Normalized defining polynomial
\( x^{16} - 5 x^{15} + 20 x^{14} - 53 x^{13} + 425 x^{12} - 1761 x^{11} + 8234 x^{10} - 26439 x^{9} + \cdots + 5607424 \)
Invariants
| Degree: | $16$ |
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| Signature: | $[0, 8]$ |
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| Discriminant: |
\(1077123334824966013513439398801\)
\(\medspace = 41^{14}\cdot 73^{4}\)
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| Root discriminant: | \(75.34\) |
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| Galois root discriminant: | $41^{7/8}73^{1/2}\approx 220.21520635937404$ | ||
| Ramified primes: |
\(41\), \(73\)
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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 a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{10}+\frac{3}{8}a^{8}-\frac{3}{8}a^{7}+\frac{3}{8}a^{6}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{16}a^{12}-\frac{1}{16}a^{11}+\frac{3}{16}a^{9}-\frac{3}{16}a^{8}+\frac{3}{16}a^{7}-\frac{1}{8}a^{6}-\frac{7}{16}a^{5}+\frac{3}{16}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{13}-\frac{1}{64}a^{12}-\frac{5}{64}a^{10}-\frac{11}{64}a^{9}+\frac{3}{64}a^{8}+\frac{3}{32}a^{7}+\frac{1}{64}a^{6}+\frac{11}{64}a^{5}-\frac{1}{32}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{468224}a^{14}-\frac{673}{468224}a^{13}-\frac{167}{14632}a^{12}+\frac{197}{15104}a^{11}+\frac{17077}{468224}a^{10}+\frac{24099}{468224}a^{9}+\frac{15491}{234112}a^{8}-\frac{199583}{468224}a^{7}+\frac{96395}{468224}a^{6}+\frac{116911}{234112}a^{5}+\frac{6463}{14632}a^{4}-\frac{655}{3658}a^{3}-\frac{5793}{29264}a^{2}+\frac{2559}{7316}a+\frac{564}{1829}$, $\frac{1}{14\cdots 28}a^{15}-\frac{77\cdots 77}{14\cdots 28}a^{14}+\frac{59\cdots 97}{37\cdots 32}a^{13}-\frac{53\cdots 17}{14\cdots 28}a^{12}+\frac{86\cdots 45}{14\cdots 28}a^{11}+\frac{15\cdots 43}{14\cdots 28}a^{10}-\frac{13\cdots 99}{74\cdots 64}a^{9}-\frac{15\cdots 91}{14\cdots 28}a^{8}+\frac{63\cdots 15}{14\cdots 28}a^{7}+\frac{76\cdots 85}{74\cdots 64}a^{6}+\frac{16\cdots 37}{18\cdots 16}a^{5}+\frac{91\cdots 89}{23\cdots 52}a^{4}+\frac{12\cdots 51}{92\cdots 08}a^{3}+\frac{99\cdots 93}{58\cdots 88}a^{2}-\frac{21\cdots 79}{58\cdots 88}a+\frac{47\cdots 60}{39\cdots 31}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{33}$, which has order $33$ (assuming GRH) |
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| Narrow class group: | $C_{33}$, which has order $33$ (assuming GRH) |
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| Relative class number: | $33$ (assuming GRH) |
Unit group
| Rank: | $7$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{78\cdots 87}{35\cdots 52}a^{15}-\frac{70\cdots 01}{35\cdots 52}a^{14}+\frac{12\cdots 17}{17\cdots 76}a^{13}-\frac{57\cdots 51}{35\cdots 52}a^{12}+\frac{30\cdots 85}{35\cdots 52}a^{11}-\frac{22\cdots 61}{35\cdots 52}a^{10}+\frac{23\cdots 43}{89\cdots 88}a^{9}-\frac{31\cdots 01}{35\cdots 52}a^{8}+\frac{77\cdots 31}{35\cdots 52}a^{7}-\frac{56\cdots 03}{89\cdots 88}a^{6}+\frac{12\cdots 69}{89\cdots 88}a^{5}-\frac{30\cdots 71}{11\cdots 36}a^{4}+\frac{76\cdots 57}{22\cdots 72}a^{3}-\frac{22\cdots 43}{36\cdots 56}a^{2}+\frac{25\cdots 21}{28\cdots 84}a-\frac{86\cdots 28}{70\cdots 71}$, $\frac{68\cdots 63}{17\cdots 76}a^{15}-\frac{36\cdots 81}{17\cdots 76}a^{14}+\frac{56\cdots 57}{89\cdots 88}a^{13}-\frac{28\cdots 79}{17\cdots 76}a^{12}+\frac{24\cdots 05}{17\cdots 76}a^{11}-\frac{11\cdots 61}{17\cdots 76}a^{10}+\frac{11\cdots 35}{44\cdots 44}a^{9}-\frac{15\cdots 61}{17\cdots 76}a^{8}+\frac{41\cdots 63}{17\cdots 76}a^{7}-\frac{29\cdots 67}{44\cdots 44}a^{6}+\frac{64\cdots 47}{44\cdots 44}a^{5}-\frac{15\cdots 29}{56\cdots 68}a^{4}+\frac{39\cdots 17}{11\cdots 36}a^{3}-\frac{11\cdots 19}{18\cdots 28}a^{2}+\frac{12\cdots 33}{14\cdots 42}a-\frac{16\cdots 17}{70\cdots 71}$, $\frac{37\cdots 87}{35\cdots 52}a^{15}-\frac{28\cdots 65}{35\cdots 52}a^{14}+\frac{49\cdots 65}{17\cdots 76}a^{13}-\frac{22\cdots 11}{35\cdots 52}a^{12}+\frac{14\cdots 65}{35\cdots 52}a^{11}-\frac{92\cdots 05}{35\cdots 52}a^{10}+\frac{94\cdots 99}{89\cdots 88}a^{9}-\frac{12\cdots 25}{35\cdots 52}a^{8}+\frac{31\cdots 19}{35\cdots 52}a^{7}-\frac{22\cdots 43}{89\cdots 88}a^{6}+\frac{50\cdots 01}{89\cdots 88}a^{5}-\frac{12\cdots 71}{11\cdots 36}a^{4}+\frac{30\cdots 05}{22\cdots 72}a^{3}-\frac{89\cdots 67}{36\cdots 56}a^{2}+\frac{10\cdots 29}{28\cdots 84}a-\frac{56\cdots 43}{70\cdots 71}$, $\frac{28\cdots 65}{74\cdots 64}a^{15}-\frac{55\cdots 63}{74\cdots 64}a^{14}+\frac{95\cdots 75}{37\cdots 32}a^{13}-\frac{50\cdots 37}{74\cdots 64}a^{12}+\frac{11\cdots 91}{74\cdots 64}a^{11}-\frac{18\cdots 19}{74\cdots 64}a^{10}+\frac{15\cdots 11}{18\cdots 16}a^{9}-\frac{25\cdots 95}{74\cdots 64}a^{8}+\frac{73\cdots 21}{74\cdots 64}a^{7}-\frac{52\cdots 75}{18\cdots 16}a^{6}+\frac{12\cdots 37}{18\cdots 16}a^{5}-\frac{71\cdots 93}{46\cdots 04}a^{4}+\frac{79\cdots 91}{46\cdots 04}a^{3}-\frac{60\cdots 95}{23\cdots 52}a^{2}+\frac{67\cdots 98}{14\cdots 47}a-\frac{26\cdots 23}{39\cdots 31}$, $\frac{40\cdots 01}{74\cdots 64}a^{15}+\frac{54\cdots 19}{74\cdots 64}a^{14}-\frac{11\cdots 65}{46\cdots 04}a^{13}+\frac{48\cdots 71}{74\cdots 64}a^{12}-\frac{93\cdots 15}{74\cdots 64}a^{11}+\frac{18\cdots 91}{74\cdots 64}a^{10}-\frac{27\cdots 61}{37\cdots 32}a^{9}+\frac{24\cdots 05}{74\cdots 64}a^{8}-\frac{68\cdots 05}{74\cdots 64}a^{7}+\frac{10\cdots 87}{37\cdots 32}a^{6}-\frac{58\cdots 81}{92\cdots 08}a^{5}+\frac{66\cdots 49}{46\cdots 04}a^{4}-\frac{73\cdots 77}{46\cdots 04}a^{3}+\frac{28\cdots 03}{11\cdots 76}a^{2}-\frac{24\cdots 65}{58\cdots 88}a+\frac{24\cdots 97}{39\cdots 31}$, $\frac{21\cdots 31}{74\cdots 64}a^{15}-\frac{41\cdots 69}{74\cdots 64}a^{14}+\frac{65\cdots 71}{37\cdots 32}a^{13}-\frac{38\cdots 79}{74\cdots 64}a^{12}+\frac{10\cdots 29}{74\cdots 64}a^{11}-\frac{12\cdots 53}{74\cdots 64}a^{10}+\frac{52\cdots 87}{92\cdots 08}a^{9}-\frac{18\cdots 49}{74\cdots 64}a^{8}+\frac{41\cdots 15}{74\cdots 64}a^{7}-\frac{64\cdots 23}{46\cdots 04}a^{6}+\frac{13\cdots 33}{59\cdots 36}a^{5}-\frac{22\cdots 71}{11\cdots 76}a^{4}-\frac{50\cdots 73}{78\cdots 56}a^{3}+\frac{67\cdots 65}{23\cdots 52}a^{2}-\frac{34\cdots 59}{58\cdots 88}a+\frac{20\cdots 95}{39\cdots 31}$, $\frac{29\cdots 23}{74\cdots 64}a^{15}-\frac{35\cdots 97}{74\cdots 64}a^{14}+\frac{87\cdots 37}{37\cdots 32}a^{13}-\frac{25\cdots 97}{12\cdots 96}a^{12}+\frac{94\cdots 17}{74\cdots 64}a^{11}-\frac{12\cdots 57}{74\cdots 64}a^{10}+\frac{26\cdots 57}{18\cdots 16}a^{9}-\frac{14\cdots 53}{74\cdots 64}a^{8}+\frac{67\cdots 15}{74\cdots 64}a^{7}-\frac{27\cdots 85}{18\cdots 16}a^{6}+\frac{26\cdots 79}{59\cdots 36}a^{5}-\frac{13\cdots 49}{11\cdots 76}a^{4}+\frac{60\cdots 01}{46\cdots 04}a^{3}+\frac{35\cdots 75}{23\cdots 52}a^{2}+\frac{15\cdots 63}{58\cdots 88}a+\frac{79\cdots 42}{39\cdots 31}$
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| Regulator: | \( 4516212.78969 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 4516212.78969 \cdot 33}{2\cdot\sqrt{1077123334824966013513439398801}}\cr\approx \mathstrut & 0.174407264376 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2:C_8$ (as 16T24):
| A solvable group of order 32 |
| The 20 conjugacy class representatives for $C_2^2 : C_8$ |
| Character table for $C_2^2 : C_8$ |
Intermediate fields
| \(\Q(\sqrt{41}) \), 4.4.68921.1, 4.0.5031233.1, 4.0.122713.1, 8.0.194754273881.1, 8.8.1037845525511849.1, 8.0.25313305500289.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 32 |
| Degree 16 sibling: | 16.0.30588408049083077668563908786045909041.1 |
| 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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(41\)
| 41.2.8.14a1.2 | $x^{16} + 304 x^{15} + 40480 x^{14} + 3085600 x^{13} + 147416080 x^{12} + 4529584192 x^{11} + 87831092608 x^{10} + 996302227840 x^{9} + 5391168776800 x^{8} + 5977813367040 x^{7} + 3161919333888 x^{6} + 978390185472 x^{5} + 191051239680 x^{4} + 23993625600 x^{3} + 1888634880 x^{2} + 85100544 x + 1679657$ | $8$ | $2$ | $14$ | $C_8\times C_2$ | $$[\ ]_{8}^{2}$$ |
|
\(73\)
| 73.4.1.0a1.1 | $x^{4} + 16 x^{2} + 56 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |
| 73.4.1.0a1.1 | $x^{4} + 16 x^{2} + 56 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
| 73.4.2.4a1.2 | $x^{8} + 32 x^{6} + 112 x^{5} + 266 x^{4} + 1792 x^{3} + 3296 x^{2} + 560 x + 98$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |