Normalized defining polynomial
\( x^{16} + 8 x^{14} - 32 x^{13} - 176 x^{10} + 224 x^{9} + 1088 x^{8} + 384 x^{7} + 1280 x^{6} + \cdots + 376 \)
Invariants
| Degree: | $16$ |
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| Signature: | $[0, 8]$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{4}a^{12}$, $\frac{1}{188}a^{13}-\frac{11}{188}a^{12}+\frac{15}{188}a^{11}+\frac{4}{47}a^{10}+\frac{6}{47}a^{9}+\frac{4}{47}a^{8}-\frac{1}{47}a^{7}-\frac{8}{47}a^{6}+\frac{23}{47}a^{5}+\frac{10}{47}a^{4}-\frac{6}{47}a^{3}+\frac{18}{47}a^{2}-\frac{16}{47}a$, $\frac{1}{3196}a^{14}-\frac{3}{3196}a^{13}+\frac{115}{3196}a^{12}+\frac{115}{1598}a^{11}-\frac{65}{1598}a^{10}+\frac{245}{1598}a^{9}+\frac{15}{1598}a^{8}-\frac{267}{1598}a^{7}-\frac{182}{799}a^{6}+\frac{241}{799}a^{5}+\frac{121}{799}a^{4}-\frac{30}{799}a^{3}-\frac{295}{799}a^{2}-\frac{222}{799}a-\frac{8}{17}$, $\frac{1}{15\cdots 28}a^{15}+\frac{3012715131625}{77\cdots 14}a^{14}-\frac{66493505310594}{38\cdots 07}a^{13}-\frac{30\cdots 78}{38\cdots 07}a^{12}-\frac{17\cdots 92}{38\cdots 07}a^{11}+\frac{45\cdots 19}{77\cdots 14}a^{10}+\frac{14\cdots 77}{38\cdots 07}a^{9}-\frac{36\cdots 30}{38\cdots 07}a^{8}-\frac{22\cdots 53}{77\cdots 14}a^{7}+\frac{81\cdots 81}{38\cdots 07}a^{6}+\frac{13\cdots 18}{38\cdots 07}a^{5}-\frac{51\cdots 05}{38\cdots 07}a^{4}+\frac{34\cdots 20}{38\cdots 07}a^{3}+\frac{15\cdots 57}{38\cdots 07}a^{2}+\frac{89\cdots 97}{38\cdots 07}a-\frac{8462698737631}{48766626872993}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
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| Narrow class group: | $C_{2}$, which has order $2$ |
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Unit group
| Rank: | $7$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{270992135559319}{77\cdots 14}a^{15}+\frac{391141027089469}{77\cdots 14}a^{14}+\frac{47\cdots 29}{15\cdots 28}a^{13}-\frac{62\cdots 05}{77\cdots 14}a^{12}-\frac{49\cdots 25}{38\cdots 07}a^{11}-\frac{13\cdots 85}{77\cdots 14}a^{10}-\frac{56\cdots 10}{38\cdots 07}a^{9}-\frac{18\cdots 71}{22\cdots 71}a^{8}+\frac{21\cdots 26}{38\cdots 07}a^{7}+\frac{60\cdots 65}{77\cdots 14}a^{6}+\frac{20\cdots 24}{38\cdots 07}a^{5}+\frac{56\cdots 12}{38\cdots 07}a^{4}+\frac{85\cdots 81}{38\cdots 07}a^{3}-\frac{44\cdots 87}{38\cdots 07}a^{2}+\frac{31\cdots 26}{38\cdots 07}a+\frac{655986105687401}{829032656840881}$, $\frac{19860755872853}{45\cdots 42}a^{15}+\frac{16977923964903}{91\cdots 84}a^{14}+\frac{443464127995533}{91\cdots 84}a^{13}-\frac{635002295936003}{45\cdots 42}a^{12}+\frac{272855725991031}{45\cdots 42}a^{11}-\frac{12\cdots 50}{22\cdots 71}a^{10}-\frac{526765862352877}{22\cdots 71}a^{9}+\frac{16\cdots 13}{45\cdots 42}a^{8}+\frac{61\cdots 40}{22\cdots 71}a^{7}+\frac{46\cdots 61}{45\cdots 42}a^{6}+\frac{689171367152847}{48766626872993}a^{5}+\frac{25\cdots 34}{22\cdots 71}a^{4}+\frac{64\cdots 73}{22\cdots 71}a^{3}+\frac{47\cdots 91}{22\cdots 71}a^{2}-\frac{29\cdots 18}{22\cdots 71}a+\frac{410087315778949}{48766626872993}$, $\frac{7593957374601}{25\cdots 94}a^{15}-\frac{2224126750444}{12\cdots 97}a^{14}+\frac{86349844662341}{25\cdots 94}a^{13}-\frac{627710820040111}{50\cdots 88}a^{12}+\frac{192780910520138}{12\cdots 97}a^{11}-\frac{11\cdots 17}{25\cdots 94}a^{10}+\frac{51246868894505}{12\cdots 97}a^{9}+\frac{10\cdots 57}{25\cdots 94}a^{8}+\frac{18\cdots 60}{12\cdots 97}a^{7}+\frac{53\cdots 74}{12\cdots 97}a^{6}+\frac{88\cdots 02}{12\cdots 97}a^{5}+\frac{51\cdots 71}{12\cdots 97}a^{4}+\frac{18\cdots 60}{12\cdots 97}a^{3}+\frac{78\cdots 66}{12\cdots 97}a^{2}+\frac{12\cdots 08}{12\cdots 97}a+\frac{102696663465231}{26742988930351}$, $\frac{729515072719561}{15\cdots 28}a^{15}-\frac{40\cdots 01}{15\cdots 28}a^{14}+\frac{99\cdots 85}{15\cdots 28}a^{13}-\frac{59\cdots 91}{15\cdots 28}a^{12}+\frac{48\cdots 11}{45\cdots 42}a^{11}-\frac{40\cdots 57}{38\cdots 07}a^{10}+\frac{45\cdots 79}{38\cdots 07}a^{9}+\frac{39\cdots 09}{77\cdots 14}a^{8}-\frac{48\cdots 69}{77\cdots 14}a^{7}-\frac{96\cdots 65}{77\cdots 14}a^{6}+\frac{18\cdots 26}{22\cdots 71}a^{5}-\frac{89\cdots 86}{38\cdots 07}a^{4}-\frac{10\cdots 47}{38\cdots 07}a^{3}+\frac{16\cdots 37}{38\cdots 07}a^{2}-\frac{43\cdots 84}{38\cdots 07}a-\frac{17\cdots 59}{829032656840881}$, $\frac{812744550836427}{15\cdots 28}a^{15}-\frac{310464380556241}{77\cdots 14}a^{14}-\frac{77\cdots 57}{15\cdots 28}a^{13}+\frac{14\cdots 01}{91\cdots 84}a^{12}+\frac{564815024309971}{38\cdots 07}a^{11}+\frac{43\cdots 87}{77\cdots 14}a^{10}-\frac{16\cdots 88}{38\cdots 07}a^{9}+\frac{12\cdots 75}{77\cdots 14}a^{8}-\frac{36\cdots 21}{45\cdots 42}a^{7}-\frac{77\cdots 91}{77\cdots 14}a^{6}-\frac{20\cdots 97}{38\cdots 07}a^{5}-\frac{36\cdots 49}{22\cdots 71}a^{4}-\frac{12\cdots 53}{38\cdots 07}a^{3}+\frac{52\cdots 91}{38\cdots 07}a^{2}-\frac{35\cdots 36}{22\cdots 71}a+\frac{719255155065997}{829032656840881}$, $\frac{15\cdots 17}{15\cdots 28}a^{15}-\frac{146117749217507}{15\cdots 28}a^{14}-\frac{12\cdots 79}{15\cdots 28}a^{13}+\frac{23\cdots 41}{77\cdots 14}a^{12}+\frac{10\cdots 32}{38\cdots 07}a^{11}+\frac{16\cdots 15}{38\cdots 07}a^{10}+\frac{67\cdots 93}{38\cdots 07}a^{9}-\frac{18\cdots 23}{77\cdots 14}a^{8}-\frac{37\cdots 38}{38\cdots 07}a^{7}-\frac{51\cdots 75}{77\cdots 14}a^{6}-\frac{52\cdots 82}{38\cdots 07}a^{5}-\frac{99\cdots 01}{38\cdots 07}a^{4}-\frac{61\cdots 53}{38\cdots 07}a^{3}-\frac{25\cdots 50}{38\cdots 07}a^{2}-\frac{12\cdots 98}{38\cdots 07}a-\frac{133448939787317}{48766626872993}$, $\frac{24\cdots 07}{77\cdots 14}a^{15}+\frac{81\cdots 59}{77\cdots 14}a^{14}-\frac{21\cdots 05}{91\cdots 84}a^{13}+\frac{16\cdots 79}{15\cdots 28}a^{12}-\frac{18\cdots 69}{77\cdots 14}a^{11}-\frac{32\cdots 97}{77\cdots 14}a^{10}+\frac{24\cdots 97}{45\cdots 42}a^{9}-\frac{33\cdots 92}{38\cdots 07}a^{8}-\frac{13\cdots 58}{38\cdots 07}a^{7}+\frac{17\cdots 51}{77\cdots 14}a^{6}-\frac{77\cdots 27}{38\cdots 07}a^{5}-\frac{27\cdots 79}{38\cdots 07}a^{4}+\frac{10\cdots 90}{22\cdots 71}a^{3}+\frac{17\cdots 89}{38\cdots 07}a^{2}-\frac{14\cdots 98}{38\cdots 07}a+\frac{21\cdots 03}{829032656840881}$
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| Regulator: | \( 139483.66642632167 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 139483.66642632167 \cdot 2}{2\cdot\sqrt{484116351470433472610304}}\cr\approx \mathstrut & 0.486953425967176 \end{aligned}\]
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
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.2.1073741824.1, 8.2.86973087744.1, 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 | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
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\(2\)
| 2.1.16.66j1.875 | $x^{16} + 8 x^{12} + 16 x^{11} + 8 x^{10} + 4 x^{8} + 8 x^{4} + 16 x^{3} + 2$ | $16$ | $1$ | $66$ | 16T259 | $$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$$ |
|
\(3\)
| 3.8.2.8a1.2 | $x^{16} + 4 x^{13} + 2 x^{12} + 8 x^{10} + 8 x^{9} + 5 x^{8} + 8 x^{7} + 12 x^{6} + 12 x^{5} + 8 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 7$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $$[\ ]_{2}^{8}$$ |