Normalized defining polynomial
\( x^{16} - 5 x^{15} + 13 x^{14} - 30 x^{13} + 73 x^{12} - 160 x^{11} + 286 x^{10} - 405 x^{9} + 455 x^{8} - 405 x^{7} + 286 x^{6} - 160 x^{5} + 73 x^{4} - 30 x^{3} + 13 x^{2} - 5 x + 1 \)
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: | \(17247076416015625=5^{14}\cdot 41^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $10.35$ | 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: | $5, 41$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7} a^{13} - \frac{1}{7} a^{12} - \frac{1}{7} a^{11} + \frac{3}{7} a^{10} - \frac{2}{7} a^{9} - \frac{1}{7} a^{8} - \frac{2}{7} a^{7} - \frac{2}{7} a^{6} - \frac{1}{7} a^{5} - \frac{2}{7} a^{4} + \frac{3}{7} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{49} a^{14} + \frac{19}{49} a^{12} + \frac{16}{49} a^{11} + \frac{22}{49} a^{10} - \frac{3}{49} a^{9} + \frac{11}{49} a^{8} + \frac{24}{49} a^{7} + \frac{4}{49} a^{6} + \frac{4}{49} a^{5} - \frac{13}{49} a^{4} - \frac{5}{49} a^{3} + \frac{19}{49} a^{2} - \frac{2}{7} a + \frac{8}{49}$, $\frac{1}{343} a^{15} + \frac{1}{343} a^{14} + \frac{19}{343} a^{13} + \frac{12}{49} a^{12} - \frac{109}{343} a^{11} - \frac{128}{343} a^{10} - \frac{139}{343} a^{9} + \frac{19}{49} a^{8} - \frac{17}{49} a^{7} - \frac{90}{343} a^{6} + \frac{89}{343} a^{5} + \frac{31}{343} a^{4} - \frac{12}{49} a^{3} + \frac{152}{343} a^{2} - \frac{104}{343} a + \frac{57}{343}$
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: | \( \frac{360}{343} a^{15} - \frac{1607}{343} a^{14} + \frac{3753}{343} a^{13} - \frac{1215}{49} a^{12} + \frac{21212}{343} a^{11} - \frac{45107}{343} a^{10} + \frac{75568}{343} a^{9} - \frac{14136}{49} a^{8} + \frac{14527}{49} a^{7} - \frac{80742}{343} a^{6} + \frac{49897}{343} a^{5} - \frac{22951}{343} a^{4} + \frac{1201}{49} a^{3} - \frac{2890}{343} a^{2} + \frac{1760}{343} a - \frac{361}{343} \) (order $10$) | 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: | \( 123.675587715 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\wr C_4$ (as 16T158):
| A solvable group of order 64 |
| The 13 conjugacy class representatives for $C_2\wr C_4$ |
| Character table for $C_2\wr C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.4.5125.1, 4.0.1025.1, 8.0.3203125.1 x2, 8.4.131328125.1 x2, 8.0.26265625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 32 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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.8.7.2 | $x^{8} - 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
| 5.8.7.2 | $x^{8} - 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ | |
| $41$ | 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |