Normalized defining polynomial
\( x^{16} - 4 x^{14} + 26 x^{12} + 12 x^{10} + 35 x^{8} + 180 x^{6} + 686 x^{4} + 632 x^{2} + 169 \)
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: | \(12877254853348294656=2^{36}\cdot 3^{8}\cdot 13^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $15.64$ | 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: | $2, 3, 13$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{8} a^{4} + \frac{1}{8}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{8} a^{5} + \frac{1}{8} a$, $\frac{1}{8} a^{10} + \frac{3}{8} a^{6} - \frac{1}{4} a^{4} + \frac{1}{8} a^{2} + \frac{1}{4}$, $\frac{1}{104} a^{11} - \frac{3}{104} a^{9} - \frac{3}{8} a^{7} - \frac{23}{104} a^{5} + \frac{25}{104} a^{3} - \frac{33}{104} a$, $\frac{1}{312} a^{12} + \frac{5}{156} a^{10} + \frac{73}{156} a^{6} - \frac{6}{13} a^{4} - \frac{5}{78} a^{2} - \frac{11}{24}$, $\frac{1}{312} a^{13} + \frac{1}{312} a^{11} - \frac{1}{26} a^{9} - \frac{49}{312} a^{7} + \frac{17}{52} a^{5} + \frac{67}{312} a^{3} + \frac{115}{312} a$, $\frac{1}{1764048} a^{14} - \frac{1}{624} a^{13} + \frac{101}{294008} a^{12} - \frac{1}{624} a^{11} + \frac{106775}{1764048} a^{10} + \frac{1}{52} a^{9} - \frac{9799}{1764048} a^{8} - \frac{263}{624} a^{7} + \frac{417319}{1764048} a^{6} - \frac{17}{104} a^{5} + \frac{855655}{1764048} a^{4} - \frac{67}{624} a^{3} + \frac{6883}{73502} a^{2} + \frac{197}{624} a - \frac{45283}{135696}$, $\frac{1}{1764048} a^{15} - \frac{2221}{1764048} a^{13} - \frac{1}{624} a^{12} + \frac{136}{110253} a^{11} + \frac{29}{624} a^{10} + \frac{108935}{1764048} a^{9} + \frac{277921}{882024} a^{7} + \frac{283}{624} a^{6} - \frac{399533}{1764048} a^{5} - \frac{41}{104} a^{4} - \frac{804469}{1764048} a^{3} + \frac{59}{624} a^{2} - \frac{210943}{882024} a - \frac{7}{48}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{331}{36751} a^{15} - \frac{39895}{882024} a^{13} + \frac{61247}{220506} a^{11} - \frac{23389}{147004} a^{9} + \frac{85927}{220506} a^{7} + \frac{201611}{147004} a^{5} + \frac{2090731}{441012} a^{3} + \frac{921431}{882024} a \) (order $24$) | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 11347.4195205 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2\times D_4$ (as 16T25):
| A solvable group of order 32 |
| The 20 conjugacy class representatives for $C_2^2 \times D_4$ |
| Character table for $C_2^2 \times D_4$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 16 siblings: | data not computed |
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 | ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | Data not computed | ||||||
| $3$ | 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $13$ | 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |