Normalized defining polynomial
\( x^{16} - 4 x^{14} - 192 x^{12} - 184 x^{10} + 2592 x^{8} + 680 x^{6} - 6256 x^{4} + 1156 \)
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: | \(178304929163297110062268416=2^{50}\cdot 3^{8}\cdot 17^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $43.72$ | 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: | $2, 3, 17$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{28} a^{10} - \frac{1}{28} a^{8} + \frac{1}{14} a^{6} - \frac{3}{14} a^{4} + \frac{1}{14} a^{2} - \frac{5}{14}$, $\frac{1}{28} a^{11} - \frac{1}{28} a^{9} + \frac{1}{14} a^{7} - \frac{3}{14} a^{5} + \frac{1}{14} a^{3} - \frac{5}{14} a$, $\frac{1}{96152} a^{12} - \frac{19}{48076} a^{10} - \frac{2595}{48076} a^{8} - \frac{4749}{12019} a^{6} - \frac{405}{6868} a^{4} - \frac{17}{202} a^{2} + \frac{352}{707}$, $\frac{1}{96152} a^{13} - \frac{19}{48076} a^{11} - \frac{2595}{48076} a^{9} - \frac{4749}{12019} a^{7} - \frac{405}{6868} a^{5} - \frac{17}{202} a^{3} + \frac{352}{707} a$, $\frac{1}{1799869288} a^{14} + \frac{2711}{1799869288} a^{12} - \frac{1913567}{224983661} a^{10} - \frac{37821705}{899934644} a^{8} - \frac{53095075}{899934644} a^{6} + \frac{411013179}{899934644} a^{4} + \frac{12250901}{26468666} a^{2} - \frac{4181635}{13234333}$, $\frac{1}{1799869288} a^{15} + \frac{2711}{1799869288} a^{13} - \frac{1913567}{224983661} a^{11} - \frac{37821705}{899934644} a^{9} - \frac{53095075}{899934644} a^{7} + \frac{411013179}{899934644} a^{5} + \frac{12250901}{26468666} a^{3} - \frac{4181635}{13234333} a$
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: | \( 15439482.4508 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^3.C_2^4.C_2$ (as 16T456):
| A solvable group of order 256 |
| The 46 conjugacy class representatives for $C_2^3.C_2^4.C_2$ |
| Character table for $C_2^3.C_2^4.C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), 4.4.4352.1, 4.4.9792.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.1534132224.1 |
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 | ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 | Data not computed | ||||||
| $3$ | 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 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $17$ | 17.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 17.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 17.8.6.2 | $x^{8} + 85 x^{4} + 2601$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |