Normalized defining polynomial
\( x^{16} - 16 x^{14} - 32 x^{13} + 36 x^{12} + 384 x^{11} + 848 x^{10} - 800 x^{9} - 6034 x^{8} + \cdots - 527 \)
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}-\frac{1}{4}a^{4}+\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{9}-\frac{1}{4}a^{5}+\frac{1}{4}a$, $\frac{1}{20}a^{14}+\frac{1}{10}a^{13}-\frac{1}{20}a^{12}-\frac{1}{10}a^{11}+\frac{1}{20}a^{10}+\frac{1}{5}a^{9}-\frac{3}{20}a^{8}-\frac{1}{10}a^{7}-\frac{1}{20}a^{6}+\frac{9}{20}a^{4}+\frac{3}{20}a^{2}-\frac{1}{10}a+\frac{7}{20}$, $\frac{1}{53\cdots 20}a^{15}+\frac{18\cdots 53}{10\cdots 04}a^{14}+\frac{12\cdots 65}{10\cdots 04}a^{13}-\frac{32\cdots 11}{10\cdots 04}a^{12}-\frac{85\cdots 69}{10\cdots 04}a^{11}+\frac{13\cdots 37}{53\cdots 20}a^{10}-\frac{22\cdots 01}{53\cdots 20}a^{9}-\frac{38\cdots 91}{53\cdots 20}a^{8}-\frac{90\cdots 07}{53\cdots 20}a^{7}+\frac{92\cdots 87}{53\cdots 20}a^{6}+\frac{41\cdots 89}{53\cdots 20}a^{5}-\frac{16\cdots 93}{53\cdots 20}a^{4}-\frac{12\cdots 77}{53\cdots 20}a^{3}+\frac{36\cdots 07}{53\cdots 20}a^{2}-\frac{25\cdots 89}{53\cdots 20}a-\frac{11\cdots 57}{31\cdots 60}$
| 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: |
$\frac{15\cdots 83}{53\cdots 20}a^{15}-\frac{15\cdots 17}{53\cdots 20}a^{14}-\frac{56\cdots 31}{13\cdots 05}a^{13}-\frac{23\cdots 63}{53\cdots 20}a^{12}+\frac{84\cdots 89}{53\cdots 20}a^{11}+\frac{49\cdots 79}{53\cdots 20}a^{10}+\frac{36\cdots 07}{26\cdots 10}a^{9}-\frac{20\cdots 37}{53\cdots 20}a^{8}-\frac{70\cdots 87}{53\cdots 20}a^{7}-\frac{21\cdots 57}{53\cdots 20}a^{6}+\frac{35\cdots 73}{13\cdots 05}a^{5}+\frac{19\cdots 13}{53\cdots 20}a^{4}+\frac{85\cdots 39}{53\cdots 20}a^{3}-\frac{29\cdots 81}{10\cdots 04}a^{2}-\frac{10\cdots 29}{26\cdots 10}a-\frac{26\cdots 73}{31\cdots 60}$, $\frac{15\cdots 21}{26\cdots 10}a^{15}-\frac{31\cdots 01}{26\cdots 01}a^{14}-\frac{76\cdots 61}{10\cdots 04}a^{13}-\frac{19\cdots 21}{10\cdots 04}a^{12}+\frac{83\cdots 02}{26\cdots 01}a^{11}+\frac{20\cdots 46}{13\cdots 05}a^{10}+\frac{69\cdots 23}{53\cdots 20}a^{9}-\frac{46\cdots 37}{53\cdots 20}a^{8}-\frac{45\cdots 67}{26\cdots 10}a^{7}+\frac{18\cdots 97}{26\cdots 10}a^{6}+\frac{21\cdots 13}{53\cdots 20}a^{5}+\frac{18\cdots 39}{53\cdots 20}a^{4}+\frac{11\cdots 94}{13\cdots 05}a^{3}-\frac{10\cdots 13}{26\cdots 10}a^{2}-\frac{20\cdots 63}{53\cdots 20}a+\frac{21\cdots 11}{31\cdots 60}$, $\frac{40\cdots 37}{15\cdots 30}a^{15}+\frac{14\cdots 88}{79\cdots 65}a^{14}+\frac{31\cdots 36}{79\cdots 65}a^{13}+\frac{17\cdots 33}{31\cdots 60}a^{12}-\frac{20\cdots 27}{15\cdots 30}a^{11}-\frac{70\cdots 19}{79\cdots 65}a^{10}-\frac{12\cdots 82}{79\cdots 65}a^{9}+\frac{10\cdots 83}{31\cdots 60}a^{8}+\frac{21\cdots 77}{15\cdots 30}a^{7}+\frac{12\cdots 80}{15\cdots 53}a^{6}-\frac{18\cdots 24}{79\cdots 65}a^{5}-\frac{27\cdots 53}{63\cdots 12}a^{4}-\frac{38\cdots 81}{15\cdots 30}a^{3}+\frac{35\cdots 57}{79\cdots 65}a^{2}+\frac{45\cdots 98}{79\cdots 65}a+\frac{10\cdots 33}{63\cdots 12}$, $\frac{34\cdots 48}{13\cdots 05}a^{15}+\frac{11\cdots 03}{53\cdots 20}a^{14}+\frac{21\cdots 01}{53\cdots 20}a^{13}+\frac{66\cdots 03}{13\cdots 05}a^{12}-\frac{37\cdots 33}{26\cdots 10}a^{11}-\frac{47\cdots 01}{53\cdots 20}a^{10}-\frac{78\cdots 01}{53\cdots 20}a^{9}+\frac{45\cdots 47}{13\cdots 05}a^{8}+\frac{35\cdots 89}{26\cdots 10}a^{7}+\frac{33\cdots 03}{53\cdots 20}a^{6}-\frac{13\cdots 83}{53\cdots 20}a^{5}-\frac{10\cdots 91}{26\cdots 10}a^{4}-\frac{28\cdots 84}{13\cdots 05}a^{3}+\frac{13\cdots 51}{10\cdots 04}a^{2}+\frac{26\cdots 87}{53\cdots 20}a+\frac{22\cdots 61}{15\cdots 30}$, $\frac{28\cdots 67}{53\cdots 20}a^{15}+\frac{52\cdots 56}{13\cdots 05}a^{14}+\frac{21\cdots 49}{26\cdots 10}a^{13}+\frac{57\cdots 91}{53\cdots 20}a^{12}-\frac{14\cdots 63}{53\cdots 20}a^{11}-\frac{97\cdots 29}{53\cdots 02}a^{10}-\frac{41\cdots 88}{13\cdots 05}a^{9}+\frac{70\cdots 99}{10\cdots 04}a^{8}+\frac{14\cdots 31}{53\cdots 20}a^{7}+\frac{39\cdots 11}{26\cdots 10}a^{6}-\frac{12\cdots 09}{26\cdots 10}a^{5}-\frac{45\cdots 53}{53\cdots 20}a^{4}-\frac{25\cdots 61}{53\cdots 20}a^{3}+\frac{49\cdots 92}{13\cdots 05}a^{2}+\frac{29\cdots 96}{26\cdots 01}a+\frac{10\cdots 83}{31\cdots 60}$, $\frac{10\cdots 03}{10\cdots 04}a^{15}+\frac{13\cdots 09}{13\cdots 05}a^{14}+\frac{37\cdots 61}{26\cdots 10}a^{13}+\frac{82\cdots 09}{53\cdots 20}a^{12}-\frac{26\cdots 87}{53\cdots 20}a^{11}-\frac{41\cdots 26}{13\cdots 05}a^{10}-\frac{63\cdots 74}{13\cdots 05}a^{9}+\frac{66\cdots 47}{53\cdots 20}a^{8}+\frac{23\cdots 83}{53\cdots 20}a^{7}+\frac{44\cdots 67}{26\cdots 10}a^{6}-\frac{44\cdots 47}{53\cdots 02}a^{5}-\frac{69\cdots 51}{53\cdots 20}a^{4}-\frac{71\cdots 37}{10\cdots 04}a^{3}+\frac{80\cdots 49}{26\cdots 10}a^{2}+\frac{21\cdots 77}{13\cdots 05}a+\frac{14\cdots 31}{31\cdots 60}$, $\frac{80\cdots 93}{53\cdots 20}a^{15}-\frac{50\cdots 63}{26\cdots 10}a^{14}-\frac{30\cdots 03}{13\cdots 05}a^{13}-\frac{25\cdots 91}{13\cdots 05}a^{12}+\frac{47\cdots 77}{53\cdots 20}a^{11}+\frac{12\cdots 06}{26\cdots 01}a^{10}+\frac{86\cdots 32}{13\cdots 05}a^{9}-\frac{11\cdots 95}{53\cdots 02}a^{8}-\frac{35\cdots 49}{53\cdots 20}a^{7}-\frac{10\cdots 17}{13\cdots 05}a^{6}+\frac{19\cdots 08}{13\cdots 05}a^{5}+\frac{45\cdots 21}{26\cdots 10}a^{4}+\frac{28\cdots 39}{53\cdots 20}a^{3}-\frac{51\cdots 51}{26\cdots 10}a^{2}-\frac{42\cdots 47}{26\cdots 01}a-\frac{16\cdots 93}{79\cdots 65}$, $\frac{68\cdots 21}{10\cdots 04}a^{15}+\frac{17\cdots 47}{26\cdots 10}a^{14}+\frac{12\cdots 82}{13\cdots 05}a^{13}+\frac{58\cdots 01}{53\cdots 20}a^{12}-\frac{17\cdots 63}{53\cdots 20}a^{11}-\frac{28\cdots 84}{13\cdots 05}a^{10}-\frac{89\cdots 17}{26\cdots 10}a^{9}+\frac{44\cdots 33}{53\cdots 20}a^{8}+\frac{16\cdots 17}{53\cdots 20}a^{7}+\frac{17\cdots 59}{13\cdots 05}a^{6}-\frac{15\cdots 01}{26\cdots 01}a^{5}-\frac{49\cdots 59}{53\cdots 20}a^{4}-\frac{53\cdots 69}{10\cdots 04}a^{3}+\frac{11\cdots 71}{26\cdots 10}a^{2}+\frac{28\cdots 61}{26\cdots 10}a+\frac{10\cdots 29}{31\cdots 60}$, $\frac{22\cdots 33}{26\cdots 10}a^{15}+\frac{35\cdots 23}{53\cdots 20}a^{14}+\frac{34\cdots 41}{53\cdots 20}a^{13}-\frac{39\cdots 43}{53\cdots 20}a^{12}-\frac{35\cdots 13}{26\cdots 10}a^{11}+\frac{31\cdots 01}{53\cdots 20}a^{10}+\frac{78\cdots 33}{53\cdots 20}a^{9}+\frac{20\cdots 77}{53\cdots 20}a^{8}-\frac{11\cdots 47}{26\cdots 10}a^{7}-\frac{27\cdots 77}{10\cdots 04}a^{6}-\frac{73\cdots 49}{53\cdots 20}a^{5}+\frac{49\cdots 57}{10\cdots 04}a^{4}+\frac{18\cdots 51}{26\cdots 10}a^{3}+\frac{12\cdots 97}{53\cdots 20}a^{2}-\frac{60\cdots 97}{53\cdots 20}a-\frac{47\cdots 03}{63\cdots 12}$
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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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.16.66j1.879 | $x^{16} + 8 x^{12} + 16 x^{11} + 8 x^{10} + 16 x^{9} + 4 x^{8} + 8 x^{4} + 16 x^{3} + 2$ | $16$ | $1$ | $66$ | 16T259 | $$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$$ |
|
\(3\)
| 3.8.2.8a1.2 | $x^{16} + 4 x^{13} + 2 x^{12} + 8 x^{10} + 8 x^{9} + 5 x^{8} + 8 x^{7} + 12 x^{6} + 12 x^{5} + 8 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 7$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $$[\ ]_{2}^{8}$$ |