Normalized defining polynomial
\( x^{16} + 10x^{14} + 28x^{12} - 86x^{8} - 100x^{6} - 33x^{4} + 1 \)
Invariants
| Degree: | $16$ |
| |
| Signature: | $[4, 6]$ |
| |
| Discriminant: |
\(6879707136000000000000\)
\(\medspace = 2^{32}\cdot 3^{8}\cdot 5^{12}\)
|
| |
| Root discriminant: | \(23.17\) |
| |
| Galois root discriminant: | $2^{21/8}3^{1/2}5^{3/4}\approx 35.72661494273978$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_4$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
| |
| Narrow class group: | $C_{2}$, which has order $2$ (assuming GRH) |
|
Unit group
| Rank: | $9$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$6a^{14}+58a^{12}+148a^{10}-53a^{8}-500a^{6}-423a^{4}-44a^{2}+17$, $4a^{14}+39a^{12}+102a^{10}-28a^{8}-342a^{6}-306a^{4}-39a^{2}+12$, $a^{15}+10a^{13}+28a^{11}-86a^{7}-100a^{5}-33a^{3}$, $a^{14}+10a^{12}+27a^{10}-6a^{8}-90a^{6}-81a^{4}-11a^{2}+2$, $11a^{15}+106a^{13}+270a^{11}-95a^{9}-909a^{7}-777a^{5}-94a^{3}+27a$, $6a^{14}+57a^{12}+141a^{10}-61a^{8}-477a^{6}-390a^{4}-43a^{2}+14$, $5a^{15}+49a^{13}+130a^{11}-29a^{9}-432a^{7}-402a^{5}-61a^{3}+12a$, $3a^{15}-7a^{14}+31a^{13}-68a^{12}+94a^{11}-177a^{10}+27a^{9}+46a^{8}-262a^{7}+577a^{6}-382a^{5}+542a^{4}-187a^{3}+114a^{2}-31a-4$, $3a^{15}+3a^{14}+29a^{13}+28a^{12}+74a^{11}+66a^{10}-28a^{9}-40a^{8}-257a^{7}-230a^{6}-211a^{5}-167a^{4}-9a^{3}-10a^{2}+14a+8$
|
| |
| Regulator: | \( 31559.3321698 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 31559.3321698 \cdot 1}{2\cdot\sqrt{6879707136000000000000}}\cr\approx \mathstrut & 0.187288894472 \end{aligned}\] (assuming GRH)
Galois group
$(C_2^3\times C_4):C_4$ (as 16T292):
| A solvable group of order 128 |
| The 26 conjugacy class representatives for $(C_2^3\times C_4):C_4$ |
| Character table for $(C_2^3\times C_4):C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{15}) \), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{15})^+\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\zeta_{60})^+\) |
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: | 16.8.1358954496000000000000.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{12}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.32b3.13 | $x^{16} + 4 x^{15} + 4 x^{13} + 20 x^{12} + 12 x^{11} + 6 x^{10} + 36 x^{9} + 44 x^{8} + 16 x^{7} + 32 x^{6} + 56 x^{5} + 33 x^{4} + 16 x^{3} + 28 x^{2} + 32 x + 9$ | $4$ | $4$ | $32$ | 16T140 | $$[2, 2, 3, 3]^{4}$$ |
|
\(3\)
| 3.8.2.8a1.2 | $x^{16} + 4 x^{13} + 2 x^{12} + 8 x^{10} + 8 x^{9} + 5 x^{8} + 8 x^{7} + 12 x^{6} + 12 x^{5} + 8 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 7$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $$[\ ]_{2}^{8}$$ |
|
\(5\)
| 5.4.4.12a1.4 | $x^{16} + 16 x^{14} + 16 x^{13} + 104 x^{12} + 192 x^{11} + 448 x^{10} + 864 x^{9} + 1432 x^{8} + 2048 x^{7} + 2624 x^{6} + 2752 x^{5} + 2208 x^{4} + 1280 x^{3} + 512 x^{2} + 128 x + 21$ | $4$ | $4$ | $12$ | $C_4^2$ | $$[\ ]_{4}^{4}$$ |