Normalized defining polynomial
\( x^{16} - 2 x^{15} - 5 x^{14} + 78 x^{13} - 475 x^{12} + 3641 x^{11} - 1085 x^{10} - 63940 x^{9} + \cdots + 3528539 \)
Invariants
| Degree: | $16$ |
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| Signature: | $[0, 8]$ |
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| Discriminant: |
\(2946406684936922037860575506582241\)
\(\medspace = 37^{14}\cdot 83^{6}\)
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| Root discriminant: | \(123.55\) |
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| Galois root discriminant: | $37^{7/8}83^{1/2}\approx 214.642419295045$ | ||
| Ramified primes: |
\(37\), \(83\)
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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: | 4.0.50653.1 | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}-\frac{3}{7}a^{5}+\frac{2}{7}a^{4}+\frac{1}{7}a^{3}-\frac{3}{7}a^{2}+\frac{2}{7}a$, $\frac{1}{7}a^{7}-\frac{1}{7}a$, $\frac{1}{7}a^{8}-\frac{1}{7}a^{2}$, $\frac{1}{7}a^{9}-\frac{1}{7}a^{3}$, $\frac{1}{7}a^{10}-\frac{1}{7}a^{4}$, $\frac{1}{49}a^{11}-\frac{1}{49}a^{10}+\frac{1}{49}a^{9}+\frac{2}{49}a^{8}+\frac{2}{49}a^{7}+\frac{1}{49}a^{6}+\frac{17}{49}a^{5}-\frac{18}{49}a^{4}+\frac{16}{49}a^{2}-\frac{3}{7}a$, $\frac{1}{49}a^{12}+\frac{3}{49}a^{9}-\frac{3}{49}a^{8}+\frac{3}{49}a^{7}-\frac{3}{49}a^{6}+\frac{13}{49}a^{5}-\frac{11}{49}a^{4}-\frac{5}{49}a^{3}+\frac{16}{49}a^{2}-\frac{2}{7}a$, $\frac{1}{49}a^{13}+\frac{3}{49}a^{10}-\frac{3}{49}a^{9}+\frac{3}{49}a^{8}-\frac{3}{49}a^{7}-\frac{1}{49}a^{6}-\frac{18}{49}a^{5}+\frac{16}{49}a^{4}+\frac{2}{49}a^{3}-\frac{3}{7}a^{2}+\frac{3}{7}a$, $\frac{1}{343}a^{14}+\frac{2}{343}a^{13}-\frac{1}{343}a^{12}+\frac{3}{343}a^{11}+\frac{3}{343}a^{10}-\frac{13}{343}a^{9}+\frac{13}{343}a^{8}-\frac{10}{343}a^{7}+\frac{11}{343}a^{6}-\frac{68}{343}a^{5}+\frac{101}{343}a^{4}-\frac{75}{343}a^{3}+\frac{166}{343}a^{2}-\frac{5}{49}a-\frac{3}{7}$, $\frac{1}{20\cdots 89}a^{15}+\frac{30\cdots 58}{29\cdots 27}a^{14}+\frac{18\cdots 51}{20\cdots 89}a^{13}+\frac{10\cdots 00}{20\cdots 89}a^{12}-\frac{20\cdots 25}{20\cdots 89}a^{11}+\frac{11\cdots 25}{20\cdots 89}a^{10}-\frac{88\cdots 89}{20\cdots 89}a^{9}-\frac{13\cdots 75}{20\cdots 89}a^{8}+\frac{21\cdots 49}{20\cdots 89}a^{7}+\frac{13\cdots 29}{20\cdots 89}a^{6}-\frac{21\cdots 74}{20\cdots 89}a^{5}-\frac{56\cdots 03}{20\cdots 89}a^{4}+\frac{75\cdots 69}{20\cdots 89}a^{3}-\frac{28\cdots 08}{20\cdots 89}a^{2}+\frac{19\cdots 74}{29\cdots 27}a-\frac{35\cdots 78}{42\cdots 61}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $7$ |
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: | $7$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{48\cdots 98}{76\cdots 89}a^{15}-\frac{17\cdots 48}{10\cdots 27}a^{14}-\frac{27\cdots 96}{76\cdots 89}a^{13}+\frac{40\cdots 82}{76\cdots 89}a^{12}-\frac{24\cdots 22}{76\cdots 89}a^{11}+\frac{18\cdots 62}{76\cdots 89}a^{10}-\frac{11\cdots 74}{76\cdots 89}a^{9}-\frac{33\cdots 56}{76\cdots 89}a^{8}+\frac{10\cdots 28}{76\cdots 89}a^{7}-\frac{14\cdots 86}{76\cdots 89}a^{6}+\frac{85\cdots 84}{76\cdots 89}a^{5}-\frac{14\cdots 80}{76\cdots 89}a^{4}+\frac{27\cdots 02}{76\cdots 89}a^{3}+\frac{15\cdots 32}{76\cdots 89}a^{2}-\frac{39\cdots 48}{10\cdots 27}a-\frac{28\cdots 49}{15\cdots 61}$, $\frac{39\cdots 14}{14\cdots 51}a^{15}-\frac{21\cdots 44}{21\cdots 93}a^{14}-\frac{83\cdots 62}{14\cdots 51}a^{13}+\frac{39\cdots 20}{14\cdots 51}a^{12}-\frac{26\cdots 78}{14\cdots 51}a^{11}+\frac{17\cdots 94}{14\cdots 51}a^{10}-\frac{24\cdots 00}{14\cdots 51}a^{9}-\frac{29\cdots 72}{14\cdots 51}a^{8}+\frac{14\cdots 98}{14\cdots 51}a^{7}-\frac{19\cdots 94}{14\cdots 51}a^{6}-\frac{15\cdots 32}{14\cdots 51}a^{5}+\frac{52\cdots 62}{14\cdots 51}a^{4}-\frac{20\cdots 04}{14\cdots 51}a^{3}+\frac{59\cdots 06}{14\cdots 51}a^{2}+\frac{10\cdots 58}{21\cdots 93}a+\frac{60\cdots 23}{30\cdots 99}$, $\frac{17\cdots 30}{20\cdots 89}a^{15}+\frac{18\cdots 52}{29\cdots 27}a^{14}-\frac{62\cdots 80}{20\cdots 89}a^{13}+\frac{11\cdots 42}{20\cdots 89}a^{12}-\frac{51\cdots 20}{20\cdots 89}a^{11}+\frac{47\cdots 70}{20\cdots 89}a^{10}+\frac{11\cdots 93}{20\cdots 89}a^{9}-\frac{81\cdots 41}{20\cdots 89}a^{8}+\frac{76\cdots 71}{20\cdots 89}a^{7}+\frac{14\cdots 32}{20\cdots 89}a^{6}+\frac{90\cdots 07}{20\cdots 89}a^{5}-\frac{26\cdots 35}{20\cdots 89}a^{4}+\frac{33\cdots 72}{20\cdots 89}a^{3}-\frac{35\cdots 77}{20\cdots 89}a^{2}-\frac{57\cdots 33}{29\cdots 27}a-\frac{38\cdots 45}{42\cdots 61}$, $\frac{28\cdots 16}{20\cdots 89}a^{15}-\frac{48\cdots 91}{29\cdots 27}a^{14}-\frac{21\cdots 57}{20\cdots 89}a^{13}+\frac{22\cdots 87}{20\cdots 89}a^{12}+\frac{11\cdots 15}{20\cdots 89}a^{11}-\frac{68\cdots 42}{20\cdots 89}a^{10}+\frac{11\cdots 45}{20\cdots 89}a^{9}+\frac{79\cdots 37}{20\cdots 89}a^{8}-\frac{28\cdots 98}{20\cdots 89}a^{7}-\frac{15\cdots 86}{20\cdots 89}a^{6}+\frac{19\cdots 81}{20\cdots 89}a^{5}-\frac{25\cdots 44}{20\cdots 89}a^{4}+\frac{23\cdots 57}{20\cdots 89}a^{3}+\frac{92\cdots 20}{20\cdots 89}a^{2}-\frac{43\cdots 17}{29\cdots 27}a-\frac{34\cdots 78}{42\cdots 61}$, $\frac{65\cdots 50}{59\cdots 61}a^{15}+\frac{10\cdots 42}{85\cdots 23}a^{14}-\frac{17\cdots 31}{59\cdots 61}a^{13}+\frac{14\cdots 12}{59\cdots 61}a^{12}-\frac{25\cdots 42}{59\cdots 61}a^{11}-\frac{42\cdots 20}{59\cdots 61}a^{10}+\frac{24\cdots 49}{59\cdots 61}a^{9}-\frac{95\cdots 05}{59\cdots 61}a^{8}+\frac{10\cdots 23}{59\cdots 61}a^{7}-\frac{29\cdots 69}{59\cdots 61}a^{6}+\frac{13\cdots 70}{59\cdots 61}a^{5}-\frac{25\cdots 89}{59\cdots 61}a^{4}-\frac{19\cdots 16}{59\cdots 61}a^{3}-\frac{35\cdots 44}{59\cdots 61}a^{2}-\frac{57\cdots 41}{85\cdots 23}a-\frac{38\cdots 74}{12\cdots 89}$, $\frac{27\cdots 24}{20\cdots 89}a^{15}-\frac{25\cdots 98}{29\cdots 27}a^{14}+\frac{70\cdots 67}{20\cdots 89}a^{13}-\frac{12\cdots 29}{20\cdots 89}a^{12}-\frac{70\cdots 37}{20\cdots 89}a^{11}+\frac{13\cdots 45}{20\cdots 89}a^{10}-\frac{64\cdots 57}{20\cdots 89}a^{9}+\frac{13\cdots 70}{20\cdots 89}a^{8}-\frac{15\cdots 95}{20\cdots 89}a^{7}+\frac{63\cdots 56}{20\cdots 89}a^{6}-\frac{22\cdots 73}{20\cdots 89}a^{5}+\frac{26\cdots 99}{20\cdots 89}a^{4}+\frac{24\cdots 18}{20\cdots 89}a^{3}-\frac{90\cdots 94}{20\cdots 89}a^{2}+\frac{47\cdots 11}{29\cdots 27}a+\frac{38\cdots 11}{42\cdots 61}$, $\frac{13\cdots 02}{20\cdots 89}a^{15}+\frac{46\cdots 12}{29\cdots 27}a^{14}-\frac{20\cdots 08}{20\cdots 89}a^{13}+\frac{74\cdots 94}{20\cdots 89}a^{12}-\frac{34\cdots 41}{20\cdots 89}a^{11}+\frac{31\cdots 38}{20\cdots 89}a^{10}+\frac{14\cdots 81}{20\cdots 89}a^{9}-\frac{48\cdots 51}{20\cdots 89}a^{8}-\frac{49\cdots 35}{20\cdots 89}a^{7}-\frac{13\cdots 11}{20\cdots 89}a^{6}+\frac{85\cdots 31}{20\cdots 89}a^{5}-\frac{18\cdots 17}{20\cdots 89}a^{4}+\frac{60\cdots 07}{20\cdots 89}a^{3}+\frac{11\cdots 77}{20\cdots 89}a^{2}-\frac{16\cdots 89}{29\cdots 27}a-\frac{54\cdots 36}{42\cdots 61}$
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| Regulator: | \( 87394636266.6 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 87394636266.6 \cdot 1}{2\cdot\sqrt{2946406684936922037860575506582241}}\cr\approx \mathstrut & 1.95545301260 \end{aligned}\] (assuming GRH)
Galois group
$\OD_{16}:C_2$ (as 16T41):
| A solvable group of order 32 |
| The 11 conjugacy class representatives for $\OD_{16}:C_2$ |
| Character table for $\OD_{16}:C_2$ |
Intermediate fields
| \(\Q(\sqrt{37}) \), 4.0.50653.1, 4.2.113627.1, 4.2.4204199.1, 8.0.653985701569237.1 x2, 8.0.17675289231601.1 |
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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.1.0.1}{1} }^{16}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(37\)
| 37.2.8.14a1.2 | $x^{16} + 264 x^{15} + 30508 x^{14} + 2016168 x^{13} + 83380486 x^{12} + 2211728904 x^{11} + 36827048860 x^{10} + 354177608664 x^{9} + 1553050257889 x^{8} + 708355217328 x^{7} + 147308195440 x^{6} + 17693831232 x^{5} + 1334087776 x^{4} + 64517376 x^{3} + 1952512 x^{2} + 33792 x + 293$ | $8$ | $2$ | $14$ | $C_8: C_2$ | $$[\ ]_{8}^{2}$$ |
|
\(83\)
| 83.2.1.0a1.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 83.1.2.1a1.2 | $x^{2} + 166$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 83.2.1.0a1.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 83.1.2.1a1.2 | $x^{2} + 166$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 83.2.2.2a1.2 | $x^{4} + 164 x^{3} + 6728 x^{2} + 328 x + 87$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 83.2.2.2a1.2 | $x^{4} + 164 x^{3} + 6728 x^{2} + 328 x + 87$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |