Normalized defining polynomial
\( x^{16} - 4 x^{14} - 24 x^{13} + 72 x^{12} - 24 x^{11} + 40 x^{10} - 280 x^{9} + 240 x^{8} + 144 x^{7} - 88 x^{6} - 192 x^{5} + 80 x^{4} + 48 x^{3} - 16 x + 4 \)
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: | \(27487790694400000000=2^{46}\cdot 5^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $16.40$ | 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: | $2, 5$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{20} a^{12} + \frac{1}{20} a^{11} + \frac{1}{10} a^{10} + \frac{1}{20} a^{9} + \frac{1}{10} a^{8} - \frac{1}{2} a^{7} + \frac{1}{10} a^{5} - \frac{3}{10} a^{4} - \frac{1}{10} a^{3} - \frac{1}{5} a^{2} - \frac{1}{10} a - \frac{1}{5}$, $\frac{1}{40} a^{13} - \frac{1}{10} a^{11} + \frac{1}{10} a^{10} - \frac{1}{10} a^{9} - \frac{1}{20} a^{8} - \frac{1}{5} a^{6} - \frac{9}{20} a^{5} + \frac{1}{10} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{3}{10} a + \frac{1}{10}$, $\frac{1}{520} a^{14} - \frac{1}{130} a^{13} - \frac{1}{65} a^{12} + \frac{9}{130} a^{11} + \frac{31}{260} a^{10} - \frac{3}{52} a^{9} - \frac{1}{13} a^{8} - \frac{31}{65} a^{7} - \frac{63}{260} a^{6} + \frac{37}{130} a^{5} + \frac{22}{65} a^{4} - \frac{14}{65} a^{3} - \frac{23}{65} a^{2} + \frac{3}{26} a - \frac{5}{13}$, $\frac{1}{425389640} a^{15} - \frac{67897}{85077928} a^{14} - \frac{2713281}{425389640} a^{13} + \frac{3234843}{212694820} a^{12} + \frac{310463}{212694820} a^{11} + \frac{7419641}{106347410} a^{10} - \frac{411577}{21269482} a^{9} + \frac{982519}{42538964} a^{8} + \frac{27036901}{212694820} a^{7} + \frac{14092517}{42538964} a^{6} + \frac{21854709}{212694820} a^{5} + \frac{17994504}{53173705} a^{4} + \frac{23904376}{53173705} a^{3} - \frac{19032837}{106347410} a^{2} - \frac{2292427}{21269482} a - \frac{37011999}{106347410}$
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{92679}{240605} a^{15} - \frac{79319}{481210} a^{14} + \frac{329302}{240605} a^{13} + \frac{470055}{48121} a^{12} - \frac{5585673}{240605} a^{11} + \frac{462146}{240605} a^{10} - \frac{4857677}{240605} a^{9} + \frac{47669207}{481210} a^{8} - \frac{13356072}{240605} a^{7} - \frac{13405983}{240605} a^{6} - \frac{30742}{240605} a^{5} + \frac{14102562}{240605} a^{4} - \frac{794594}{240605} a^{3} - \frac{2631782}{240605} a^{2} - \frac{1315626}{240605} a + \frac{655312}{240605} \) (order $4$) | 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: | \( 2548.59228871 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2\wr C_2$ (as 16T39):
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $C_2^2\wr C_2$ |
| Character table for $C_2^2\wr C_2$ |
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 8 siblings: | 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 | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 | ||||||
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |