Normalized defining polynomial
\( x^{16} - 6 x^{15} + 7 x^{14} + 27 x^{13} - 95 x^{12} + 83 x^{11} + 143 x^{10} - 409 x^{9} + 287 x^{8} + 177 x^{7} - 321 x^{6} + 31 x^{5} + 95 x^{4} - 21 x^{3} - x^{2} + 4 x - 1 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(718749457809384765625=5^{10}\cdot 29^{4}\cdot 101^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $20.12$ | 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: | $5, 29, 101$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} - \frac{5}{11} a^{11} + \frac{4}{11} a^{10} + \frac{2}{11} a^{9} + \frac{3}{11} a^{8} + \frac{5}{11} a^{7} + \frac{3}{11} a^{5} - \frac{2}{11} a^{4} + \frac{3}{11} a^{3} - \frac{5}{11} a^{2} + \frac{5}{11} a + \frac{3}{11}$, $\frac{1}{11} a^{13} + \frac{1}{11} a^{11} + \frac{2}{11} a^{9} - \frac{2}{11} a^{8} + \frac{3}{11} a^{7} + \frac{3}{11} a^{6} + \frac{2}{11} a^{5} + \frac{4}{11} a^{4} - \frac{1}{11} a^{3} + \frac{2}{11} a^{2} - \frac{5}{11} a + \frac{4}{11}$, $\frac{1}{132} a^{14} + \frac{5}{132} a^{13} + \frac{1}{44} a^{12} + \frac{17}{132} a^{11} - \frac{1}{132} a^{10} - \frac{43}{132} a^{9} + \frac{65}{132} a^{8} - \frac{49}{132} a^{7} + \frac{17}{132} a^{6} - \frac{19}{44} a^{5} - \frac{7}{132} a^{4} - \frac{19}{132} a^{3} - \frac{49}{132} a^{2} + \frac{1}{4} a + \frac{37}{132}$, $\frac{1}{868956} a^{15} - \frac{403}{217239} a^{14} - \frac{1901}{144826} a^{13} + \frac{11953}{434478} a^{12} + \frac{48835}{434478} a^{11} + \frac{6365}{39498} a^{10} + \frac{5545}{19749} a^{9} + \frac{49891}{434478} a^{8} + \frac{33097}{434478} a^{7} + \frac{55105}{144826} a^{6} - \frac{92531}{434478} a^{5} + \frac{1762}{7491} a^{4} + \frac{122413}{434478} a^{3} + \frac{3673}{13166} a^{2} + \frac{10348}{217239} a - \frac{125215}{289652}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 28245.8280672 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^4.C_2^3.C_2$ (as 16T542):
| A solvable group of order 256 |
| The 40 conjugacy class representatives for $C_2^4.C_2^3.C_2$ |
| Character table for $C_2^4.C_2^3.C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.73225.1, 4.4.2525.1, 4.4.725.1, 8.4.924465625.1, 8.4.924465625.2, 8.8.5361900625.2 |
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 | ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $29$ | 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.4.2.1 | $x^{4} + 145 x^{2} + 7569$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $101$ | 101.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 101.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 101.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 101.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 101.4.2.1 | $x^{4} + 505 x^{2} + 91809$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 101.4.2.1 | $x^{4} + 505 x^{2} + 91809$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |