Normalized defining polynomial
\( x^{16} + 8 x^{14} - 16 x^{13} - 212 x^{12} + 368 x^{11} - 224 x^{10} + 592 x^{9} + 2964 x^{8} - 13056 x^{7} + \cdots - 2 \)
Invariants
| Degree: | $16$ |
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| Signature: | $[8, 4]$ |
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| Discriminant: |
\(106341808682864896865468416\)
\(\medspace = 2^{64}\cdot 7^{8}\)
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| Root discriminant: | \(42.33\) |
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| Galois root discriminant: | $2^{137/32}7^{3/4}\approx 83.67740413998479$ | ||
| Ramified primes: |
\(2\), \(7\)
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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. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{36\cdots 77}a^{15}+\frac{60\cdots 35}{36\cdots 77}a^{14}+\frac{49\cdots 83}{36\cdots 77}a^{13}-\frac{57\cdots 79}{36\cdots 77}a^{12}-\frac{16\cdots 89}{36\cdots 77}a^{11}+\frac{31\cdots 71}{36\cdots 77}a^{10}-\frac{14\cdots 11}{36\cdots 77}a^{9}+\frac{23\cdots 70}{36\cdots 77}a^{8}+\frac{12\cdots 73}{36\cdots 77}a^{7}+\frac{12\cdots 82}{36\cdots 77}a^{6}-\frac{14\cdots 23}{36\cdots 77}a^{5}-\frac{21\cdots 27}{36\cdots 77}a^{4}-\frac{26\cdots 32}{12\cdots 59}a^{3}-\frac{15\cdots 01}{36\cdots 77}a^{2}-\frac{55\cdots 12}{12\cdots 59}a+\frac{14\cdots 18}{36\cdots 77}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{2}$, which has order $4$ (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: |
$\frac{19\cdots 84}{11\cdots 91}a^{15}+\frac{31\cdots 54}{36\cdots 97}a^{14}+\frac{15\cdots 20}{11\cdots 91}a^{13}-\frac{77\cdots 28}{36\cdots 97}a^{12}-\frac{42\cdots 12}{11\cdots 91}a^{11}+\frac{49\cdots 96}{11\cdots 91}a^{10}-\frac{11\cdots 00}{11\cdots 91}a^{9}+\frac{10\cdots 09}{11\cdots 91}a^{8}+\frac{62\cdots 72}{11\cdots 91}a^{7}-\frac{22\cdots 96}{11\cdots 91}a^{6}+\frac{33\cdots 16}{11\cdots 91}a^{5}-\frac{92\cdots 02}{36\cdots 97}a^{4}+\frac{10\cdots 92}{11\cdots 91}a^{3}-\frac{32\cdots 72}{36\cdots 97}a^{2}-\frac{18\cdots 12}{11\cdots 91}a-\frac{26\cdots 99}{11\cdots 91}$, $\frac{59\cdots 58}{36\cdots 77}a^{15}+\frac{87\cdots 77}{12\cdots 59}a^{14}+\frac{14\cdots 52}{12\cdots 59}a^{13}+\frac{96\cdots 13}{36\cdots 77}a^{12}-\frac{17\cdots 38}{36\cdots 77}a^{11}-\frac{11\cdots 45}{12\cdots 59}a^{10}+\frac{93\cdots 52}{36\cdots 77}a^{9}+\frac{65\cdots 56}{12\cdots 59}a^{8}+\frac{31\cdots 20}{36\cdots 77}a^{7}-\frac{18\cdots 60}{36\cdots 77}a^{6}-\frac{74\cdots 90}{12\cdots 59}a^{5}+\frac{15\cdots 11}{12\cdots 59}a^{4}-\frac{16\cdots 00}{12\cdots 59}a^{3}+\frac{22\cdots 18}{36\cdots 77}a^{2}-\frac{77\cdots 84}{12\cdots 59}a-\frac{38\cdots 41}{36\cdots 77}$, $\frac{17\cdots 52}{36\cdots 77}a^{15}-\frac{14\cdots 85}{36\cdots 77}a^{14}+\frac{12\cdots 12}{36\cdots 77}a^{13}-\frac{39\cdots 78}{36\cdots 77}a^{12}-\frac{34\cdots 22}{36\cdots 77}a^{11}+\frac{93\cdots 22}{36\cdots 77}a^{10}-\frac{73\cdots 02}{36\cdots 77}a^{9}+\frac{12\cdots 39}{36\cdots 77}a^{8}+\frac{14\cdots 90}{12\cdots 59}a^{7}-\frac{27\cdots 59}{36\cdots 77}a^{6}+\frac{56\cdots 74}{36\cdots 77}a^{5}-\frac{69\cdots 71}{36\cdots 77}a^{4}+\frac{51\cdots 08}{36\cdots 77}a^{3}-\frac{70\cdots 28}{12\cdots 59}a^{2}+\frac{10\cdots 66}{12\cdots 59}a+\frac{26\cdots 55}{36\cdots 77}$, $\frac{19\cdots 30}{12\cdots 59}a^{15}+\frac{16\cdots 61}{12\cdots 59}a^{14}+\frac{46\cdots 84}{36\cdots 77}a^{13}-\frac{85\cdots 61}{36\cdots 77}a^{12}-\frac{12\cdots 26}{36\cdots 77}a^{11}+\frac{19\cdots 47}{36\cdots 77}a^{10}-\frac{43\cdots 84}{12\cdots 59}a^{9}+\frac{33\cdots 55}{36\cdots 77}a^{8}+\frac{57\cdots 88}{12\cdots 59}a^{7}-\frac{72\cdots 52}{36\cdots 77}a^{6}+\frac{13\cdots 22}{36\cdots 77}a^{5}-\frac{45\cdots 05}{12\cdots 59}a^{4}+\frac{83\cdots 24}{36\cdots 77}a^{3}-\frac{25\cdots 70}{36\cdots 77}a^{2}+\frac{17\cdots 88}{36\cdots 77}a+\frac{61\cdots 19}{36\cdots 77}$, $\frac{15\cdots 14}{36\cdots 77}a^{15}+\frac{88\cdots 19}{36\cdots 77}a^{14}+\frac{13\cdots 73}{36\cdots 77}a^{13}-\frac{60\cdots 19}{12\cdots 59}a^{12}-\frac{34\cdots 52}{36\cdots 77}a^{11}+\frac{39\cdots 19}{36\cdots 77}a^{10}-\frac{42\cdots 74}{12\cdots 59}a^{9}+\frac{84\cdots 92}{36\cdots 77}a^{8}+\frac{17\cdots 48}{12\cdots 59}a^{7}-\frac{17\cdots 15}{36\cdots 77}a^{6}+\frac{28\cdots 84}{36\cdots 77}a^{5}-\frac{23\cdots 39}{36\cdots 77}a^{4}+\frac{30\cdots 82}{12\cdots 59}a^{3}-\frac{54\cdots 95}{36\cdots 77}a^{2}-\frac{21\cdots 88}{12\cdots 59}a-\frac{33\cdots 39}{12\cdots 59}$, $\frac{49\cdots 38}{36\cdots 77}a^{15}+\frac{67\cdots 97}{36\cdots 77}a^{14}+\frac{43\cdots 61}{36\cdots 77}a^{13}-\frac{74\cdots 56}{12\cdots 59}a^{12}-\frac{11\cdots 92}{36\cdots 77}a^{11}+\frac{33\cdots 77}{36\cdots 77}a^{10}+\frac{47\cdots 18}{36\cdots 77}a^{9}+\frac{24\cdots 70}{36\cdots 77}a^{8}+\frac{60\cdots 24}{12\cdots 59}a^{7}-\frac{42\cdots 53}{36\cdots 77}a^{6}+\frac{43\cdots 68}{36\cdots 77}a^{5}-\frac{22\cdots 93}{12\cdots 59}a^{4}-\frac{67\cdots 94}{12\cdots 59}a^{3}+\frac{55\cdots 79}{12\cdots 59}a^{2}-\frac{37\cdots 72}{36\cdots 77}a-\frac{10\cdots 77}{12\cdots 59}$, $\frac{54\cdots 44}{12\cdots 59}a^{15}+\frac{58\cdots 09}{36\cdots 77}a^{14}+\frac{44\cdots 81}{12\cdots 59}a^{13}-\frac{21\cdots 47}{36\cdots 77}a^{12}-\frac{35\cdots 54}{36\cdots 77}a^{11}+\frac{47\cdots 65}{36\cdots 77}a^{10}-\frac{70\cdots 46}{12\cdots 59}a^{9}+\frac{30\cdots 67}{12\cdots 59}a^{8}+\frac{51\cdots 98}{36\cdots 77}a^{7}-\frac{19\cdots 38}{36\cdots 77}a^{6}+\frac{32\cdots 66}{36\cdots 77}a^{5}-\frac{99\cdots 70}{12\cdots 59}a^{4}+\frac{15\cdots 90}{36\cdots 77}a^{3}-\frac{39\cdots 43}{36\cdots 77}a^{2}+\frac{10\cdots 66}{12\cdots 59}a+\frac{68\cdots 57}{12\cdots 59}$, $\frac{58\cdots 40}{36\cdots 77}a^{15}+\frac{95\cdots 63}{36\cdots 77}a^{14}+\frac{15\cdots 62}{12\cdots 59}a^{13}-\frac{85\cdots 18}{36\cdots 77}a^{12}-\frac{41\cdots 79}{12\cdots 59}a^{11}+\frac{19\cdots 43}{36\cdots 77}a^{10}-\frac{35\cdots 37}{12\cdots 59}a^{9}+\frac{33\cdots 73}{36\cdots 77}a^{8}+\frac{59\cdots 26}{12\cdots 59}a^{7}-\frac{24\cdots 18}{12\cdots 59}a^{6}+\frac{12\cdots 64}{36\cdots 77}a^{5}-\frac{12\cdots 56}{36\cdots 77}a^{4}+\frac{69\cdots 33}{36\cdots 77}a^{3}-\frac{18\cdots 30}{36\cdots 77}a^{2}+\frac{53\cdots 52}{36\cdots 77}a+\frac{44\cdots 27}{36\cdots 77}$, $\frac{35\cdots 93}{36\cdots 77}a^{15}+\frac{14\cdots 14}{12\cdots 59}a^{14}+\frac{10\cdots 40}{12\cdots 59}a^{13}-\frac{20\cdots 79}{36\cdots 77}a^{12}-\frac{80\cdots 68}{36\cdots 77}a^{11}+\frac{11\cdots 07}{12\cdots 59}a^{10}+\frac{92\cdots 64}{36\cdots 77}a^{9}+\frac{62\cdots 59}{12\cdots 59}a^{8}+\frac{12\cdots 70}{36\cdots 77}a^{7}-\frac{31\cdots 37}{36\cdots 77}a^{6}+\frac{13\cdots 12}{12\cdots 59}a^{5}-\frac{63\cdots 70}{12\cdots 59}a^{4}-\frac{45\cdots 17}{12\cdots 59}a^{3}+\frac{50\cdots 06}{36\cdots 77}a^{2}-\frac{35\cdots 70}{12\cdots 59}a-\frac{13\cdots 53}{36\cdots 77}$, $\frac{63\cdots 74}{36\cdots 77}a^{15}+\frac{21\cdots 96}{12\cdots 59}a^{14}+\frac{50\cdots 80}{36\cdots 77}a^{13}-\frac{31\cdots 71}{12\cdots 59}a^{12}-\frac{13\cdots 65}{36\cdots 77}a^{11}+\frac{21\cdots 11}{36\cdots 77}a^{10}-\frac{12\cdots 73}{36\cdots 77}a^{9}+\frac{36\cdots 51}{36\cdots 77}a^{8}+\frac{19\cdots 65}{36\cdots 77}a^{7}-\frac{80\cdots 10}{36\cdots 77}a^{6}+\frac{14\cdots 88}{36\cdots 77}a^{5}-\frac{48\cdots 85}{12\cdots 59}a^{4}+\frac{83\cdots 15}{36\cdots 77}a^{3}-\frac{82\cdots 20}{12\cdots 59}a^{2}+\frac{15\cdots 28}{36\cdots 77}a+\frac{56\cdots 21}{36\cdots 77}$, $\frac{66\cdots 35}{12\cdots 59}a^{15}-\frac{51\cdots 30}{12\cdots 59}a^{14}+\frac{16\cdots 53}{36\cdots 77}a^{13}-\frac{33\cdots 51}{36\cdots 77}a^{12}-\frac{42\cdots 25}{36\cdots 77}a^{11}+\frac{25\cdots 16}{12\cdots 59}a^{10}-\frac{51\cdots 42}{36\cdots 77}a^{9}+\frac{12\cdots 84}{36\cdots 77}a^{8}+\frac{19\cdots 27}{12\cdots 59}a^{7}-\frac{26\cdots 01}{36\cdots 77}a^{6}+\frac{16\cdots 16}{12\cdots 59}a^{5}-\frac{53\cdots 15}{36\cdots 77}a^{4}+\frac{34\cdots 75}{36\cdots 77}a^{3}-\frac{12\cdots 80}{36\cdots 77}a^{2}+\frac{64\cdots 46}{12\cdots 59}a-\frac{39\cdots 03}{36\cdots 77}$
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| Regulator: | \( 17878940.9848 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 17878940.9848 \cdot 1}{2\cdot\sqrt{106341808682864896865468416}}\cr\approx \mathstrut & 0.345875032239 \end{aligned}\] (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
| \(\Q(\sqrt{2}) \), 4.4.7168.1, \(\Q(\zeta_{16})^+\), 4.4.14336.1, 8.8.3288334336.1 |
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: | 16.8.1302687156365094986601988096.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.16.64l1.148 | $x^{16} + 4 x^{12} + 8 x^{10} + 8 x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 16 x + 2$ | $16$ | $1$ | $64$ | 16T153 | $$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]$$ |
|
\(7\)
| 7.4.1.0a1.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |
| 7.2.2.2a1.1 | $x^{4} + 12 x^{3} + 42 x^{2} + 43 x + 9$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ | |
| 7.2.4.6a1.3 | $x^{8} + 24 x^{7} + 228 x^{6} + 1080 x^{5} + 2646 x^{4} + 3240 x^{3} + 2052 x^{2} + 655 x + 109$ | $4$ | $2$ | $6$ | $Q_8$ | $$[\ ]_{4}^{2}$$ |