Normalized defining polynomial
\( x^{16} - x^{15} + 7 x^{14} + 15 x^{13} - 62 x^{12} + 43 x^{11} - 310 x^{10} - 629 x^{9} - 268 x^{8} + \cdots + 16 \)
Invariants
| Degree: | $16$ |
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| Signature: | $[8, 4]$ |
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| Discriminant: |
\(575650281819106455466129\)
\(\medspace = 17^{14}\cdot 43^{4}\)
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| Root discriminant: | \(30.55\) |
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| Galois root discriminant: | $17^{7/8}43^{1/2}\approx 78.23066716385114$ | ||
| Ramified primes: |
\(17\), \(43\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2\times C_4$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{11\cdots 04}a^{15}+\frac{10\cdots 29}{11\cdots 04}a^{14}-\frac{14\cdots 27}{11\cdots 04}a^{13}-\frac{68\cdots 31}{11\cdots 04}a^{12}-\frac{75\cdots 35}{29\cdots 26}a^{11}+\frac{69\cdots 19}{11\cdots 04}a^{10}+\frac{45\cdots 71}{29\cdots 26}a^{9}+\frac{13\cdots 87}{11\cdots 04}a^{8}+\frac{14\cdots 81}{58\cdots 52}a^{7}-\frac{87\cdots 09}{11\cdots 04}a^{6}-\frac{51\cdots 61}{11\cdots 04}a^{5}-\frac{24\cdots 41}{58\cdots 52}a^{4}+\frac{35\cdots 35}{11\cdots 04}a^{3}-\frac{38\cdots 90}{14\cdots 63}a^{2}+\frac{18\cdots 07}{14\cdots 63}a-\frac{45\cdots 60}{14\cdots 63}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
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| Narrow class group: | Trivial group, which has order $1$ (assuming GRH) |
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Unit group
| Rank: | $11$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{15\cdots 05}{58\cdots 52}a^{15}-\frac{17\cdots 15}{29\cdots 26}a^{14}+\frac{24\cdots 13}{14\cdots 63}a^{13}+\frac{80\cdots 71}{14\cdots 63}a^{12}-\frac{79\cdots 61}{58\cdots 52}a^{11}-\frac{62\cdots 97}{58\cdots 52}a^{10}-\frac{42\cdots 33}{58\cdots 52}a^{9}-\frac{13\cdots 27}{58\cdots 52}a^{8}-\frac{11\cdots 43}{58\cdots 52}a^{7}-\frac{34\cdots 55}{58\cdots 52}a^{6}-\frac{79\cdots 91}{29\cdots 26}a^{5}+\frac{57\cdots 01}{58\cdots 52}a^{4}-\frac{18\cdots 79}{58\cdots 52}a^{3}+\frac{25\cdots 01}{58\cdots 52}a^{2}+\frac{29\cdots 03}{29\cdots 26}a-\frac{26\cdots 50}{14\cdots 63}$, $\frac{47\cdots 19}{58\cdots 52}a^{15}+\frac{12\cdots 35}{58\cdots 52}a^{14}+\frac{39\cdots 79}{58\cdots 52}a^{13}+\frac{10\cdots 13}{58\cdots 52}a^{12}-\frac{29\cdots 16}{14\cdots 63}a^{11}+\frac{74\cdots 29}{58\cdots 52}a^{10}-\frac{91\cdots 67}{29\cdots 26}a^{9}-\frac{46\cdots 17}{58\cdots 52}a^{8}-\frac{43\cdots 93}{29\cdots 26}a^{7}-\frac{20\cdots 13}{58\cdots 52}a^{6}-\frac{13\cdots 39}{58\cdots 52}a^{5}-\frac{58\cdots 69}{14\cdots 63}a^{4}-\frac{29\cdots 89}{58\cdots 52}a^{3}-\frac{91\cdots 31}{29\cdots 26}a^{2}-\frac{79\cdots 89}{29\cdots 26}a+\frac{26\cdots 37}{14\cdots 63}$, $\frac{11\cdots 93}{11\cdots 04}a^{15}-\frac{62\cdots 53}{11\cdots 04}a^{14}+\frac{58\cdots 55}{11\cdots 04}a^{13}+\frac{20\cdots 55}{11\cdots 04}a^{12}-\frac{35\cdots 89}{58\cdots 52}a^{11}-\frac{10\cdots 69}{11\cdots 04}a^{10}-\frac{12\cdots 77}{58\cdots 52}a^{9}-\frac{86\cdots 89}{11\cdots 04}a^{8}-\frac{33\cdots 53}{29\cdots 26}a^{7}-\frac{10\cdots 49}{11\cdots 04}a^{6}+\frac{19\cdots 61}{11\cdots 04}a^{5}+\frac{42\cdots 27}{14\cdots 63}a^{4}-\frac{62\cdots 33}{11\cdots 04}a^{3}+\frac{18\cdots 91}{58\cdots 52}a^{2}+\frac{18\cdots 77}{29\cdots 26}a-\frac{17\cdots 16}{14\cdots 63}$, $\frac{68\cdots 93}{11\cdots 04}a^{15}+\frac{14\cdots 71}{11\cdots 04}a^{14}+\frac{58\cdots 19}{11\cdots 04}a^{13}+\frac{24\cdots 15}{11\cdots 04}a^{12}+\frac{52\cdots 73}{58\cdots 52}a^{11}-\frac{34\cdots 33}{11\cdots 04}a^{10}-\frac{13\cdots 77}{58\cdots 52}a^{9}-\frac{10\cdots 61}{11\cdots 04}a^{8}-\frac{30\cdots 30}{14\cdots 63}a^{7}-\frac{47\cdots 49}{11\cdots 04}a^{6}-\frac{61\cdots 39}{11\cdots 04}a^{5}-\frac{14\cdots 23}{29\cdots 26}a^{4}-\frac{36\cdots 77}{11\cdots 04}a^{3}-\frac{12\cdots 67}{58\cdots 52}a^{2}-\frac{41\cdots 23}{14\cdots 63}a+\frac{19\cdots 43}{14\cdots 63}$, $\frac{21\cdots 69}{29\cdots 26}a^{15}+\frac{14\cdots 49}{58\cdots 52}a^{14}+\frac{17\cdots 97}{58\cdots 52}a^{13}+\frac{10\cdots 17}{58\cdots 52}a^{12}-\frac{22\cdots 73}{58\cdots 52}a^{11}-\frac{68\cdots 23}{14\cdots 63}a^{10}-\frac{72\cdots 19}{58\cdots 52}a^{9}-\frac{23\cdots 69}{29\cdots 26}a^{8}-\frac{24\cdots 19}{58\cdots 52}a^{7}-\frac{26\cdots 07}{29\cdots 26}a^{6}+\frac{18\cdots 31}{58\cdots 52}a^{5}+\frac{23\cdots 19}{58\cdots 52}a^{4}-\frac{38\cdots 83}{29\cdots 26}a^{3}+\frac{25\cdots 07}{58\cdots 52}a^{2}+\frac{13\cdots 06}{14\cdots 63}a-\frac{15\cdots 71}{14\cdots 63}$, $\frac{19\cdots 65}{58\cdots 52}a^{15}+\frac{14\cdots 60}{14\cdots 63}a^{14}+\frac{50\cdots 85}{14\cdots 63}a^{13}+\frac{20\cdots 75}{14\cdots 63}a^{12}+\frac{80\cdots 25}{58\cdots 52}a^{11}-\frac{11\cdots 65}{58\cdots 52}a^{10}-\frac{10\cdots 09}{58\cdots 52}a^{9}-\frac{37\cdots 25}{58\cdots 52}a^{8}-\frac{92\cdots 59}{58\cdots 52}a^{7}-\frac{17\cdots 55}{58\cdots 52}a^{6}-\frac{53\cdots 97}{14\cdots 63}a^{5}-\frac{17\cdots 07}{58\cdots 52}a^{4}-\frac{11\cdots 75}{58\cdots 52}a^{3}+\frac{10\cdots 91}{58\cdots 52}a^{2}+\frac{93\cdots 39}{29\cdots 26}a+\frac{54\cdots 69}{14\cdots 63}$, $\frac{56\cdots 69}{11\cdots 04}a^{15}-\frac{21\cdots 07}{11\cdots 04}a^{14}+\frac{38\cdots 69}{11\cdots 04}a^{13}+\frac{10\cdots 81}{11\cdots 04}a^{12}-\frac{35\cdots 52}{14\cdots 63}a^{11}+\frac{67\cdots 19}{11\cdots 04}a^{10}-\frac{21\cdots 23}{14\cdots 63}a^{9}-\frac{46\cdots 33}{11\cdots 04}a^{8}-\frac{22\cdots 07}{58\cdots 52}a^{7}-\frac{13\cdots 49}{11\cdots 04}a^{6}+\frac{48\cdots 91}{11\cdots 04}a^{5}-\frac{21\cdots 49}{58\cdots 52}a^{4}-\frac{24\cdots 93}{11\cdots 04}a^{3}+\frac{95\cdots 39}{14\cdots 63}a^{2}+\frac{36\cdots 37}{29\cdots 26}a-\frac{69\cdots 11}{14\cdots 63}$, $\frac{13\cdots 47}{11\cdots 04}a^{15}-\frac{11\cdots 55}{11\cdots 04}a^{14}+\frac{90\cdots 93}{11\cdots 04}a^{13}+\frac{20\cdots 25}{11\cdots 04}a^{12}-\frac{38\cdots 65}{58\cdots 52}a^{11}+\frac{45\cdots 01}{11\cdots 04}a^{10}-\frac{20\cdots 61}{58\cdots 52}a^{9}-\frac{88\cdots 19}{11\cdots 04}a^{8}-\frac{12\cdots 31}{29\cdots 26}a^{7}-\frac{26\cdots 67}{11\cdots 04}a^{6}+\frac{16\cdots 87}{11\cdots 04}a^{5}-\frac{11\cdots 83}{14\cdots 63}a^{4}-\frac{93\cdots 55}{11\cdots 04}a^{3}+\frac{10\cdots 93}{58\cdots 52}a^{2}-\frac{56\cdots 02}{14\cdots 63}a-\frac{24\cdots 13}{14\cdots 63}$, $\frac{41\cdots 65}{58\cdots 52}a^{15}-\frac{60\cdots 15}{14\cdots 63}a^{14}+\frac{59\cdots 66}{14\cdots 63}a^{13}+\frac{19\cdots 44}{14\cdots 63}a^{12}-\frac{26\cdots 45}{58\cdots 52}a^{11}+\frac{21\cdots 79}{58\cdots 52}a^{10}-\frac{10\cdots 97}{58\cdots 52}a^{9}-\frac{33\cdots 73}{58\cdots 52}a^{8}-\frac{91\cdots 21}{58\cdots 52}a^{7}-\frac{64\cdots 67}{58\cdots 52}a^{6}+\frac{23\cdots 05}{29\cdots 26}a^{5}+\frac{77\cdots 87}{58\cdots 52}a^{4}-\frac{70\cdots 45}{58\cdots 52}a^{3}+\frac{15\cdots 43}{58\cdots 52}a^{2}-\frac{20\cdots 27}{29\cdots 26}a-\frac{53\cdots 32}{14\cdots 63}$, $\frac{27\cdots 99}{11\cdots 04}a^{15}-\frac{30\cdots 69}{11\cdots 04}a^{14}+\frac{19\cdots 19}{11\cdots 04}a^{13}+\frac{38\cdots 19}{11\cdots 04}a^{12}-\frac{43\cdots 81}{29\cdots 26}a^{11}+\frac{13\cdots 45}{11\cdots 04}a^{10}-\frac{21\cdots 23}{29\cdots 26}a^{9}-\frac{16\cdots 55}{11\cdots 04}a^{8}-\frac{28\cdots 07}{58\cdots 52}a^{7}-\frac{51\cdots 07}{11\cdots 04}a^{6}+\frac{44\cdots 65}{11\cdots 04}a^{5}-\frac{13\cdots 25}{58\cdots 52}a^{4}-\frac{89\cdots 55}{11\cdots 04}a^{3}+\frac{60\cdots 38}{14\cdots 63}a^{2}-\frac{35\cdots 93}{14\cdots 63}a+\frac{60\cdots 27}{14\cdots 63}$, $\frac{61\cdots 45}{29\cdots 26}a^{15}-\frac{21\cdots 65}{29\cdots 26}a^{14}+\frac{47\cdots 43}{29\cdots 26}a^{13}+\frac{11\cdots 29}{29\cdots 26}a^{12}-\frac{12\cdots 44}{14\cdots 63}a^{11}+\frac{66\cdots 65}{14\cdots 63}a^{10}-\frac{11\cdots 25}{14\cdots 63}a^{9}-\frac{45\cdots 59}{29\cdots 26}a^{8}-\frac{35\cdots 05}{14\cdots 63}a^{7}-\frac{86\cdots 22}{14\cdots 63}a^{6}-\frac{11\cdots 25}{29\cdots 26}a^{5}-\frac{17\cdots 33}{29\cdots 26}a^{4}+\frac{71\cdots 50}{14\cdots 63}a^{3}-\frac{11\cdots 79}{29\cdots 26}a^{2}+\frac{70\cdots 87}{14\cdots 63}a-\frac{25\cdots 65}{14\cdots 63}$
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| Regulator: | \( 549740.860993 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 549740.860993 \cdot 1}{2\cdot\sqrt{575650281819106455466129}}\cr\approx \mathstrut & 0.144546673168 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2:C_8$ (as 16T24):
| A solvable group of order 32 |
| The 20 conjugacy class representatives for $C_2^2 : C_8$ |
| Character table for $C_2^2 : C_8$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.4.4913.1, 4.2.12427.1, 4.2.211259.1, \(\Q(\zeta_{17})^+\), 8.4.758716206377.1, 8.4.44630365081.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 32 |
| Degree 16 sibling: | 16.0.1968033759133442969054057291329.4 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.2.8.14a1.2 | $x^{16} + 128 x^{15} + 7192 x^{14} + 232064 x^{13} + 4716796 x^{12} + 62185088 x^{11} + 525781480 x^{10} + 2696730752 x^{9} + 7365142054 x^{8} + 8090192256 x^{7} + 4732033320 x^{6} + 1678997376 x^{5} + 382060476 x^{4} + 56391552 x^{3} + 5242968 x^{2} + 279936 x + 6578$ | $8$ | $2$ | $14$ | $C_8\times C_2$ | $$[\ ]_{8}^{2}$$ |
|
\(43\)
| 43.4.1.0a1.1 | $x^{4} + 5 x^{2} + 42 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |
| 43.4.1.0a1.1 | $x^{4} + 5 x^{2} + 42 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
| 43.4.2.4a1.2 | $x^{8} + 10 x^{6} + 84 x^{5} + 31 x^{4} + 420 x^{3} + 1794 x^{2} + 252 x + 52$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |