Normalized defining polynomial
\( x^{16} - 2 x^{15} - 10 x^{14} + 42 x^{13} + 122 x^{12} - 708 x^{11} + 749 x^{10} + 2585 x^{9} + \cdots + 848 \)
Invariants
| Degree: | $16$ |
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| Signature: | $[0, 8]$ |
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| Discriminant: |
\(5823129643646193092027401\)
\(\medspace = 7^{12}\cdot 29^{10}\)
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| Root discriminant: | \(35.30\) |
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| Galois root discriminant: | $7^{3/4}29^{3/4}\approx 53.78015238265253$ | ||
| Ramified primes: |
\(7\), \(29\)
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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. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-7}, \sqrt{29})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{80}a^{14}-\frac{9}{80}a^{13}-\frac{9}{80}a^{12}-\frac{17}{80}a^{11}-\frac{1}{80}a^{10}-\frac{7}{80}a^{9}-\frac{1}{4}a^{8}-\frac{1}{80}a^{7}+\frac{9}{80}a^{6}+\frac{31}{80}a^{5}-\frac{3}{20}a^{4}+\frac{31}{80}a^{3}+\frac{9}{20}a^{2}-\frac{1}{5}a+\frac{3}{10}$, $\frac{1}{50\cdots 60}a^{15}+\frac{46\cdots 91}{12\cdots 40}a^{14}+\frac{25\cdots 37}{25\cdots 80}a^{13}+\frac{30\cdots 13}{25\cdots 80}a^{12}+\frac{44\cdots 29}{25\cdots 80}a^{11}-\frac{60\cdots 45}{25\cdots 68}a^{10}+\frac{84\cdots 29}{50\cdots 60}a^{9}+\frac{74\cdots 19}{50\cdots 60}a^{8}-\frac{49\cdots 71}{12\cdots 40}a^{7}-\frac{40\cdots 33}{12\cdots 40}a^{6}+\frac{68\cdots 91}{50\cdots 60}a^{5}+\frac{45\cdots 51}{10\cdots 72}a^{4}-\frac{76\cdots 81}{50\cdots 60}a^{3}-\frac{56\cdots 57}{12\cdots 40}a^{2}-\frac{16\cdots 83}{63\cdots 20}a+\frac{29\cdots 83}{11\cdots 40}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{5}$, which has order $5$ (assuming GRH) |
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| Narrow class group: | $C_{5}$, which has order $5$ (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: |
$\frac{22\cdots 51}{50\cdots 60}a^{15}-\frac{36\cdots 09}{25\cdots 80}a^{14}-\frac{77\cdots 87}{12\cdots 40}a^{13}+\frac{34\cdots 61}{12\cdots 40}a^{12}+\frac{76\cdots 53}{12\cdots 40}a^{11}-\frac{11\cdots 59}{25\cdots 80}a^{10}+\frac{16\cdots 93}{50\cdots 60}a^{9}+\frac{11\cdots 89}{50\cdots 60}a^{8}-\frac{12\cdots 51}{25\cdots 80}a^{7}-\frac{76\cdots 57}{50\cdots 36}a^{6}+\frac{54\cdots 99}{50\cdots 60}a^{5}-\frac{57\cdots 31}{50\cdots 60}a^{4}-\frac{21\cdots 13}{50\cdots 60}a^{3}+\frac{15\cdots 53}{25\cdots 68}a^{2}+\frac{17\cdots 41}{63\cdots 20}a+\frac{79\cdots 11}{11\cdots 40}$, $\frac{19\cdots 89}{80\cdots 70}a^{15}-\frac{17\cdots 41}{64\cdots 60}a^{14}-\frac{17\cdots 31}{64\cdots 60}a^{13}+\frac{50\cdots 33}{64\cdots 60}a^{12}+\frac{23\cdots 89}{64\cdots 60}a^{11}-\frac{91\cdots 51}{64\cdots 60}a^{10}+\frac{35\cdots 11}{64\cdots 60}a^{9}+\frac{11\cdots 27}{16\cdots 40}a^{8}-\frac{58\cdots 59}{64\cdots 60}a^{7}-\frac{36\cdots 21}{12\cdots 32}a^{6}+\frac{15\cdots 13}{64\cdots 60}a^{5}-\frac{86\cdots 09}{80\cdots 70}a^{4}-\frac{69\cdots 91}{64\cdots 60}a^{3}+\frac{21\cdots 95}{32\cdots 08}a^{2}+\frac{50\cdots 57}{80\cdots 70}a-\frac{45\cdots 83}{15\cdots 90}$, $\frac{18\cdots 89}{14\cdots 16}a^{15}-\frac{47\cdots 37}{29\cdots 32}a^{14}-\frac{39\cdots 19}{29\cdots 32}a^{13}+\frac{12\cdots 53}{29\cdots 32}a^{12}+\frac{53\cdots 13}{29\cdots 32}a^{11}-\frac{21\cdots 03}{29\cdots 32}a^{10}+\frac{11\cdots 25}{29\cdots 32}a^{9}+\frac{50\cdots 49}{14\cdots 16}a^{8}-\frac{15\cdots 55}{29\cdots 32}a^{7}-\frac{29\cdots 17}{29\cdots 32}a^{6}+\frac{37\cdots 71}{29\cdots 32}a^{5}-\frac{99\cdots 89}{14\cdots 16}a^{4}-\frac{18\cdots 37}{29\cdots 32}a^{3}+\frac{11\cdots 71}{73\cdots 08}a^{2}+\frac{12\cdots 33}{36\cdots 54}a-\frac{95\cdots 69}{69\cdots 18}$, $\frac{98\cdots 59}{50\cdots 60}a^{15}-\frac{75\cdots 33}{25\cdots 80}a^{14}-\frac{26\cdots 79}{12\cdots 40}a^{13}+\frac{91\cdots 83}{12\cdots 40}a^{12}+\frac{34\cdots 49}{12\cdots 40}a^{11}-\frac{31\cdots 79}{25\cdots 80}a^{10}+\frac{85\cdots 53}{10\cdots 72}a^{9}+\frac{28\cdots 41}{50\cdots 60}a^{8}-\frac{24\cdots 27}{25\cdots 80}a^{7}-\frac{74\cdots 73}{25\cdots 80}a^{6}+\frac{11\cdots 07}{50\cdots 60}a^{5}-\frac{91\cdots 71}{50\cdots 60}a^{4}-\frac{42\cdots 81}{50\cdots 60}a^{3}+\frac{10\cdots 19}{12\cdots 40}a^{2}+\frac{14\cdots 07}{63\cdots 20}a-\frac{38\cdots 85}{23\cdots 28}$, $\frac{61\cdots 39}{50\cdots 60}a^{15}-\frac{10\cdots 41}{25\cdots 80}a^{14}-\frac{15\cdots 63}{12\cdots 40}a^{13}+\frac{94\cdots 69}{12\cdots 40}a^{12}+\frac{14\cdots 17}{12\cdots 40}a^{11}-\frac{30\cdots 51}{25\cdots 80}a^{10}+\frac{83\cdots 77}{50\cdots 60}a^{9}+\frac{20\cdots 01}{50\cdots 60}a^{8}-\frac{34\cdots 19}{25\cdots 80}a^{7}+\frac{23\cdots 91}{50\cdots 36}a^{6}+\frac{11\cdots 11}{50\cdots 60}a^{5}-\frac{18\cdots 19}{50\cdots 60}a^{4}-\frac{49\cdots 57}{50\cdots 60}a^{3}+\frac{55\cdots 09}{25\cdots 68}a^{2}-\frac{21\cdots 71}{63\cdots 20}a-\frac{52\cdots 01}{11\cdots 40}$, $\frac{41\cdots 61}{50\cdots 60}a^{15}-\frac{77\cdots 49}{12\cdots 40}a^{14}-\frac{23\cdots 43}{25\cdots 80}a^{13}+\frac{58\cdots 73}{25\cdots 80}a^{12}+\frac{33\cdots 29}{25\cdots 80}a^{11}-\frac{10\cdots 03}{25\cdots 68}a^{10}+\frac{11\cdots 49}{50\cdots 60}a^{9}+\frac{12\cdots 59}{50\cdots 60}a^{8}-\frac{28\cdots 51}{12\cdots 40}a^{7}-\frac{25\cdots 23}{12\cdots 40}a^{6}+\frac{41\cdots 51}{50\cdots 60}a^{5}-\frac{75\cdots 53}{10\cdots 72}a^{4}-\frac{24\cdots 21}{50\cdots 60}a^{3}+\frac{37\cdots 53}{12\cdots 40}a^{2}+\frac{12\cdots 67}{63\cdots 20}a-\frac{91\cdots 37}{11\cdots 40}$, $\frac{76\cdots 49}{25\cdots 80}a^{15}+\frac{24\cdots 01}{31\cdots 10}a^{14}-\frac{45\cdots 71}{12\cdots 40}a^{13}-\frac{33\cdots 57}{12\cdots 40}a^{12}+\frac{89\cdots 69}{12\cdots 40}a^{11}+\frac{22\cdots 43}{15\cdots 05}a^{10}-\frac{10\cdots 23}{25\cdots 80}a^{9}+\frac{17\cdots 71}{25\cdots 80}a^{8}+\frac{58\cdots 19}{31\cdots 10}a^{7}-\frac{23\cdots 09}{12\cdots 84}a^{6}+\frac{18\cdots 51}{25\cdots 80}a^{5}+\frac{19\cdots 51}{25\cdots 80}a^{4}+\frac{32\cdots 43}{25\cdots 80}a^{3}-\frac{26\cdots 61}{12\cdots 84}a^{2}-\frac{73\cdots 33}{15\cdots 05}a+\frac{55\cdots 19}{59\cdots 70}$
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| Regulator: | \( 343368.044177 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 343368.044177 \cdot 5}{2\cdot\sqrt{5823129643646193092027401}}\cr\approx \mathstrut & 0.864093226339 \end{aligned}\] (assuming GRH)
Galois group
$C_4\wr C_2$ (as 16T42):
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $C_4\wr C_2$ |
| Character table for $C_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{29}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-203}) \), 4.2.5887.1 x2, 4.0.1421.1 x2, \(\Q(\sqrt{-7}, \sqrt{29})\), 8.0.2869341461.1, 8.0.2413116168701.1, 8.0.1698181681.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 32 |
| Degree 8 siblings: | 8.0.2413116168701.1, 8.0.2869341461.1 |
| Degree 16 sibling: | 16.4.4897252030306448390395044241.1 |
| Minimal sibling: | 8.0.2869341461.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.1.0.1}{1} }^{16}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.2.4.6a1.3 | $x^{8} + 24 x^{7} + 228 x^{6} + 1080 x^{5} + 2646 x^{4} + 3240 x^{3} + 2052 x^{2} + 655 x + 109$ | $4$ | $2$ | $6$ | $Q_8$ | $$[\ ]_{4}^{2}$$ |
| 7.2.4.6a1.3 | $x^{8} + 24 x^{7} + 228 x^{6} + 1080 x^{5} + 2646 x^{4} + 3240 x^{3} + 2052 x^{2} + 655 x + 109$ | $4$ | $2$ | $6$ | $Q_8$ | $$[\ ]_{4}^{2}$$ | |
|
\(29\)
| 29.2.4.6a1.4 | $x^{8} + 96 x^{7} + 3464 x^{6} + 55872 x^{5} + 345624 x^{4} + 111744 x^{3} + 13856 x^{2} + 913 x + 799$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ |
| 29.4.2.4a1.2 | $x^{8} + 4 x^{6} + 30 x^{5} + 8 x^{4} + 60 x^{3} + 233 x^{2} + 60 x + 33$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |