Normalized defining polynomial
\( x^{16} - 7 x^{15} + 17 x^{14} - 41 x^{13} + 151 x^{12} - 72 x^{11} - 1195 x^{10} + 2912 x^{9} + \cdots + 11421 \)
Invariants
| Degree: | $16$ |
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| Signature: | $[0, 8]$ |
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| Discriminant: |
\(2159514057650567877197265625\)
\(\medspace = 5^{14}\cdot 29^{12}\)
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| Root discriminant: | \(51.10\) |
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| Galois root discriminant: | $5^{7/8}29^{3/4}\approx 51.09721761228082$ | ||
| Ramified primes: |
\(5\), \(29\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_4$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | 8.0.9294114390625.1 | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6}a^{9}+\frac{1}{6}a^{8}+\frac{1}{3}a^{6}-\frac{1}{2}a^{5}-\frac{1}{6}a^{4}-\frac{1}{2}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{2}$, $\frac{1}{6}a^{10}-\frac{1}{6}a^{8}+\frac{1}{3}a^{7}+\frac{1}{6}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{6}a^{3}-\frac{1}{3}a^{2}+\frac{1}{6}a-\frac{1}{2}$, $\frac{1}{6}a^{11}-\frac{1}{2}a^{8}+\frac{1}{6}a^{7}-\frac{1}{3}a^{6}+\frac{1}{6}a^{5}+\frac{1}{6}a^{3}-\frac{1}{6}a^{2}-\frac{1}{6}a-\frac{1}{2}$, $\frac{1}{18}a^{12}+\frac{1}{18}a^{11}-\frac{1}{18}a^{10}+\frac{1}{18}a^{9}+\frac{1}{6}a^{8}-\frac{1}{6}a^{7}-\frac{1}{18}a^{5}-\frac{7}{18}a^{4}-\frac{1}{18}a^{3}+\frac{2}{9}a^{2}+\frac{1}{6}a$, $\frac{1}{18}a^{13}+\frac{1}{18}a^{11}-\frac{1}{18}a^{10}-\frac{1}{18}a^{9}+\frac{1}{6}a^{8}+\frac{1}{9}a^{6}-\frac{1}{6}a^{4}-\frac{2}{9}a^{3}+\frac{4}{9}a^{2}+\frac{1}{6}a-\frac{1}{2}$, $\frac{1}{3132}a^{14}+\frac{55}{3132}a^{13}-\frac{43}{1566}a^{12}+\frac{10}{261}a^{11}-\frac{185}{3132}a^{10}+\frac{79}{1566}a^{9}-\frac{217}{522}a^{8}+\frac{595}{1566}a^{7}-\frac{22}{783}a^{6}+\frac{119}{522}a^{5}-\frac{487}{3132}a^{4}-\frac{1061}{3132}a^{3}-\frac{226}{783}a^{2}+\frac{64}{261}a-\frac{3}{116}$, $\frac{1}{14\cdots 12}a^{15}-\frac{31\cdots 03}{24\cdots 52}a^{14}-\frac{33\cdots 49}{16\cdots 68}a^{13}+\frac{20\cdots 57}{74\cdots 56}a^{12}-\frac{75\cdots 03}{14\cdots 12}a^{11}-\frac{26\cdots 87}{51\cdots 28}a^{10}-\frac{59\cdots 23}{74\cdots 56}a^{9}-\frac{19\cdots 79}{74\cdots 56}a^{8}-\frac{35\cdots 97}{12\cdots 26}a^{7}+\frac{79\cdots 58}{18\cdots 89}a^{6}+\frac{67\cdots 71}{14\cdots 12}a^{5}+\frac{71\cdots 95}{37\cdots 78}a^{4}+\frac{40\cdots 55}{14\cdots 12}a^{3}-\frac{11\cdots 27}{74\cdots 56}a^{2}-\frac{19\cdots 19}{49\cdots 04}a-\frac{17\cdots 27}{55\cdots 56}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$ |
Class group and class number
| Ideal class group: | $C_{2}\times C_{24}$, which has order $48$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{24}$, which has order $48$ (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: |
$\frac{26\cdots 70}{18\cdots 89}a^{15}-\frac{37\cdots 67}{37\cdots 78}a^{14}+\frac{14\cdots 93}{64\cdots 41}a^{13}-\frac{28\cdots 18}{62\cdots 63}a^{12}+\frac{22\cdots 93}{12\cdots 82}a^{11}+\frac{10\cdots 09}{37\cdots 78}a^{10}-\frac{25\cdots 33}{12\cdots 26}a^{9}+\frac{84\cdots 24}{18\cdots 89}a^{8}+\frac{24\cdots 34}{18\cdots 89}a^{7}-\frac{26\cdots 43}{12\cdots 26}a^{6}+\frac{47\cdots 72}{18\cdots 89}a^{5}+\frac{61\cdots 79}{18\cdots 89}a^{4}-\frac{13\cdots 56}{18\cdots 89}a^{3}-\frac{14\cdots 17}{62\cdots 63}a^{2}+\frac{14\cdots 22}{20\cdots 21}a+\frac{49\cdots 75}{13\cdots 14}$, $\frac{21\cdots 67}{37\cdots 78}a^{15}-\frac{33\cdots 35}{74\cdots 56}a^{14}+\frac{99\cdots 37}{74\cdots 56}a^{13}-\frac{42\cdots 47}{12\cdots 26}a^{12}+\frac{42\cdots 21}{37\cdots 78}a^{11}-\frac{96\cdots 25}{74\cdots 56}a^{10}-\frac{36\cdots 03}{62\cdots 63}a^{9}+\frac{27\cdots 59}{12\cdots 82}a^{8}-\frac{61\cdots 79}{37\cdots 78}a^{7}-\frac{22\cdots 33}{41\cdots 42}a^{6}+\frac{15\cdots 27}{12\cdots 82}a^{5}+\frac{13\cdots 43}{74\cdots 56}a^{4}-\frac{17\cdots 67}{74\cdots 56}a^{3}+\frac{25\cdots 43}{62\cdots 63}a^{2}+\frac{15\cdots 16}{68\cdots 07}a+\frac{14\cdots 19}{27\cdots 28}$, $\frac{38\cdots 11}{14\cdots 12}a^{15}-\frac{10\cdots 02}{62\cdots 63}a^{14}+\frac{18\cdots 09}{49\cdots 04}a^{13}-\frac{77\cdots 47}{74\cdots 56}a^{12}+\frac{56\cdots 63}{14\cdots 12}a^{11}-\frac{12\cdots 99}{14\cdots 12}a^{10}-\frac{20\cdots 83}{74\cdots 56}a^{9}+\frac{45\cdots 05}{74\cdots 56}a^{8}+\frac{13\cdots 17}{62\cdots 63}a^{7}-\frac{43\cdots 39}{18\cdots 89}a^{6}+\frac{38\cdots 05}{14\cdots 12}a^{5}+\frac{28\cdots 65}{74\cdots 56}a^{4}-\frac{69\cdots 13}{14\cdots 12}a^{3}-\frac{22\cdots 97}{74\cdots 56}a^{2}+\frac{17\cdots 67}{49\cdots 04}a+\frac{11\cdots 57}{55\cdots 56}$, $\frac{23\cdots 23}{14\cdots 12}a^{15}-\frac{10\cdots 07}{82\cdots 84}a^{14}+\frac{18\cdots 15}{49\cdots 04}a^{13}-\frac{73\cdots 21}{74\cdots 56}a^{12}+\frac{49\cdots 91}{14\cdots 12}a^{11}-\frac{58\cdots 85}{14\cdots 12}a^{10}-\frac{11\cdots 85}{74\cdots 56}a^{9}+\frac{42\cdots 95}{74\cdots 56}a^{8}-\frac{20\cdots 79}{41\cdots 42}a^{7}-\frac{21\cdots 99}{18\cdots 89}a^{6}+\frac{45\cdots 57}{14\cdots 12}a^{5}-\frac{15\cdots 79}{37\cdots 78}a^{4}-\frac{70\cdots 91}{14\cdots 12}a^{3}+\frac{10\cdots 39}{74\cdots 56}a^{2}+\frac{21\cdots 63}{49\cdots 04}a+\frac{84\cdots 15}{55\cdots 56}$, $\frac{31\cdots 03}{55\cdots 56}a^{15}-\frac{55\cdots 33}{74\cdots 56}a^{14}+\frac{49\cdots 19}{14\cdots 12}a^{13}-\frac{65\cdots 99}{74\cdots 56}a^{12}+\frac{13\cdots 15}{49\cdots 04}a^{11}-\frac{34\cdots 83}{51\cdots 28}a^{10}-\frac{10\cdots 87}{74\cdots 56}a^{9}+\frac{11\cdots 83}{24\cdots 52}a^{8}-\frac{17\cdots 31}{18\cdots 89}a^{7}+\frac{48\cdots 03}{37\cdots 78}a^{6}+\frac{15\cdots 53}{55\cdots 56}a^{5}-\frac{11\cdots 61}{37\cdots 78}a^{4}-\frac{45\cdots 65}{14\cdots 12}a^{3}+\frac{26\cdots 95}{74\cdots 56}a^{2}+\frac{15\cdots 45}{49\cdots 04}a+\frac{32\cdots 85}{55\cdots 56}$, $\frac{13\cdots 73}{14\cdots 12}a^{15}-\frac{50\cdots 05}{74\cdots 56}a^{14}+\frac{31\cdots 91}{14\cdots 12}a^{13}-\frac{49\cdots 41}{74\cdots 56}a^{12}+\frac{33\cdots 61}{14\cdots 12}a^{11}-\frac{59\cdots 51}{16\cdots 68}a^{10}-\frac{21\cdots 29}{74\cdots 56}a^{9}+\frac{14\cdots 59}{74\cdots 56}a^{8}-\frac{61\cdots 69}{37\cdots 78}a^{7}-\frac{18\cdots 47}{37\cdots 78}a^{6}+\frac{15\cdots 99}{14\cdots 12}a^{5}+\frac{32\cdots 46}{68\cdots 07}a^{4}-\frac{99\cdots 03}{49\cdots 04}a^{3}-\frac{90\cdots 09}{74\cdots 56}a^{2}+\frac{40\cdots 33}{17\cdots 76}a+\frac{49\cdots 61}{55\cdots 56}$, $\frac{23082125309571}{78\cdots 11}a^{15}-\frac{181335779865926}{78\cdots 11}a^{14}+\frac{546148021263256}{78\cdots 11}a^{13}-\frac{13\cdots 33}{78\cdots 11}a^{12}+\frac{46\cdots 92}{78\cdots 11}a^{11}-\frac{54\cdots 70}{78\cdots 11}a^{10}-\frac{23\cdots 01}{78\cdots 11}a^{9}+\frac{88\cdots 79}{78\cdots 11}a^{8}-\frac{71\cdots 06}{78\cdots 11}a^{7}-\frac{20\cdots 49}{78\cdots 11}a^{6}+\frac{50\cdots 72}{78\cdots 11}a^{5}+\frac{10\cdots 35}{269319553305059}a^{4}-\frac{31\cdots 43}{269319553305059}a^{3}+\frac{21\cdots 27}{78\cdots 11}a^{2}+\frac{80\cdots 75}{78\cdots 11}a+\frac{99\cdots 28}{78\cdots 11}$
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| Regulator: | \( 1810504.50906 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1810504.50906 \cdot 48}{2\cdot\sqrt{2159514057650567877197265625}}\cr\approx \mathstrut & 2.27128577844 \end{aligned}\] (assuming GRH)
Galois group
$C_2^3.D_4$ (as 16T153):
| A solvable group of order 64 |
| The 13 conjugacy class representatives for $C_2^3.D_4$ |
| Character table for $C_2^3.D_4$ |
Intermediate fields
| \(\Q(\sqrt{145}) \), 4.0.105125.1, 4.4.3048625.2, 4.0.609725.1, 8.0.9294114390625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 16.8.2159514057650567877197265625.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.8.7a1.4 | $x^{8} + 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $$[\ ]_{8}^{2}$$ |
| 5.1.8.7a1.4 | $x^{8} + 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $$[\ ]_{8}^{2}$$ | |
|
\(29\)
| 29.1.4.3a1.3 | $x^{4} + 116$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 29.1.4.3a1.3 | $x^{4} + 116$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 29.1.4.3a1.3 | $x^{4} + 116$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 29.1.4.3a1.3 | $x^{4} + 116$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |