Normalized defining polynomial
\( x^{16} + 8 x^{14} + 68 x^{12} - 192 x^{11} + 104 x^{10} - 384 x^{9} + 662 x^{8} - 384 x^{7} + \cdots + 9297 \)
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{4}-\frac{3}{8}$, $\frac{1}{16}a^{9}-\frac{1}{16}a^{8}-\frac{1}{8}a^{5}+\frac{1}{8}a^{4}+\frac{5}{16}a-\frac{5}{16}$, $\frac{1}{16}a^{10}-\frac{1}{16}a^{8}-\frac{1}{8}a^{6}+\frac{1}{8}a^{4}+\frac{5}{16}a^{2}-\frac{5}{16}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{8}-\frac{1}{8}a^{7}+\frac{1}{8}a^{4}+\frac{5}{16}a^{3}-\frac{5}{16}$, $\frac{1}{32}a^{12}-\frac{1}{32}a^{8}+\frac{3}{32}a^{4}+\frac{5}{32}$, $\frac{1}{64}a^{13}-\frac{1}{64}a^{12}-\frac{1}{32}a^{11}-\frac{1}{32}a^{10}+\frac{1}{64}a^{9}-\frac{1}{64}a^{8}+\frac{1}{16}a^{7}+\frac{1}{16}a^{6}-\frac{1}{64}a^{5}+\frac{1}{64}a^{4}-\frac{5}{32}a^{3}-\frac{5}{32}a^{2}+\frac{15}{64}a-\frac{15}{64}$, $\frac{1}{192}a^{14}-\frac{1}{192}a^{12}-\frac{1}{192}a^{10}+\frac{5}{192}a^{8}-\frac{13}{192}a^{6}+\frac{5}{192}a^{4}-\frac{11}{192}a^{2}-\frac{11}{64}$, $\frac{1}{40\cdots 16}a^{15}-\frac{35\cdots 85}{45\cdots 24}a^{14}-\frac{17\cdots 51}{40\cdots 16}a^{13}+\frac{60\cdots 41}{45\cdots 24}a^{12}+\frac{73\cdots 71}{40\cdots 16}a^{11}+\frac{22\cdots 43}{13\cdots 72}a^{10}+\frac{67\cdots 83}{40\cdots 16}a^{9}+\frac{74\cdots 49}{13\cdots 72}a^{8}+\frac{26\cdots 99}{40\cdots 16}a^{7}-\frac{12\cdots 57}{13\cdots 72}a^{6}-\frac{39\cdots 85}{40\cdots 16}a^{5}-\frac{13\cdots 19}{13\cdots 72}a^{4}-\frac{15\cdots 99}{40\cdots 16}a^{3}-\frac{10\cdots 53}{45\cdots 24}a^{2}+\frac{44\cdots 15}{13\cdots 72}a+\frac{20\cdots 01}{45\cdots 24}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: |
$\frac{28\cdots 29}{50\cdots 52}a^{15}-\frac{26\cdots 21}{13\cdots 72}a^{14}-\frac{18\cdots 01}{20\cdots 08}a^{13}-\frac{32\cdots 25}{13\cdots 72}a^{12}-\frac{90\cdots 51}{10\cdots 04}a^{11}-\frac{16\cdots 27}{13\cdots 72}a^{10}-\frac{31\cdots 63}{20\cdots 08}a^{9}-\frac{90\cdots 45}{45\cdots 24}a^{8}+\frac{21\cdots 09}{25\cdots 76}a^{7}+\frac{31\cdots 35}{45\cdots 24}a^{6}+\frac{14\cdots 53}{20\cdots 08}a^{5}+\frac{58\cdots 05}{13\cdots 72}a^{4}-\frac{28\cdots 51}{10\cdots 04}a^{3}-\frac{21\cdots 65}{13\cdots 72}a^{2}-\frac{14\cdots 51}{67\cdots 36}a-\frac{59\cdots 11}{45\cdots 24}$, $\frac{78\cdots 21}{67\cdots 36}a^{15}-\frac{17\cdots 71}{13\cdots 72}a^{14}-\frac{22\cdots 59}{21\cdots 73}a^{13}-\frac{16\cdots 39}{13\cdots 72}a^{12}-\frac{62\cdots 29}{67\cdots 36}a^{11}+\frac{16\cdots 79}{13\cdots 72}a^{10}+\frac{13\cdots 29}{84\cdots 92}a^{9}+\frac{62\cdots 75}{13\cdots 72}a^{8}-\frac{16\cdots 59}{67\cdots 36}a^{7}+\frac{24\cdots 63}{13\cdots 72}a^{6}-\frac{15\cdots 59}{42\cdots 46}a^{5}+\frac{11\cdots 51}{13\cdots 72}a^{4}-\frac{14\cdots 87}{67\cdots 36}a^{3}+\frac{12\cdots 05}{13\cdots 72}a^{2}-\frac{58\cdots 73}{28\cdots 64}a+\frac{43\cdots 27}{45\cdots 24}$, $\frac{16\cdots 87}{45\cdots 24}a^{15}-\frac{61\cdots 49}{13\cdots 72}a^{14}-\frac{15\cdots 05}{45\cdots 24}a^{13}-\frac{58\cdots 75}{13\cdots 72}a^{12}-\frac{13\cdots 05}{45\cdots 24}a^{11}+\frac{43\cdots 17}{13\cdots 72}a^{10}+\frac{18\cdots 29}{45\cdots 24}a^{9}+\frac{18\cdots 11}{13\cdots 72}a^{8}-\frac{30\cdots 41}{45\cdots 24}a^{7}+\frac{67\cdots 61}{13\cdots 72}a^{6}-\frac{31\cdots 59}{45\cdots 24}a^{5}+\frac{36\cdots 55}{13\cdots 72}a^{4}-\frac{28\cdots 59}{45\cdots 24}a^{3}+\frac{31\cdots 07}{13\cdots 72}a^{2}-\frac{31\cdots 49}{45\cdots 24}a+\frac{12\cdots 99}{45\cdots 24}$, $\frac{21\cdots 31}{40\cdots 16}a^{15}+\frac{20\cdots 75}{13\cdots 72}a^{14}+\frac{23\cdots 33}{40\cdots 16}a^{13}+\frac{18\cdots 45}{13\cdots 72}a^{12}+\frac{20\cdots 01}{40\cdots 16}a^{11}+\frac{25\cdots 85}{13\cdots 72}a^{10}-\frac{45\cdots 53}{40\cdots 16}a^{9}-\frac{95\cdots 07}{45\cdots 24}a^{8}-\frac{75\cdots 59}{40\cdots 16}a^{7}+\frac{39\cdots 31}{45\cdots 24}a^{6}+\frac{44\cdots 51}{40\cdots 16}a^{5}-\frac{56\cdots 49}{13\cdots 72}a^{4}+\frac{12\cdots 63}{40\cdots 16}a^{3}+\frac{19\cdots 87}{13\cdots 72}a^{2}-\frac{12\cdots 41}{13\cdots 72}a-\frac{25\cdots 37}{45\cdots 24}$, $\frac{10\cdots 97}{10\cdots 04}a^{15}-\frac{17\cdots 01}{13\cdots 72}a^{14}-\frac{51\cdots 49}{50\cdots 52}a^{13}-\frac{17\cdots 75}{13\cdots 72}a^{12}-\frac{11\cdots 01}{12\cdots 38}a^{11}+\frac{12\cdots 97}{13\cdots 72}a^{10}+\frac{96\cdots 29}{12\cdots 38}a^{9}+\frac{18\cdots 45}{45\cdots 24}a^{8}-\frac{18\cdots 77}{10\cdots 04}a^{7}+\frac{80\cdots 63}{45\cdots 24}a^{6}-\frac{12\cdots 39}{50\cdots 52}a^{5}+\frac{10\cdots 47}{13\cdots 72}a^{4}-\frac{96\cdots 35}{50\cdots 52}a^{3}+\frac{10\cdots 03}{13\cdots 72}a^{2}-\frac{10\cdots 25}{42\cdots 46}a+\frac{36\cdots 95}{45\cdots 24}$, $\frac{37\cdots 21}{50\cdots 52}a^{15}-\frac{12\cdots 43}{13\cdots 72}a^{14}-\frac{71\cdots 07}{10\cdots 04}a^{13}-\frac{11\cdots 99}{13\cdots 72}a^{12}-\frac{62\cdots 25}{10\cdots 04}a^{11}+\frac{88\cdots 43}{13\cdots 72}a^{10}-\frac{17\cdots 89}{10\cdots 04}a^{9}+\frac{12\cdots 73}{45\cdots 24}a^{8}-\frac{34\cdots 87}{25\cdots 76}a^{7}+\frac{65\cdots 53}{45\cdots 24}a^{6}-\frac{16\cdots 33}{10\cdots 04}a^{5}+\frac{73\cdots 67}{13\cdots 72}a^{4}-\frac{13\cdots 09}{10\cdots 04}a^{3}+\frac{80\cdots 97}{13\cdots 72}a^{2}-\frac{69\cdots 69}{33\cdots 68}a+\frac{22\cdots 35}{45\cdots 24}$, $\frac{25\cdots 93}{33\cdots 68}a^{15}-\frac{46\cdots 59}{45\cdots 24}a^{14}-\frac{52\cdots 87}{67\cdots 36}a^{13}-\frac{45\cdots 33}{45\cdots 24}a^{12}-\frac{22\cdots 45}{33\cdots 68}a^{11}+\frac{25\cdots 15}{45\cdots 24}a^{10}-\frac{13\cdots 55}{67\cdots 36}a^{9}+\frac{15\cdots 93}{45\cdots 24}a^{8}-\frac{47\cdots 95}{33\cdots 68}a^{7}+\frac{10\cdots 95}{45\cdots 24}a^{6}-\frac{20\cdots 33}{67\cdots 36}a^{5}+\frac{25\cdots 53}{45\cdots 24}a^{4}-\frac{44\cdots 23}{33\cdots 68}a^{3}+\frac{32\cdots 49}{45\cdots 24}a^{2}-\frac{21\cdots 91}{22\cdots 12}a+\frac{53\cdots 95}{45\cdots 24}$
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| Regulator: | \( 2488210.386391306 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2488210.386391306 \cdot 2}{2\cdot\sqrt{4357047163233901253492736}}\cr\approx \mathstrut & 2.89554230786435 \end{aligned}\] (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, 8.0.260919263232.15, 8.0.21743271936.5 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.16.66j1.123 | $x^{16} + 8 x^{14} + 8 x^{12} + 16 x^{11} + 8 x^{10} + 16 x^{3} + 2$ | $16$ | $1$ | $66$ | 16T969 | $$[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$$ |
|
\(3\)
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.2.4.6a1.2 | $x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 64 x + 19$ | $4$ | $2$ | $6$ | $D_4$ | $$[\ ]_{4}^{2}$$ |