Normalized defining polynomial
\( x^{16} - 16 x^{14} - 16 x^{13} + 108 x^{12} + 288 x^{11} - 208 x^{10} - 1840 x^{9} - 2710 x^{8} + \cdots - 47 \)
Invariants
| Degree: | $16$ |
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| Signature: | $[4, 6]$ |
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| Discriminant: |
\(484116351470433472610304\)
\(\medspace = 2^{66}\cdot 3^{8}\)
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| Root discriminant: | \(30.22\) |
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| Galois root discriminant: | $2^{137/32}3^{1/2}\approx 33.677922740001875$ | ||
| Ramified primes: |
\(2\), \(3\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2^2$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}+\frac{1}{4}a^{4}-\frac{1}{4}$, $\frac{1}{20}a^{13}-\frac{1}{10}a^{12}+\frac{1}{20}a^{10}-\frac{1}{20}a^{9}-\frac{3}{20}a^{8}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}-\frac{7}{20}a^{5}+\frac{1}{10}a^{4}+\frac{2}{5}a^{3}+\frac{9}{20}a^{2}-\frac{1}{20}a-\frac{3}{20}$, $\frac{1}{100}a^{14}+\frac{1}{50}a^{13}+\frac{3}{25}a^{12}+\frac{3}{50}a^{11}+\frac{2}{25}a^{10}-\frac{1}{50}a^{9}+\frac{1}{20}a^{8}-\frac{7}{25}a^{7}+\frac{29}{100}a^{6}+\frac{17}{50}a^{5}+\frac{4}{25}a^{4}-\frac{7}{50}a^{3}-\frac{1}{5}a^{2}-\frac{1}{50}a-\frac{47}{100}$, $\frac{1}{56\cdots 00}a^{15}-\frac{29\cdots 56}{14\cdots 25}a^{14}-\frac{19\cdots 47}{28\cdots 25}a^{13}-\frac{16\cdots 14}{14\cdots 25}a^{12}-\frac{30\cdots 23}{56\cdots 00}a^{11}+\frac{12\cdots 42}{28\cdots 25}a^{10}-\frac{39\cdots 43}{56\cdots 00}a^{9}-\frac{13\cdots 33}{56\cdots 00}a^{8}-\frac{19\cdots 93}{56\cdots 00}a^{7}-\frac{55\cdots 27}{56\cdots 50}a^{6}-\frac{98\cdots 59}{28\cdots 50}a^{5}-\frac{13\cdots 73}{56\cdots 50}a^{4}-\frac{24\cdots 81}{56\cdots 00}a^{3}-\frac{98\cdots 91}{28\cdots 50}a^{2}-\frac{43\cdots 49}{11\cdots 00}a+\frac{14\cdots 97}{56\cdots 00}$
| 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: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{47\cdots 64}{14\cdots 25}a^{15}-\frac{10\cdots 47}{28\cdots 50}a^{14}-\frac{14\cdots 32}{28\cdots 25}a^{13}+\frac{30\cdots 89}{56\cdots 00}a^{12}+\frac{10\cdots 81}{28\cdots 50}a^{11}+\frac{65\cdots 43}{11\cdots 00}a^{10}-\frac{39\cdots 29}{28\cdots 50}a^{9}-\frac{13\cdots 49}{28\cdots 50}a^{8}-\frac{53\cdots 02}{14\cdots 25}a^{7}+\frac{66\cdots 93}{56\cdots 50}a^{6}+\frac{51\cdots 98}{14\cdots 25}a^{5}+\frac{48\cdots 69}{11\cdots 00}a^{4}+\frac{73\cdots 07}{28\cdots 50}a^{3}+\frac{38\cdots 83}{56\cdots 00}a^{2}-\frac{77\cdots 47}{56\cdots 50}a-\frac{29\cdots 59}{28\cdots 50}$, $\frac{10\cdots 88}{14\cdots 25}a^{15}-\frac{25\cdots 23}{56\cdots 00}a^{14}-\frac{31\cdots 24}{28\cdots 25}a^{13}-\frac{71\cdots 78}{14\cdots 25}a^{12}+\frac{11\cdots 76}{14\cdots 25}a^{11}+\frac{92\cdots 03}{56\cdots 50}a^{10}-\frac{33\cdots 84}{14\cdots 25}a^{9}-\frac{67\cdots 41}{56\cdots 00}a^{8}-\frac{18\cdots 59}{14\cdots 25}a^{7}+\frac{25\cdots 77}{11\cdots 00}a^{6}+\frac{12\cdots 91}{14\cdots 25}a^{5}+\frac{36\cdots 12}{28\cdots 25}a^{4}+\frac{13\cdots 22}{14\cdots 25}a^{3}+\frac{87\cdots 93}{28\cdots 50}a^{2}+\frac{33\cdots 18}{28\cdots 25}a-\frac{15\cdots 81}{56\cdots 00}$, $\frac{13\cdots 87}{16\cdots 50}a^{15}+\frac{88\cdots 13}{16\cdots 50}a^{14}+\frac{21\cdots 68}{16\cdots 25}a^{13}+\frac{13\cdots 19}{33\cdots 00}a^{12}-\frac{15\cdots 99}{16\cdots 50}a^{11}-\frac{28\cdots 58}{16\cdots 25}a^{10}+\frac{26\cdots 83}{82\cdots 25}a^{9}+\frac{44\cdots 67}{33\cdots 00}a^{8}+\frac{19\cdots 41}{16\cdots 50}a^{7}-\frac{96\cdots 87}{33\cdots 50}a^{6}-\frac{83\cdots 42}{82\cdots 25}a^{5}-\frac{80\cdots 21}{66\cdots 00}a^{4}-\frac{11\cdots 03}{16\cdots 50}a^{3}-\frac{14\cdots 58}{82\cdots 25}a^{2}+\frac{15\cdots 29}{16\cdots 25}a+\frac{58\cdots 47}{33\cdots 00}$, $\frac{97\cdots 37}{56\cdots 00}a^{15}+\frac{60\cdots 29}{28\cdots 50}a^{14}+\frac{70\cdots 71}{28\cdots 25}a^{13}-\frac{19\cdots 63}{56\cdots 00}a^{12}-\frac{10\cdots 29}{56\cdots 00}a^{11}-\frac{30\cdots 09}{11\cdots 00}a^{10}+\frac{19\cdots 63}{28\cdots 50}a^{9}+\frac{64\cdots 23}{28\cdots 50}a^{8}+\frac{10\cdots 31}{56\cdots 00}a^{7}-\frac{30\cdots 13}{56\cdots 50}a^{6}-\frac{24\cdots 51}{14\cdots 25}a^{5}-\frac{24\cdots 59}{11\cdots 00}a^{4}-\frac{95\cdots 33}{56\cdots 00}a^{3}-\frac{50\cdots 41}{56\cdots 00}a^{2}-\frac{37\cdots 89}{11\cdots 10}a-\frac{15\cdots 27}{28\cdots 50}$, $\frac{12\cdots 37}{56\cdots 00}a^{15}-\frac{52\cdots 72}{14\cdots 25}a^{14}-\frac{36\cdots 61}{11\cdots 00}a^{13}+\frac{36\cdots 32}{14\cdots 25}a^{12}+\frac{70\cdots 37}{28\cdots 50}a^{11}+\frac{47\cdots 69}{28\cdots 25}a^{10}-\frac{34\cdots 83}{28\cdots 50}a^{9}-\frac{68\cdots 73}{28\cdots 50}a^{8}+\frac{24\cdots 09}{56\cdots 00}a^{7}+\frac{26\cdots 18}{28\cdots 25}a^{6}+\frac{94\cdots 59}{56\cdots 00}a^{5}+\frac{18\cdots 92}{28\cdots 25}a^{4}-\frac{11\cdots 11}{28\cdots 50}a^{3}-\frac{11\cdots 46}{14\cdots 25}a^{2}+\frac{20\cdots 71}{56\cdots 50}a-\frac{13\cdots 93}{28\cdots 50}$, $\frac{16\cdots 61}{56\cdots 00}a^{15}-\frac{19\cdots 51}{56\cdots 00}a^{14}+\frac{61\cdots 77}{11\cdots 00}a^{13}+\frac{52\cdots 11}{56\cdots 00}a^{12}-\frac{10\cdots 81}{28\cdots 50}a^{11}-\frac{13\cdots 07}{11\cdots 00}a^{10}+\frac{10\cdots 89}{28\cdots 50}a^{9}+\frac{40\cdots 63}{56\cdots 00}a^{8}+\frac{64\cdots 43}{56\cdots 00}a^{7}-\frac{68\cdots 83}{11\cdots 00}a^{6}-\frac{32\cdots 87}{56\cdots 00}a^{5}-\frac{11\cdots 37}{11\cdots 00}a^{4}-\frac{23\cdots 87}{28\cdots 50}a^{3}-\frac{20\cdots 23}{56\cdots 00}a^{2}-\frac{56\cdots 47}{11\cdots 10}a+\frac{23\cdots 63}{56\cdots 00}$, $\frac{50\cdots 67}{28\cdots 50}a^{15}+\frac{25\cdots 91}{56\cdots 00}a^{14}+\frac{17\cdots 51}{56\cdots 50}a^{13}+\frac{92\cdots 79}{56\cdots 00}a^{12}-\frac{67\cdots 09}{28\cdots 50}a^{11}-\frac{24\cdots 51}{56\cdots 50}a^{10}+\frac{22\cdots 31}{28\cdots 50}a^{9}+\frac{48\cdots 93}{14\cdots 25}a^{8}+\frac{73\cdots 81}{28\cdots 50}a^{7}-\frac{89\cdots 49}{11\cdots 00}a^{6}-\frac{71\cdots 69}{28\cdots 50}a^{5}-\frac{30\cdots 41}{11\cdots 00}a^{4}-\frac{22\cdots 23}{28\cdots 50}a^{3}+\frac{71\cdots 19}{28\cdots 50}a^{2}+\frac{17\cdots 53}{56\cdots 50}a+\frac{10\cdots 88}{14\cdots 25}$, $\frac{33\cdots 39}{56\cdots 00}a^{15}+\frac{10\cdots 21}{56\cdots 00}a^{14}+\frac{53\cdots 23}{56\cdots 50}a^{13}+\frac{37\cdots 29}{56\cdots 00}a^{12}-\frac{93\cdots 67}{14\cdots 25}a^{11}-\frac{85\cdots 53}{56\cdots 50}a^{10}+\frac{97\cdots 07}{56\cdots 00}a^{9}+\frac{14\cdots 53}{14\cdots 25}a^{8}+\frac{73\cdots 97}{56\cdots 00}a^{7}-\frac{19\cdots 51}{11\cdots 00}a^{6}-\frac{23\cdots 29}{28\cdots 50}a^{5}-\frac{13\cdots 39}{11\cdots 00}a^{4}-\frac{13\cdots 89}{14\cdots 25}a^{3}-\frac{11\cdots 51}{28\cdots 50}a^{2}-\frac{76\cdots 53}{11\cdots 00}a-\frac{19\cdots 57}{14\cdots 25}$, $\frac{88\cdots 31}{28\cdots 25}a^{15}+\frac{12\cdots 19}{22\cdots 20}a^{14}-\frac{29\cdots 19}{56\cdots 50}a^{13}-\frac{61\cdots 77}{11\cdots 00}a^{12}+\frac{10\cdots 56}{28\cdots 25}a^{11}+\frac{26\cdots 77}{28\cdots 25}a^{10}-\frac{41\cdots 27}{56\cdots 50}a^{9}-\frac{34\cdots 01}{56\cdots 50}a^{8}-\frac{47\cdots 13}{56\cdots 05}a^{7}+\frac{84\cdots 99}{11\cdots 00}a^{6}+\frac{26\cdots 61}{56\cdots 50}a^{5}+\frac{85\cdots 51}{11\cdots 00}a^{4}+\frac{18\cdots 63}{28\cdots 25}a^{3}+\frac{10\cdots 33}{28\cdots 25}a^{2}+\frac{75\cdots 99}{56\cdots 50}a+\frac{14\cdots 13}{56\cdots 50}$
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| Regulator: | \( 705835.1154494614 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 705835.1154494614 \cdot 1}{2\cdot\sqrt{484116351470433472610304}}\cr\approx \mathstrut & 0.499341408734770 \end{aligned}\] (assuming GRH)
Galois group
$C_2^5:C_4$ (as 16T259):
| A solvable group of order 128 |
| The 26 conjugacy class representatives for $C_2^5:C_4$ |
| Character table for $C_2^5:C_4$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.6.86973087744.1, 8.2.1073741824.1, 8.4.21743271936.4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
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 | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.16.66j1.880 | $x^{16} + 24 x^{12} + 16 x^{11} + 8 x^{10} + 16 x^{9} + 4 x^{8} + 8 x^{4} + 16 x^{3} + 2$ | $16$ | $1$ | $66$ | 16T259 | $$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$$ |
|
\(3\)
| 3.8.2.8a1.2 | $x^{16} + 4 x^{13} + 2 x^{12} + 8 x^{10} + 8 x^{9} + 5 x^{8} + 8 x^{7} + 12 x^{6} + 12 x^{5} + 8 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 7$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $$[\ ]_{2}^{8}$$ |