Normalized defining polynomial
\( x^{16} - 2 x^{15} - 28 x^{14} + 2 x^{13} + 244 x^{12} + 152 x^{11} - 862 x^{10} - 314 x^{9} + 2403 x^{8} + \cdots + 103 \)
Invariants
| Degree: | $16$ |
| |
| Signature: | $[12, 2]$ |
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| Discriminant: |
\(371353068328608385984364544\)
\(\medspace = 2^{24}\cdot 3^{2}\cdot 199^{8}\)
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| Root discriminant: | \(45.77\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(199\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{10}+\frac{1}{4}a^{6}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}+\frac{1}{4}a^{7}-\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{1}{4}a^{6}+\frac{1}{4}a^{4}-\frac{1}{4}$, $\frac{1}{20\cdots 16}a^{15}+\frac{235715717193499}{30\cdots 37}a^{14}-\frac{193181018550235}{51\cdots 29}a^{13}+\frac{28\cdots 82}{51\cdots 29}a^{12}+\frac{790225702371783}{12\cdots 48}a^{11}+\frac{37\cdots 02}{51\cdots 29}a^{10}+\frac{17\cdots 01}{15\cdots 32}a^{9}+\frac{11\cdots 55}{51\cdots 29}a^{8}+\frac{74\cdots 73}{20\cdots 16}a^{7}-\frac{76\cdots 00}{51\cdots 29}a^{6}-\frac{52\cdots 01}{15\cdots 32}a^{5}-\frac{27\cdots 89}{51\cdots 29}a^{4}-\frac{10\cdots 13}{51\cdots 29}a^{3}-\frac{20\cdots 31}{51\cdots 29}a^{2}+\frac{20\cdots 33}{20\cdots 16}a+\frac{13\cdots 06}{51\cdots 29}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
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: | $13$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{20\cdots 16}{51\cdots 29}a^{15}-\frac{16\cdots 57}{12\cdots 48}a^{14}-\frac{18\cdots 09}{20\cdots 16}a^{13}+\frac{69\cdots 38}{51\cdots 29}a^{12}+\frac{93\cdots 35}{12\cdots 48}a^{11}-\frac{98\cdots 35}{20\cdots 16}a^{10}-\frac{21\cdots 33}{79\cdots 66}a^{9}+\frac{53\cdots 43}{20\cdots 16}a^{8}+\frac{12\cdots 49}{20\cdots 16}a^{7}-\frac{18\cdots 73}{20\cdots 16}a^{6}-\frac{93\cdots 09}{79\cdots 66}a^{5}+\frac{19\cdots 97}{20\cdots 16}a^{4}+\frac{21\cdots 19}{20\cdots 16}a^{3}-\frac{18\cdots 88}{51\cdots 29}a^{2}-\frac{59\cdots 99}{20\cdots 16}a+\frac{68\cdots 91}{20\cdots 16}$, $\frac{69\cdots 95}{20\cdots 16}a^{15}-\frac{69\cdots 79}{60\cdots 74}a^{14}-\frac{16\cdots 51}{20\cdots 16}a^{13}+\frac{24\cdots 97}{20\cdots 16}a^{12}+\frac{40\cdots 51}{60\cdots 74}a^{11}-\frac{86\cdots 69}{20\cdots 16}a^{10}-\frac{37\cdots 23}{15\cdots 32}a^{9}+\frac{23\cdots 67}{10\cdots 58}a^{8}+\frac{53\cdots 99}{10\cdots 58}a^{7}-\frac{16\cdots 79}{20\cdots 16}a^{6}-\frac{16\cdots 09}{15\cdots 32}a^{5}+\frac{89\cdots 73}{10\cdots 58}a^{4}+\frac{18\cdots 25}{20\cdots 16}a^{3}-\frac{65\cdots 07}{20\cdots 16}a^{2}-\frac{26\cdots 45}{10\cdots 58}a+\frac{54\cdots 35}{20\cdots 16}$, $\frac{48\cdots 59}{20\cdots 16}a^{15}-\frac{23\cdots 36}{30\cdots 37}a^{14}-\frac{56\cdots 35}{10\cdots 58}a^{13}+\frac{16\cdots 19}{20\cdots 16}a^{12}+\frac{56\cdots 85}{12\cdots 48}a^{11}-\frac{55\cdots 81}{20\cdots 16}a^{10}-\frac{26\cdots 27}{15\cdots 32}a^{9}+\frac{76\cdots 42}{51\cdots 29}a^{8}+\frac{74\cdots 91}{20\cdots 16}a^{7}-\frac{11\cdots 03}{20\cdots 16}a^{6}-\frac{11\cdots 65}{15\cdots 32}a^{5}+\frac{28\cdots 60}{51\cdots 29}a^{4}+\frac{32\cdots 53}{51\cdots 29}a^{3}-\frac{40\cdots 73}{20\cdots 16}a^{2}-\frac{35\cdots 65}{20\cdots 16}a+\frac{36\cdots 59}{20\cdots 16}$, $\frac{44\cdots 61}{20\cdots 16}a^{15}-\frac{89\cdots 87}{12\cdots 48}a^{14}-\frac{10\cdots 97}{20\cdots 16}a^{13}+\frac{15\cdots 23}{20\cdots 16}a^{12}+\frac{12\cdots 62}{30\cdots 37}a^{11}-\frac{28\cdots 29}{10\cdots 58}a^{10}-\frac{23\cdots 27}{15\cdots 32}a^{9}+\frac{30\cdots 01}{20\cdots 16}a^{8}+\frac{16\cdots 29}{51\cdots 29}a^{7}-\frac{53\cdots 89}{10\cdots 58}a^{6}-\frac{95\cdots 25}{15\cdots 32}a^{5}+\frac{11\cdots 95}{20\cdots 16}a^{4}+\frac{10\cdots 85}{20\cdots 16}a^{3}-\frac{38\cdots 45}{20\cdots 16}a^{2}-\frac{67\cdots 90}{51\cdots 29}a+\frac{70\cdots 76}{51\cdots 29}$, $\frac{26\cdots 23}{20\cdots 16}a^{15}-\frac{13\cdots 27}{30\cdots 37}a^{14}-\frac{15\cdots 34}{51\cdots 29}a^{13}+\frac{22\cdots 40}{51\cdots 29}a^{12}+\frac{30\cdots 51}{12\cdots 48}a^{11}-\frac{15\cdots 49}{10\cdots 58}a^{10}-\frac{13\cdots 83}{15\cdots 32}a^{9}+\frac{85\cdots 81}{10\cdots 58}a^{8}+\frac{38\cdots 57}{20\cdots 16}a^{7}-\frac{30\cdots 61}{10\cdots 58}a^{6}-\frac{58\cdots 65}{15\cdots 32}a^{5}+\frac{31\cdots 21}{10\cdots 58}a^{4}+\frac{32\cdots 35}{10\cdots 58}a^{3}-\frac{11\cdots 65}{10\cdots 58}a^{2}-\frac{18\cdots 19}{20\cdots 16}a+\frac{90\cdots 37}{10\cdots 58}$, $\frac{41\cdots 79}{51\cdots 29}a^{15}-\frac{35\cdots 19}{12\cdots 48}a^{14}-\frac{35\cdots 41}{20\cdots 16}a^{13}+\frac{59\cdots 21}{20\cdots 16}a^{12}+\frac{16\cdots 99}{12\cdots 48}a^{11}-\frac{53\cdots 71}{51\cdots 29}a^{10}-\frac{31\cdots 19}{79\cdots 66}a^{9}+\frac{11\cdots 89}{20\cdots 16}a^{8}+\frac{13\cdots 25}{20\cdots 16}a^{7}-\frac{82\cdots 02}{51\cdots 29}a^{6}-\frac{90\cdots 29}{79\cdots 66}a^{5}+\frac{15\cdots 51}{20\cdots 16}a^{4}+\frac{16\cdots 83}{20\cdots 16}a^{3}+\frac{80\cdots 01}{20\cdots 16}a^{2}+\frac{55\cdots 93}{20\cdots 16}a-\frac{47\cdots 69}{10\cdots 58}$, $\frac{24\cdots 89}{20\cdots 16}a^{15}-\frac{48\cdots 95}{12\cdots 48}a^{14}-\frac{57\cdots 43}{20\cdots 16}a^{13}+\frac{83\cdots 57}{20\cdots 16}a^{12}+\frac{71\cdots 21}{30\cdots 37}a^{11}-\frac{71\cdots 78}{51\cdots 29}a^{10}-\frac{13\cdots 59}{15\cdots 32}a^{9}+\frac{15\cdots 51}{20\cdots 16}a^{8}+\frac{94\cdots 62}{51\cdots 29}a^{7}-\frac{14\cdots 40}{51\cdots 29}a^{6}-\frac{57\cdots 25}{15\cdots 32}a^{5}+\frac{59\cdots 65}{20\cdots 16}a^{4}+\frac{65\cdots 21}{20\cdots 16}a^{3}-\frac{21\cdots 99}{20\cdots 16}a^{2}-\frac{92\cdots 87}{10\cdots 58}a+\frac{46\cdots 42}{51\cdots 29}$, $\frac{155816065675877}{932388892343396}a^{15}-\frac{597188084445183}{932388892343396}a^{14}-\frac{16\cdots 41}{466194446171698}a^{13}+\frac{67\cdots 57}{932388892343396}a^{12}+\frac{27\cdots 23}{932388892343396}a^{11}-\frac{74\cdots 14}{233097223085849}a^{10}-\frac{95\cdots 21}{932388892343396}a^{9}+\frac{13\cdots 35}{932388892343396}a^{8}+\frac{17\cdots 65}{932388892343396}a^{7}-\frac{10\cdots 66}{233097223085849}a^{6}-\frac{29\cdots 95}{932388892343396}a^{5}+\frac{45\cdots 61}{932388892343396}a^{4}+\frac{67\cdots 66}{233097223085849}a^{3}-\frac{16\cdots 23}{932388892343396}a^{2}-\frac{80\cdots 75}{932388892343396}a+\frac{21\cdots 07}{233097223085849}$, $a+1$, $\frac{11\cdots 74}{51\cdots 29}a^{15}-\frac{93\cdots 29}{12\cdots 48}a^{14}-\frac{11\cdots 91}{20\cdots 16}a^{13}+\frac{16\cdots 69}{20\cdots 16}a^{12}+\frac{54\cdots 09}{12\cdots 48}a^{11}-\frac{28\cdots 63}{10\cdots 58}a^{10}-\frac{63\cdots 71}{39\cdots 33}a^{9}+\frac{30\cdots 39}{20\cdots 16}a^{8}+\frac{71\cdots 83}{20\cdots 16}a^{7}-\frac{55\cdots 21}{10\cdots 58}a^{6}-\frac{26\cdots 82}{39\cdots 33}a^{5}+\frac{11\cdots 33}{20\cdots 16}a^{4}+\frac{12\cdots 77}{20\cdots 16}a^{3}-\frac{40\cdots 19}{20\cdots 16}a^{2}-\frac{33\cdots 15}{20\cdots 16}a+\frac{86\cdots 96}{51\cdots 29}$, $\frac{10\cdots 27}{20\cdots 16}a^{15}-\frac{97\cdots 35}{60\cdots 74}a^{14}-\frac{25\cdots 65}{20\cdots 16}a^{13}+\frac{78\cdots 28}{51\cdots 29}a^{12}+\frac{64\cdots 39}{60\cdots 74}a^{11}-\frac{46\cdots 99}{10\cdots 58}a^{10}-\frac{61\cdots 29}{15\cdots 32}a^{9}+\frac{29\cdots 59}{10\cdots 58}a^{8}+\frac{92\cdots 53}{10\cdots 58}a^{7}-\frac{11\cdots 83}{10\cdots 58}a^{6}-\frac{29\cdots 95}{15\cdots 32}a^{5}+\frac{12\cdots 03}{10\cdots 58}a^{4}+\frac{32\cdots 73}{20\cdots 16}a^{3}-\frac{23\cdots 68}{51\cdots 29}a^{2}-\frac{45\cdots 85}{10\cdots 58}a+\frac{56\cdots 35}{10\cdots 58}$, $\frac{27\cdots 27}{10\cdots 58}a^{15}-\frac{54\cdots 69}{60\cdots 74}a^{14}-\frac{31\cdots 67}{51\cdots 29}a^{13}+\frac{46\cdots 75}{51\cdots 29}a^{12}+\frac{31\cdots 91}{60\cdots 74}a^{11}-\frac{16\cdots 36}{51\cdots 29}a^{10}-\frac{73\cdots 86}{39\cdots 33}a^{9}+\frac{17\cdots 57}{10\cdots 58}a^{8}+\frac{41\cdots 51}{10\cdots 58}a^{7}-\frac{31\cdots 43}{51\cdots 29}a^{6}-\frac{31\cdots 95}{39\cdots 33}a^{5}+\frac{65\cdots 57}{10\cdots 58}a^{4}+\frac{35\cdots 07}{51\cdots 29}a^{3}-\frac{23\cdots 11}{10\cdots 58}a^{2}-\frac{99\cdots 78}{51\cdots 29}a+\frac{20\cdots 07}{10\cdots 58}$, $\frac{11\cdots 11}{20\cdots 16}a^{15}-\frac{21\cdots 75}{12\cdots 48}a^{14}-\frac{13\cdots 57}{10\cdots 58}a^{13}+\frac{36\cdots 41}{20\cdots 16}a^{12}+\frac{13\cdots 59}{12\cdots 48}a^{11}-\frac{62\cdots 03}{10\cdots 58}a^{10}-\frac{63\cdots 31}{15\cdots 32}a^{9}+\frac{69\cdots 45}{20\cdots 16}a^{8}+\frac{18\cdots 61}{20\cdots 16}a^{7}-\frac{13\cdots 07}{10\cdots 58}a^{6}-\frac{28\cdots 29}{15\cdots 32}a^{5}+\frac{29\cdots 91}{20\cdots 16}a^{4}+\frac{16\cdots 95}{10\cdots 58}a^{3}-\frac{11\cdots 45}{20\cdots 16}a^{2}-\frac{10\cdots 13}{20\cdots 16}a+\frac{54\cdots 31}{10\cdots 58}$
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| Regulator: | \( 97496804.5048 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{2}\cdot 97496804.5048 \cdot 1}{2\cdot\sqrt{371353068328608385984364544}}\cr\approx \mathstrut & 0.409059574678 \end{aligned}\] (assuming GRH)
Galois group
$C_2^7:F_8:C_3$ (as 16T1800):
| A solvable group of order 21504 |
| The 36 conjugacy class representatives for $C_2^7:F_8:C_3$ |
| Character table for $C_2^7:F_8:C_3$ |
Intermediate fields
| 8.8.6423507767296.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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 | ${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.8.24a1.1 | $x^{16} + 8 x^{15} + 36 x^{14} + 112 x^{13} + 266 x^{12} + 504 x^{11} + 786 x^{10} + 1026 x^{9} + 1137 x^{8} + 1076 x^{7} + 874 x^{6} + 606 x^{5} + 356 x^{4} + 172 x^{3} + 66 x^{2} + 18 x + 5$ | $8$ | $2$ | $24$ | 16T712 | $$[\frac{12}{7}, \frac{12}{7}, \frac{12}{7}]_{7}^{6}$$ |
|
\(3\)
| 3.2.2.2a1.1 | $x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
| 3.12.1.0a1.1 | $x^{12} + x^{6} + x^{5} + x^{4} + x^{2} + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $$[\ ]^{12}$$ | |
|
\(199\)
| 199.2.1.0a1.1 | $x^{2} + 193 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 199.2.1.0a1.1 | $x^{2} + 193 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 199.2.3.4a1.2 | $x^{6} + 579 x^{5} + 111756 x^{4} + 7192531 x^{3} + 335268 x^{2} + 5211 x + 226$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ | |
| 199.2.3.4a1.2 | $x^{6} + 579 x^{5} + 111756 x^{4} + 7192531 x^{3} + 335268 x^{2} + 5211 x + 226$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ |