a^{10}-\frac{58\cdots 21}{13\cdots 50}a^{9}-\frac{11\cdots 43}{13\cdots 50}a^{8}+\frac{24\cdots 28}{68\cdots 25}a^{7}+\frac{15\cdots 65}{54\cdots 22}a^{6}+\frac{33\cdots 89}{13\cdots 50}a^{5}-\frac{30\cdots 82}{68\cdots 25}a^{4}-\frac{31\cdots 92}{68\cdots 25}a^{3}+\frac{52\cdots 69}{13\cdots 50}a^{2}+\frac{13\cdots 49}{13\cdots 50}a-\frac{47\cdots 59}{13\cdots 50}$, $\frac{23\cdots 99}{68\cdots 25}a^{15}-\frac{12\cdots 51}{68\cdots 25}a^{14}-\frac{19\cdots 33}{13\cdots 50}a^{13}+\frac{25\cdots 69}{13\cdots 50}a^{12}+\frac{26\cdots 79}{13\cdots 05}a^{11}+\frac{44\cdots 56}{68\cdots 25}a^{10}-\frac{13\cdots 03}{13\cdots 50}a^{9}-\frac{68\cdots 77}{68\cdots 25}a^{8}+\frac{86\cdots 94}{68\cdots 25}a^{7}+\frac{13\cdots 64}{27\cdots 61}a^{6}+\frac{29\cdots 57}{13\cdots 50}a^{5}-\frac{14\cdots 47}{13\cdots 50}a^{4}-\frac{89\cdots 96}{68\cdots 25}a^{3}+\frac{48\cdots 56}{68\cdots 25}a^{2}-\frac{50\cdots 13}{13\cdots 50}a-\frac{16\cdots 91}{68\cdots 25}$, $\frac{19\cdots 67}{68\cdots 25}a^{15}-\frac{23\cdots 88}{68\cdots 25}a^{14}-\frac{81\cdots 92}{68\cdots 25}a^{13}-\frac{37\cdots 83}{13\cdots 50}a^{12}+\frac{22\cdots 83}{13\cdots 05}a^{11}+\frac{72\cdots 28}{68\cdots 25}a^{10}-\frac{57\cdots 52}{68\cdots 25}a^{9}-\frac{70\cdots 76}{68\cdots 25}a^{8}+\frac{66\cdots 07}{68\cdots 25}a^{7}+\frac{11\cdots 55}{27\cdots 61}a^{6}+\frac{18\cdots 78}{68\cdots 25}a^{5}-\frac{12\cdots 41}{13\cdots 50}a^{4}-\frac{16\cdots 93}{68\cdots 25}a^{3}+\frac{58\cdots 43}{68\cdots 25}a^{2}-\frac{32\cdots 52}{68\cdots 25}a-\frac{90\cdots 73}{68\cdots 25}$, $\frac{14\cdots 06}{68\cdots 25}a^{15}+\frac{35\cdots 26}{68\cdots 25}a^{14}-\frac{62\cdots 86}{68\cdots 25}a^{13}-\frac{75\cdots 99}{13\cdots 50}a^{12}+\frac{17\cdots 97}{13\cdots 05}a^{11}+\frac{17\cdots 63}{13\cdots 50}a^{10}-\frac{41\cdots 81}{68\cdots 25}a^{9}-\frac{71\cdots 48}{68\cdots 25}a^{8}+\frac{32\cdots 66}{68\cdots 25}a^{7}+\frac{10\cdots 96}{27\cdots 61}a^{6}+\frac{20\cdots 79}{68\cdots 25}a^{5}-\frac{86\cdots 33}{13\cdots 50}a^{4}-\frac{27\cdots 99}{68\cdots 25}a^{3}+\frac{74\cdots 93}{13\cdots 50}a^{2}-\frac{35\cdots 86}{68\cdots 25}a-\frac{22\cdots 99}{68\cdots 25}$, $\frac{25\cdots 49}{68\cdots 25}a^{15}-\frac{53\cdots 61}{68\cdots 25}a^{14}-\frac{10\cdots 49}{68\cdots 25}a^{13}-\frac{40\cdots 51}{13\cdots 50}a^{12}+\frac{59\cdots 87}{27\cdots 10}a^{11}+\frac{20\cdots 57}{13\cdots 50}a^{10}-\frac{79\cdots 19}{68\cdots 25}a^{9}-\frac{10\cdots 22}{68\cdots 25}a^{8}+\frac{95\cdots 54}{68\cdots 25}a^{7}+\frac{18\cdots 51}{27\cdots 61}a^{6}+\frac{32\cdots 91}{68\cdots 25}a^{5}-\frac{16\cdots 27}{13\cdots 50}a^{4}-\frac{11\cdots 17}{13\cdots 50}a^{3}+\frac{13\cdots 17}{13\cdots 50}a^{2}-\frac{97\cdots 94}{68\cdots 25}a-\frac{48\cdots 06}{68\cdots 25}$, $\frac{51\cdots 41}{13\cdots 50}a^{15}-\frac{32\cdots 87}{68\cdots 25}a^{14}-\frac{10\cdots 08}{68\cdots 25}a^{13}+\frac{94\cdots 29}{68\cdots 25}a^{12}+\frac{28\cdots 32}{13\cdots 05}a^{11}-\frac{65\cdots 78}{68\cdots 25}a^{10}-\frac{75\cdots 73}{68\cdots 25}a^{9}-\frac{10\cdots 24}{68\cdots 25}a^{8}+\frac{29\cdots 11}{13\cdots 50}a^{7}+\frac{10\cdots 55}{27\cdots 61}a^{6}-\frac{16\cdots 28}{68\cdots 25}a^{5}-\frac{86\cdots 92}{68\cdots 25}a^{4}+\frac{70\cdots 93}{68\cdots 25}a^{3}+\frac{78\cdots 07}{68\cdots 25}a^{2}-\frac{96\cdots 98}{68\cdots 25}a+\frac{46\cdots 48}{68\cdots 25}$, $\frac{71\cdots 51}{13\cdots 50}a^{15}-\frac{17\cdots 17}{68\cdots 25}a^{14}-\frac{14\cdots 58}{68\cdots 25}a^{13}+\frac{41\cdots 93}{13\cdots 50}a^{12}+\frac{78\cdots 73}{27\cdots 10}a^{11}+\frac{10\cdots 29}{13\cdots 50}a^{10}-\frac{96\cdots 08}{68\cdots 25}a^{9}-\frac{16\cdots 93}{13\cdots 50}a^{8}+\frac{25\cdots 41}{13\cdots 50}a^{7}+\frac{16\cdots 96}{27\cdots 61}a^{6}+\frac{79\cdots 67}{68\cdots 25}a^{5}-\frac{21\cdots 29}{13\cdots 50}a^{4}+\frac{41\cdots 71}{13\cdots 50}a^{3}+\frac{19\cdots 59}{13\cdots 50}a^{2}-\frac{63\cdots 53}{68\cdots 25}a-\frac{19\cdots 49}{13\cdots 50}$, $\frac{56\cdots 19}{68\cdots 25}a^{15}+\frac{41\cdots 99}{68\cdots 25}a^{14}-\frac{43\cdots 03}{13\cdots 50}a^{13}-\frac{51\cdots 01}{13\cdots 50}a^{12}+\frac{49\cdots 38}{13\cdots 05}a^{11}+\frac{89\cdots 37}{13\cdots 50}a^{10}-\frac{11\cdots 63}{13\cdots 50}a^{9}-\frac{23\cdots 77}{68\cdots 25}a^{8}-\frac{28\cdots 66}{68\cdots 25}a^{7}+\frac{97\cdots 90}{27\cdots 61}a^{6}+\frac{15\cdots 17}{13\cdots 50}a^{5}+\frac{73\cdots 83}{13\cdots 50}a^{4}+\frac{37\cdots 24}{68\cdots 25}a^{3}+\frac{37\cdots 07}{13\cdots 50}a^{2}-\frac{33\cdots 53}{13\cdots 50}a+\frac{13\cdots 74}{68\cdots 25}$, $\frac{13\cdots 23}{68\cdots 25}a^{15}-\frac{26\cdots 57}{68\cdots 25}a^{14}-\frac{56\cdots 93}{68\cdots 25}a^{13}-\frac{10\cdots 61}{68\cdots 25}a^{12}+\frac{31\cdots 33}{27\cdots 10}a^{11}+\frac{52\cdots 67}{68\cdots 25}a^{10}-\frac{42\cdots 43}{68\cdots 25}a^{9}-\frac{56\cdots 14}{68\cdots 25}a^{8}+\frac{54\cdots 43}{68\cdots 25}a^{7}+\frac{97\cdots 88}{27\cdots 61}a^{6}+\frac{15\cdots 07}{68\cdots 25}a^{5}-\frac{46\cdots 42}{68\cdots 25}a^{4}-\frac{57\cdots 59}{13\cdots 50}a^{3}+\frac{39\cdots 57}{68\cdots 25}a^{2}-\frac{64\cdots 63}{68\cdots 25}a-\frac{23\cdots 02}{68\cdots 25}$, $\frac{82\cdots 23}{68\cdots 25}a^{15}+\frac{20\cdots 13}{68\cdots 25}a^{14}-\frac{75\cdots 31}{13\cdots 50}a^{13}-\frac{96\cdots 16}{68\cdots 25}a^{12}+\frac{22\cdots 88}{27\cdots 61}a^{11}+\frac{15\cdots 72}{68\cdots 25}a^{10}-\frac{28\cdots 33}{68\cdots 25}a^{9}-\frac{95\cdots 99}{68\cdots 25}a^{8}-\frac{17\cdots 77}{68\cdots 25}a^{7}+\frac{98\cdots 25}{27\cdots 61}a^{6}+\frac{89\cdots 89}{13\cdots 50}a^{5}-\frac{24\cdots 12}{68\cdots 25}a^{4}-\frac{81\cdots 17}{68\cdots 25}a^{3}+\frac{11\cdots 52}{68\cdots 25}a^{2}+\frac{28\cdots 87}{68\cdots 25}a-\frac{17\cdots 47}{68\cdots 25}$
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| Regulator: | \( 147673601.947 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{2}\cdot 147673601.947 \cdot 1}{2\cdot\sqrt{1272272789919032934400000000}}\cr\approx \mathstrut & 0.334736011855 \end{aligned}\] (assuming GRH)
Galois group
$C_2^7.A_4\wr C_2$ (as 16T1824):
| A solvable group of order 36864 |
| The 94 conjugacy class representatives for $C_2^7.A_4\wr C_2$ |
| Character table for $C_2^7.A_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 8.8.22982560000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 16 sibling: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | R | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.8.24b3.5 | $x^{16} + 8 x^{15} + 38 x^{14} + 126 x^{13} + 322 x^{12} + 658 x^{11} + 1108 x^{10} + 1558 x^{9} + 1851 x^{8} + 1862 x^{7} + 1590 x^{6} + 1146 x^{5} + 694 x^{4} + 346 x^{3} + 138 x^{2} + 40 x + 9$ | $8$ | $2$ | $24$ | 16T1286 | $$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2]_{3}^{6}$$ |
|
\(5\)
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 5.4.2.4a1.2 | $x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ | |
|
\(97\)
| 97.2.2.2a1.1 | $x^{4} + 192 x^{3} + 9226 x^{2} + 1057 x + 25$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
| 97.12.1.0a1.1 | $x^{12} + 30 x^{7} + 59 x^{6} + 81 x^{5} + 86 x^{3} + 78 x^{2} + 94 x + 5$ | $1$ | $12$ | $0$ | $C_{12}$ | $$[\ ]^{12}$$ | |
|
\(379\)
| $\Q_{379}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{379}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $3$ | $3$ | $1$ | $2$ | ||||
| Deg $3$ | $3$ | $1$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ |