Normalized defining polynomial
\( x^{16} + 49 x^{14} + 987 x^{12} + 10477 x^{10} + 62235 x^{8} + 201538 x^{6} + 317672 x^{4} + 190896 x^{2} + 26896 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1996961573765376000000000000=2^{20}\cdot 3^{8}\cdot 5^{12}\cdot 29^{4}\cdot 41^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $50.85$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 3, 5, 29, 41$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{6} - \frac{1}{5} a^{4} - \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{7} - \frac{1}{5} a^{5} - \frac{2}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{10} a^{10} - \frac{1}{10} a^{8} + \frac{3}{10} a^{6} + \frac{1}{10} a^{4} - \frac{3}{10} a^{2} + \frac{1}{5}$, $\frac{1}{10} a^{11} - \frac{1}{10} a^{9} + \frac{3}{10} a^{7} + \frac{1}{10} a^{5} - \frac{3}{10} a^{3} + \frac{1}{5} a$, $\frac{1}{20} a^{12} - \frac{1}{20} a^{10} + \frac{1}{20} a^{8} + \frac{7}{20} a^{6} + \frac{9}{20} a^{4} - \frac{1}{5} a^{2} + \frac{2}{5}$, $\frac{1}{20} a^{13} - \frac{1}{20} a^{11} + \frac{1}{20} a^{9} + \frac{7}{20} a^{7} + \frac{9}{20} a^{5} - \frac{1}{5} a^{3} + \frac{2}{5} a$, $\frac{1}{97899800} a^{14} - \frac{2174509}{97899800} a^{12} - \frac{3556911}{97899800} a^{10} - \frac{295581}{19579960} a^{8} - \frac{1138259}{19579960} a^{6} - \frac{7565673}{24474950} a^{4} - \frac{3911584}{12237475} a^{2} - \frac{37241}{298475}$, $\frac{1}{97899800} a^{15} - \frac{2174509}{97899800} a^{13} - \frac{3556911}{97899800} a^{11} - \frac{295581}{19579960} a^{9} - \frac{1138259}{19579960} a^{7} - \frac{7565673}{24474950} a^{5} - \frac{3911584}{12237475} a^{3} - \frac{37241}{298475} a$
Class group and class number
$C_{2}\times C_{2}\times C_{296}$, which has order $1184$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 6317.46461709 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 512 |
| The 62 conjugacy class representatives for t16n781 are not computed |
| Character table for t16n781 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.725.1, \(\Q(\zeta_{15})^+\), 4.4.32625.1, 8.8.1064390625.1 |
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 | R | R | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.12.6 | $x^{8} + 2 x^{6} + 8 x^{4} + 80$ | $2$ | $4$ | $12$ | $C_8$ | $[3]^{4}$ |
| 2.8.8.4 | $x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$ | $2$ | $4$ | $8$ | $C_8$ | $[2]^{4}$ | |
| $3$ | 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $29$ | 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.2.1 | $x^{4} + 145 x^{2} + 7569$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $41$ | 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 41.4.0.1 | $x^{4} - x + 17$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 41.4.0.1 | $x^{4} - x + 17$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |