Normalized defining polynomial
\( x^{16} - 8 x^{14} - 16 x^{12} - 32 x^{11} + 288 x^{10} + 192 x^{9} + 816 x^{8} - 224 x^{7} + 192 x^{6} + \cdots + 264 \)
Invariants
| Degree: | $16$ |
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| Signature: | $[0, 8]$ |
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| Discriminant: |
\(4357047163233901253492736\)
\(\medspace = 2^{66}\cdot 3^{10}\)
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| Root discriminant: | \(34.67\) |
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| Galois root discriminant: | $2^{555/128}3^{3/4}\approx 46.035004279893066$ | ||
| 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. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-2}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{6}a^{10}+\frac{1}{6}a^{8}+\frac{1}{6}a^{7}-\frac{1}{6}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{12}a^{11}-\frac{1}{6}a^{9}-\frac{1}{6}a^{8}+\frac{1}{6}a^{7}+\frac{1}{6}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}$, $\frac{1}{12}a^{12}-\frac{1}{6}a^{9}-\frac{1}{6}a^{8}-\frac{1}{6}a^{7}-\frac{1}{6}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{36}a^{13}+\frac{1}{36}a^{11}+\frac{1}{18}a^{10}+\frac{2}{9}a^{9}-\frac{1}{18}a^{7}-\frac{2}{9}a^{6}+\frac{1}{3}a^{4}-\frac{1}{9}a^{3}-\frac{2}{9}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3852}a^{14}+\frac{31}{3852}a^{13}-\frac{125}{3852}a^{12}-\frac{19}{1284}a^{11}-\frac{62}{963}a^{10}-\frac{101}{1926}a^{9}+\frac{19}{963}a^{8}+\frac{49}{1926}a^{7}+\frac{125}{1926}a^{6}+\frac{116}{321}a^{5}-\frac{43}{963}a^{4}-\frac{34}{321}a^{3}-\frac{473}{963}a^{2}+\frac{4}{107}a-\frac{131}{321}$, $\frac{1}{50\cdots 60}a^{15}-\frac{15\cdots 37}{12\cdots 65}a^{14}+\frac{32\cdots 69}{55\cdots 40}a^{13}+\frac{71\cdots 01}{25\cdots 30}a^{12}-\frac{12\cdots 07}{50\cdots 60}a^{11}-\frac{94\cdots 19}{12\cdots 65}a^{10}-\frac{51\cdots 47}{25\cdots 30}a^{9}+\frac{34\cdots 13}{27\cdots 70}a^{8}-\frac{51\cdots 94}{12\cdots 65}a^{7}-\frac{11\cdots 03}{25\cdots 30}a^{6}+\frac{97\cdots 46}{25\cdots 33}a^{5}-\frac{10\cdots 27}{25\cdots 33}a^{4}+\frac{10\cdots 88}{13\cdots 85}a^{3}-\frac{13\cdots 86}{12\cdots 65}a^{2}-\frac{24\cdots 99}{27\cdots 37}a+\frac{11\cdots 39}{41\cdots 55}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ (assuming GRH) |
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| Narrow class group: | $C_{2}$, which has order $2$ (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: |
$\frac{25\cdots 13}{83\cdots 10}a^{15}-\frac{40\cdots 49}{50\cdots 60}a^{14}-\frac{12\cdots 07}{50\cdots 60}a^{13}+\frac{10\cdots 09}{12\cdots 65}a^{12}-\frac{59\cdots 46}{13\cdots 85}a^{11}-\frac{12\cdots 47}{12\cdots 65}a^{10}+\frac{11\cdots 92}{12\cdots 65}a^{9}+\frac{41\cdots 68}{12\cdots 65}a^{8}+\frac{25\cdots 23}{12\cdots 65}a^{7}-\frac{10\cdots 67}{12\cdots 65}a^{6}+\frac{63\cdots 84}{83\cdots 11}a^{5}-\frac{19\cdots 30}{25\cdots 33}a^{4}+\frac{28\cdots 57}{41\cdots 55}a^{3}+\frac{35\cdots 27}{12\cdots 65}a^{2}+\frac{22\cdots 34}{27\cdots 37}a+\frac{20\cdots 77}{41\cdots 55}$, $\frac{76\cdots 07}{25\cdots 30}a^{15}-\frac{58\cdots 37}{50\cdots 60}a^{14}-\frac{11\cdots 51}{50\cdots 60}a^{13}+\frac{22\cdots 39}{25\cdots 30}a^{12}-\frac{89\cdots 87}{12\cdots 65}a^{11}-\frac{66\cdots 59}{83\cdots 10}a^{10}+\frac{12\cdots 54}{13\cdots 85}a^{9}+\frac{10\cdots 88}{41\cdots 55}a^{8}+\frac{45\cdots 31}{13\cdots 85}a^{7}-\frac{18\cdots 51}{12\cdots 65}a^{6}+\frac{45\cdots 32}{25\cdots 33}a^{5}-\frac{28\cdots 23}{25\cdots 33}a^{4}-\frac{13\cdots 37}{12\cdots 65}a^{3}-\frac{74\cdots 99}{12\cdots 65}a^{2}-\frac{33\cdots 36}{83\cdots 11}a-\frac{12\cdots 39}{41\cdots 55}$, $\frac{36\cdots 82}{41\cdots 55}a^{15}-\frac{34\cdots 83}{50\cdots 60}a^{14}+\frac{73\cdots 39}{12\cdots 65}a^{13}+\frac{53\cdots 78}{12\cdots 65}a^{12}+\frac{96\cdots 14}{41\cdots 55}a^{11}+\frac{62\cdots 81}{12\cdots 65}a^{10}-\frac{27\cdots 91}{12\cdots 65}a^{9}-\frac{39\cdots 74}{12\cdots 65}a^{8}-\frac{13\cdots 89}{12\cdots 65}a^{7}-\frac{11\cdots 19}{12\cdots 65}a^{6}-\frac{33\cdots 22}{27\cdots 37}a^{5}+\frac{50\cdots 29}{25\cdots 33}a^{4}+\frac{32\cdots 64}{41\cdots 55}a^{3}+\frac{15\cdots 74}{12\cdots 65}a^{2}-\frac{58\cdots 70}{27\cdots 37}a+\frac{11\cdots 09}{41\cdots 55}$, $\frac{69\cdots 93}{50\cdots 60}a^{15}+\frac{33\cdots 79}{55\cdots 40}a^{14}-\frac{18\cdots 19}{16\cdots 20}a^{13}-\frac{45\cdots 67}{25\cdots 30}a^{12}-\frac{59\cdots 23}{25\cdots 30}a^{11}-\frac{22\cdots 49}{25\cdots 30}a^{10}+\frac{10\cdots 99}{25\cdots 30}a^{9}+\frac{13\cdots 21}{25\cdots 30}a^{8}+\frac{26\cdots 11}{25\cdots 30}a^{7}+\frac{13\cdots 93}{12\cdots 65}a^{6}+\frac{10\cdots 81}{25\cdots 33}a^{5}-\frac{15\cdots 66}{25\cdots 33}a^{4}-\frac{56\cdots 49}{12\cdots 65}a^{3}-\frac{31\cdots 23}{12\cdots 65}a^{2}+\frac{12\cdots 70}{83\cdots 11}a+\frac{15\cdots 87}{41\cdots 55}$, $\frac{22\cdots 84}{41\cdots 55}a^{15}+\frac{79\cdots 19}{50\cdots 60}a^{14}+\frac{12\cdots 29}{27\cdots 70}a^{13}-\frac{18\cdots 14}{12\cdots 65}a^{12}+\frac{10\cdots 14}{12\cdots 65}a^{11}+\frac{20\cdots 62}{12\cdots 65}a^{10}-\frac{13\cdots 53}{83\cdots 10}a^{9}-\frac{75\cdots 28}{12\cdots 65}a^{8}-\frac{96\cdots 21}{25\cdots 30}a^{7}+\frac{87\cdots 34}{41\cdots 55}a^{6}-\frac{54\cdots 30}{83\cdots 11}a^{5}+\frac{41\cdots 89}{25\cdots 33}a^{4}-\frac{13\cdots 36}{12\cdots 65}a^{3}-\frac{43\cdots 38}{13\cdots 85}a^{2}-\frac{48\cdots 86}{83\cdots 11}a-\frac{68\cdots 09}{13\cdots 85}$, $\frac{15\cdots 79}{41\cdots 55}a^{15}-\frac{12\cdots 01}{50\cdots 60}a^{14}+\frac{13\cdots 71}{41\cdots 55}a^{13}+\frac{61\cdots 17}{25\cdots 30}a^{12}+\frac{39\cdots 14}{12\cdots 65}a^{11}+\frac{24\cdots 59}{25\cdots 30}a^{10}-\frac{82\cdots 03}{83\cdots 10}a^{9}-\frac{16\cdots 13}{12\cdots 65}a^{8}-\frac{70\cdots 91}{25\cdots 30}a^{7}+\frac{91\cdots 48}{13\cdots 85}a^{6}+\frac{10\cdots 92}{27\cdots 37}a^{5}+\frac{26\cdots 75}{25\cdots 33}a^{4}+\frac{33\cdots 14}{12\cdots 65}a^{3}-\frac{12\cdots 28}{13\cdots 85}a^{2}-\frac{57\cdots 46}{83\cdots 11}a-\frac{91\cdots 79}{13\cdots 85}$, $\frac{77\cdots 81}{25\cdots 30}a^{15}+\frac{57\cdots 49}{50\cdots 60}a^{14}-\frac{35\cdots 52}{12\cdots 65}a^{13}-\frac{22\cdots 98}{41\cdots 55}a^{12}-\frac{44\cdots 17}{25\cdots 30}a^{11}-\frac{19\cdots 68}{12\cdots 65}a^{10}+\frac{75\cdots 67}{83\cdots 10}a^{9}+\frac{13\cdots 87}{12\cdots 65}a^{8}+\frac{37\cdots 19}{25\cdots 30}a^{7}+\frac{68\cdots 77}{12\cdots 65}a^{6}-\frac{36\cdots 99}{25\cdots 33}a^{5}-\frac{61\cdots 06}{83\cdots 11}a^{4}-\frac{25\cdots 04}{13\cdots 85}a^{3}+\frac{16\cdots 18}{12\cdots 65}a^{2}-\frac{55\cdots 52}{27\cdots 37}a+\frac{63\cdots 93}{41\cdots 55}$
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| Regulator: | \( 1599747.4952237268 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1599747.4952237268 \cdot 2}{2\cdot\sqrt{4357047163233901253492736}}\cr\approx \mathstrut & 1.86163379899655 \end{aligned}\] (assuming GRH)
Galois group
$C_2^6:D_4$ (as 16T969):
| A solvable group of order 512 |
| The 44 conjugacy class representatives for $C_2^6:D_4$ |
| Character table for $C_2^6:D_4$ |
Intermediate fields
| \(\Q(\sqrt{-2}) \), 4.0.6144.1, 8.0.28991029248.10, 8.0.260919263232.14, 8.0.21743271936.4 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.16.66j1.36 | $x^{16} + 16 x^{12} + 16 x^{11} + 8 x^{10} + 16 x^{9} + 16 x^{5} + 16 x^{4} + 16 x^{3} + 2$ | $16$ | $1$ | $66$ | 16T969 | $$[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$$ |
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\(3\)
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 3.2.4.6a1.2 | $x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 64 x + 19$ | $4$ | $2$ | $6$ | $D_4$ | $$[\ ]_{4}^{2}$$ |