Normalized defining polynomial
\( x^{16} - 7x^{14} + 22x^{12} - 40x^{10} + 43x^{8} - 22x^{6} - 2x^{4} + 5x^{2} + 1 \)
Invariants
| Degree: | $16$ |
| |
| Signature: | $[0, 8]$ |
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| Discriminant: |
\(10457596138654531584\)
\(\medspace = 2^{24}\cdot 3^{14}\cdot 19^{4}\)
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| Root discriminant: | \(15.44\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(19\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-3}) \) | ||
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 $1$ |
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| Narrow class group: | Trivial group, which has order $1$ |
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Unit group
| Rank: | $7$ |
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| Torsion generator: |
\( -a^{12} + 5 a^{10} - 12 a^{8} + 17 a^{6} - 13 a^{4} + 3 a^{2} + 2 \)
(order $6$)
|
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| Fundamental units: |
$a^{14}-6a^{12}+17a^{10}-29a^{8}+30a^{6}-16a^{4}+2a^{2}+1$, $a^{15}-7a^{13}+22a^{11}-41a^{9}+47a^{7}-30a^{5}+7a^{3}+a$, $a^{12}-5a^{10}+12a^{8}-16a^{6}+11a^{4}-a^{2}-2$, $a^{11}+a^{10}-4a^{9}-4a^{8}+8a^{7}+8a^{6}-9a^{5}-9a^{4}+4a^{3}+4a^{2}+a+1$, $2a^{15}-2a^{14}-12a^{13}+12a^{12}+33a^{11}-33a^{10}-53a^{9}+53a^{8}+49a^{7}-49a^{6}-19a^{5}+20a^{4}-4a^{3}+2a^{2}+2a-1$, $2a^{15}-14a^{13}-a^{12}+44a^{11}+6a^{10}-81a^{9}-16a^{8}+90a^{7}+25a^{6}-52a^{5}-22a^{4}+4a^{3}+8a^{2}+7a+2$, $a^{15}+2a^{14}-6a^{13}-10a^{12}+17a^{11}+22a^{10}-28a^{9}-25a^{8}+27a^{7}+8a^{6}-12a^{5}+13a^{4}-a^{3}-10a^{2}+a-3$
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| Regulator: | \( 2556.07087636 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2556.07087636 \cdot 1}{6\cdot\sqrt{10457596138654531584}}\cr\approx \mathstrut & 0.319996078112 \end{aligned}\]
Galois group
$(C_2^2\times D_4^2).D_4$ (as 16T1347):
| A solvable group of order 2048 |
| The 71 conjugacy class representatives for $(C_2^2\times D_4^2).D_4$ |
| Character table for $(C_2^2\times D_4^2).D_4$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 4.0.513.1, 8.0.12632112.1 |
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: | 16.0.10457596138654531584.2 |
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 | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.16b3.8 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 59 x^{4} + 64 x^{3} + 54 x^{2} + 28 x + 17$ | $4$ | $2$ | $16$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $$[2, 2, 3, 3]^{4}$$ |
| 2.4.2.8a2.1 | $x^{8} + 4 x^{5} + 2 x^{4} + 3 x^{2} + 4 x + 3$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $$[2, 2, 2, 2]^{4}$$ | |
|
\(3\)
| 3.2.8.14a1.2 | $x^{16} + 16 x^{15} + 128 x^{14} + 672 x^{13} + 2576 x^{12} + 7616 x^{11} + 17920 x^{10} + 34176 x^{9} + 53344 x^{8} + 68352 x^{7} + 71680 x^{6} + 60928 x^{5} + 41216 x^{4} + 21504 x^{3} + 8192 x^{2} + 2048 x + 259$ | $8$ | $2$ | $14$ | $QD_{16}$ | $$[\ ]_{8}^{2}$$ |
|
\(19\)
| 19.4.1.0a1.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |
| 19.4.1.0a1.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
| 19.4.2.4a1.2 | $x^{8} + 4 x^{6} + 22 x^{5} + 8 x^{4} + 44 x^{3} + 129 x^{2} + 44 x + 23$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |