Normalized defining polynomial
\( x^{16} - 8 x^{14} + 22 x^{12} + 140 x^{10} - 249 x^{8} - 620 x^{6} - 402 x^{4} - 84 x^{2} + 1 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(121393709056000000000000=2^{28}\cdot 5^{12}\cdot 1361^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $27.72$ | 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: | $2, 5, 1361$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{8} a^{2} + \frac{1}{8} a - \frac{3}{8}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{8}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} + \frac{3}{16} a^{6} - \frac{1}{8} a^{5} + \frac{7}{16} a^{2} + \frac{1}{8} a - \frac{5}{16}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{3}{16} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{3}{16} a^{3} + \frac{1}{4} a^{2} - \frac{3}{16} a - \frac{1}{8}$, $\frac{1}{523184} a^{14} - \frac{6191}{523184} a^{12} - \frac{5777}{261592} a^{10} + \frac{56605}{523184} a^{8} - \frac{1}{4} a^{7} - \frac{22133}{261592} a^{6} + \frac{103525}{523184} a^{4} - \frac{1}{2} a^{3} + \frac{216341}{523184} a^{2} - \frac{1}{4} a + \frac{44425}{130796}$, $\frac{1}{523184} a^{15} - \frac{6191}{523184} a^{13} - \frac{5777}{261592} a^{11} + \frac{56605}{523184} a^{9} - \frac{22133}{261592} a^{7} + \frac{103525}{523184} a^{5} - \frac{1}{4} a^{4} + \frac{216341}{523184} a^{3} - \frac{1}{4} a^{2} + \frac{44425}{130796} a + \frac{1}{4}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 435590.936634 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 2048 |
| The 71 conjugacy class representatives for t16n1385 are not computed |
| Character table for t16n1385 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.8000.1, 8.4.87104000000.1 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.16.2 | $x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 20$ | $4$ | $2$ | $16$ | $C_4\times C_2$ | $[2, 3]^{2}$ |
| 2.8.12.2 | $x^{8} + 2 x^{6} + 8 x^{4} + 16$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
| $5$ | 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 1361 | Data not computed | ||||||