Normalized defining polynomial
\( x^{16} - 6 x^{15} - 6 x^{14} + 746 x^{13} + 11012 x^{12} - 109002 x^{11} - 760306 x^{10} + \cdots + 9729481076736 \)
Invariants
| Degree: | $16$ |
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| Signature: | $[0, 8]$ |
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| Discriminant: |
\(516750138955503312702912555388547984539201\)
\(\medspace = 41^{12}\cdot 73^{12}\)
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| Root discriminant: | \(404.65\) |
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| Galois root discriminant: | $41^{3/4}73^{3/4}\approx 404.65045925689327$ | ||
| Ramified primes: |
\(41\), \(73\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_4$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | 8.0.427634354612630929.2 | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{16}a^{5}-\frac{1}{16}a^{4}-\frac{3}{16}a^{3}+\frac{1}{16}a^{2}-\frac{3}{8}a-\frac{1}{2}$, $\frac{1}{32}a^{6}-\frac{1}{16}a^{3}+\frac{7}{32}a^{2}-\frac{7}{16}a+\frac{1}{4}$, $\frac{1}{64}a^{7}-\frac{1}{32}a^{5}-\frac{1}{16}a^{4}+\frac{5}{64}a^{3}-\frac{3}{16}a^{2}-\frac{1}{16}a+\frac{1}{4}$, $\frac{1}{128}a^{8}-\frac{1}{64}a^{6}+\frac{1}{128}a^{4}-\frac{3}{16}a^{3}-\frac{1}{4}a^{2}+\frac{3}{16}a+\frac{1}{4}$, $\frac{1}{512}a^{9}+\frac{1}{512}a^{8}-\frac{1}{256}a^{7}-\frac{3}{256}a^{6}+\frac{9}{512}a^{5}+\frac{17}{512}a^{4}-\frac{13}{64}a^{3}-\frac{19}{128}a^{2}+\frac{3}{16}a+\frac{1}{8}$, $\frac{1}{2048}a^{10}-\frac{3}{2048}a^{8}-\frac{3}{512}a^{7}+\frac{31}{2048}a^{6}+\frac{3}{256}a^{5}+\frac{103}{2048}a^{4}+\frac{21}{512}a^{3}+\frac{111}{512}a^{2}+\frac{29}{64}a+\frac{7}{32}$, $\frac{1}{4096}a^{11}-\frac{3}{4096}a^{9}-\frac{3}{1024}a^{8}+\frac{31}{4096}a^{7}+\frac{3}{512}a^{6}-\frac{25}{4096}a^{5}-\frac{11}{1024}a^{4}-\frac{177}{1024}a^{3}+\frac{1}{128}a^{2}+\frac{27}{64}a+\frac{1}{4}$, $\frac{1}{671744}a^{12}-\frac{9}{335872}a^{11}-\frac{131}{671744}a^{10}-\frac{295}{335872}a^{9}-\frac{769}{671744}a^{8}-\frac{1267}{335872}a^{7}+\frac{1287}{671744}a^{6}-\frac{7697}{335872}a^{5}+\frac{7159}{167936}a^{4}-\frac{16291}{83968}a^{3}-\frac{1147}{20992}a^{2}-\frac{909}{5248}a-\frac{471}{1312}$, $\frac{1}{10747904}a^{13}-\frac{7}{10747904}a^{12}-\frac{329}{10747904}a^{11}+\frac{593}{10747904}a^{10}+\frac{8485}{10747904}a^{9}+\frac{7375}{10747904}a^{8}-\frac{26587}{10747904}a^{7}-\frac{14357}{10747904}a^{6}-\frac{135949}{5373952}a^{5}-\frac{100121}{2686976}a^{4}-\frac{174605}{1343488}a^{3}-\frac{82181}{335872}a^{2}+\frac{35677}{83968}a+\frac{9251}{20992}$, $\frac{1}{2622488576}a^{14}+\frac{107}{2622488576}a^{13}-\frac{1799}{2622488576}a^{12}-\frac{250481}{2622488576}a^{11}+\frac{237591}{2622488576}a^{10}-\frac{1352567}{2622488576}a^{9}-\frac{9406477}{2622488576}a^{8}+\frac{224649}{42991616}a^{7}-\frac{1121077}{655622144}a^{6}+\frac{4124633}{327811072}a^{5}-\frac{9082487}{163905536}a^{4}+\frac{30225953}{163905536}a^{3}+\frac{7056285}{40976384}a^{2}-\frac{4215605}{10244096}a-\frac{911797}{2561024}$, $\frac{1}{59\cdots 08}a^{15}-\frac{56\cdots 65}{29\cdots 04}a^{14}-\frac{81\cdots 79}{29\cdots 04}a^{13}+\frac{10\cdots 05}{29\cdots 04}a^{12}+\frac{15\cdots 57}{14\cdots 52}a^{11}+\frac{71\cdots 15}{29\cdots 04}a^{10}+\frac{11\cdots 21}{99\cdots 68}a^{9}+\frac{34\cdots 91}{29\cdots 04}a^{8}+\frac{23\cdots 99}{59\cdots 08}a^{7}+\frac{60\cdots 65}{24\cdots 32}a^{6}+\frac{91\cdots 81}{24\cdots 92}a^{5}+\frac{25\cdots 83}{18\cdots 44}a^{4}+\frac{32\cdots 55}{37\cdots 88}a^{3}-\frac{60\cdots 23}{31\cdots 24}a^{2}-\frac{51\cdots 71}{23\cdots 68}a-\frac{69\cdots 57}{19\cdots 64}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$ |
Class group and class number
| Ideal class group: | $C_{4}\times C_{1188}$, which has order $4752$ (assuming GRH) |
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| Narrow class group: | $C_{4}\times C_{1188}$, which has order $4752$ (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: |
$\frac{85\cdots 75}{56\cdots 52}a^{15}-\frac{44\cdots 75}{28\cdots 76}a^{14}-\frac{34\cdots 25}{28\cdots 76}a^{13}+\frac{39\cdots 75}{28\cdots 76}a^{12}+\frac{16\cdots 25}{14\cdots 88}a^{11}-\frac{73\cdots 75}{28\cdots 76}a^{10}-\frac{25\cdots 75}{28\cdots 76}a^{9}+\frac{28\cdots 25}{28\cdots 76}a^{8}+\frac{13\cdots 25}{56\cdots 52}a^{7}-\frac{16\cdots 25}{14\cdots 88}a^{6}-\frac{64\cdots 75}{70\cdots 44}a^{5}+\frac{43\cdots 75}{17\cdots 36}a^{4}+\frac{96\cdots 25}{35\cdots 72}a^{3}-\frac{13\cdots 75}{88\cdots 68}a^{2}-\frac{88\cdots 25}{22\cdots 92}a-\frac{38\cdots 89}{55\cdots 48}$, $\frac{11\cdots 23}{64\cdots 48}a^{15}+\frac{96\cdots 05}{32\cdots 24}a^{14}+\frac{84\cdots 19}{32\cdots 24}a^{13}+\frac{92\cdots 11}{32\cdots 24}a^{12}+\frac{82\cdots 95}{16\cdots 12}a^{11}+\frac{21\cdots 49}{32\cdots 24}a^{10}+\frac{27\cdots 75}{10\cdots 08}a^{9}-\frac{36\cdots 43}{32\cdots 24}a^{8}-\frac{10\cdots 19}{64\cdots 48}a^{7}+\frac{99\cdots 43}{16\cdots 12}a^{6}+\frac{25\cdots 39}{26\cdots 52}a^{5}+\frac{62\cdots 23}{20\cdots 64}a^{4}-\frac{16\cdots 67}{40\cdots 28}a^{3}-\frac{25\cdots 33}{33\cdots 44}a^{2}+\frac{33\cdots 99}{25\cdots 08}a+\frac{63\cdots 21}{20\cdots 84}$, $\frac{12\cdots 99}{18\cdots 88}a^{15}-\frac{14\cdots 31}{93\cdots 44}a^{14}+\frac{10\cdots 87}{93\cdots 44}a^{13}+\frac{10\cdots 39}{93\cdots 44}a^{12}-\frac{73\cdots 71}{46\cdots 72}a^{11}-\frac{95\cdots 51}{93\cdots 44}a^{10}+\frac{21\cdots 13}{93\cdots 44}a^{9}-\frac{57\cdots 07}{93\cdots 44}a^{8}-\frac{84\cdots 59}{18\cdots 88}a^{7}+\frac{24\cdots 07}{46\cdots 72}a^{6}+\frac{46\cdots 09}{23\cdots 36}a^{5}-\frac{32\cdots 09}{58\cdots 84}a^{4}-\frac{24\cdots 11}{11\cdots 68}a^{3}+\frac{23\cdots 49}{29\cdots 92}a^{2}+\frac{38\cdots 11}{72\cdots 48}a-\frac{98\cdots 37}{18\cdots 12}$, $\frac{30\cdots 39}{36\cdots 72}a^{15}-\frac{22\cdots 31}{74\cdots 76}a^{14}-\frac{12\cdots 89}{74\cdots 76}a^{13}+\frac{44\cdots 95}{74\cdots 76}a^{12}+\frac{40\cdots 93}{37\cdots 88}a^{11}-\frac{50\cdots 07}{74\cdots 76}a^{10}-\frac{21\cdots 09}{24\cdots 92}a^{9}-\frac{10\cdots 03}{74\cdots 76}a^{8}+\frac{42\cdots 33}{36\cdots 72}a^{7}+\frac{31\cdots 45}{37\cdots 88}a^{6}-\frac{21\cdots 87}{62\cdots 48}a^{5}-\frac{26\cdots 53}{23\cdots 68}a^{4}+\frac{86\cdots 41}{93\cdots 72}a^{3}+\frac{27\cdots 15}{78\cdots 56}a^{2}-\frac{21\cdots 65}{58\cdots 92}a-\frac{90\cdots 43}{48\cdots 16}$, $\frac{92\cdots 65}{74\cdots 76}a^{15}-\frac{60\cdots 07}{37\cdots 88}a^{14}+\frac{27\cdots 63}{37\cdots 88}a^{13}+\frac{43\cdots 23}{37\cdots 88}a^{12}-\frac{20\cdots 43}{46\cdots 36}a^{11}-\frac{11\cdots 95}{37\cdots 88}a^{10}-\frac{16\cdots 65}{12\cdots 96}a^{9}+\frac{66\cdots 17}{37\cdots 88}a^{8}+\frac{23\cdots 99}{74\cdots 76}a^{7}-\frac{70\cdots 75}{18\cdots 44}a^{6}-\frac{48\cdots 55}{31\cdots 24}a^{5}+\frac{36\cdots 21}{23\cdots 68}a^{4}+\frac{87\cdots 19}{46\cdots 36}a^{3}-\frac{63\cdots 27}{39\cdots 28}a^{2}+\frac{28\cdots 25}{29\cdots 96}a+\frac{30\cdots 91}{24\cdots 08}$, $\frac{35\cdots 01}{29\cdots 04}a^{15}-\frac{47\cdots 15}{14\cdots 52}a^{14}-\frac{32\cdots 01}{14\cdots 52}a^{13}+\frac{73\cdots 91}{14\cdots 52}a^{12}-\frac{12\cdots 41}{37\cdots 88}a^{11}+\frac{32\cdots 97}{14\cdots 52}a^{10}-\frac{50\cdots 01}{12\cdots 24}a^{9}+\frac{48\cdots 33}{14\cdots 52}a^{8}+\frac{49\cdots 43}{29\cdots 04}a^{7}-\frac{35\cdots 83}{74\cdots 76}a^{6}-\frac{51\cdots 11}{12\cdots 96}a^{5}+\frac{24\cdots 01}{93\cdots 72}a^{4}+\frac{95\cdots 71}{18\cdots 44}a^{3}-\frac{48\cdots 27}{15\cdots 12}a^{2}+\frac{84\cdots 77}{11\cdots 84}a+\frac{23\cdots 75}{97\cdots 32}$, $\frac{40\cdots 91}{29\cdots 04}a^{15}-\frac{20\cdots 09}{14\cdots 52}a^{14}-\frac{44\cdots 03}{14\cdots 52}a^{13}+\frac{13\cdots 45}{14\cdots 52}a^{12}+\frac{72\cdots 09}{37\cdots 88}a^{11}-\frac{88\cdots 01}{14\cdots 52}a^{10}-\frac{76\cdots 55}{49\cdots 84}a^{9}-\frac{92\cdots 81}{14\cdots 52}a^{8}+\frac{50\cdots 65}{29\cdots 04}a^{7}+\frac{42\cdots 91}{74\cdots 76}a^{6}-\frac{49\cdots 61}{12\cdots 96}a^{5}-\frac{26\cdots 91}{93\cdots 72}a^{4}+\frac{14\cdots 73}{18\cdots 44}a^{3}+\frac{11\cdots 67}{15\cdots 12}a^{2}+\frac{15\cdots 71}{11\cdots 84}a+\frac{13\cdots 13}{97\cdots 32}$
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| Regulator: | \( 9410025743600000 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 9410025743600000 \cdot 4752}{2\cdot\sqrt{516750138955503312702912555388547984539201}}\cr\approx \mathstrut & 75550.2478961930 \end{aligned}\] (assuming GRH)
Galois group
$C_2^3.D_4$ (as 16T153):
| A solvable group of order 64 |
| The 13 conjugacy class representatives for $C_2^3.D_4$ |
| Character table for $C_2^3.D_4$ |
Intermediate fields
| \(\Q(\sqrt{73}) \), 4.0.218489.1, 4.4.653937577.1, 4.0.15949697.1, 8.0.427634354612630929.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 16.8.516750138955503312702912555388547984539201.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.1.0.1}{1} }^{16}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(41\)
| 41.2.4.6a1.2 | $x^{8} + 152 x^{7} + 8688 x^{6} + 222224 x^{5} + 2189320 x^{4} + 1333344 x^{3} + 312768 x^{2} + 32832 x + 1337$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ |
| 41.2.4.6a1.2 | $x^{8} + 152 x^{7} + 8688 x^{6} + 222224 x^{5} + 2189320 x^{4} + 1333344 x^{3} + 312768 x^{2} + 32832 x + 1337$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ | |
|
\(73\)
| 73.1.4.3a1.1 | $x^{4} + 73$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 73.1.4.3a1.1 | $x^{4} + 73$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 73.1.4.3a1.1 | $x^{4} + 73$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 73.1.4.3a1.1 | $x^{4} + 73$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |